Computational methods in finance - KU Leuvennikos/papers/methodsinfinance.pdf · 2012-01-01 ·...

Nikos Skantzos IAE Toulouse 2011-12 1

Computational Methods in Finance

Nikos SkantzosIAE Toulouse 2011-12

N1

∆ιαφάνεια 1

N1 Nikos, 10/20/2010


Course Organisation� Introduction

� Organisation inside the dealing room� Why do we need numerical methods inside a dealingroom?

� Some reminders …� Derivative products� Mathematics used in finance� Introduction to stochastic processes and probability� Introduction to VBA programming


Course Organisation� Evaluation of financial assets:

� Historical background� Brownian motion: motivation and examples� Black & Scholes model� Greeks� Other Models – Numerical methods – Payouts� Numerical methods

� Analytical solutions� Monte Carlo� Binomial Tree� Partial differential equations (PDE)

� Introduction to interest rate derivative products


� Volatility smile and market models

� Risk Management� Calculation of VAR� Introduction to credit risk

� Real world markets� Stylised facts� Pairs trading: an example strategy� Kelly’s criterion

Course Organisation


Introduction� Pictures from a dealing room


Introduction

� A more realistic picture of the dealing room

Cartoon by Adam Zyglis


Introduction� The presence and interaction of differentunits in a dealing room

TraderQuant

ITClient

Sales

Structurer

Risk Management Quant, IT


Inside the dealing room: Sales

� Sales� In touch with customers� They sell options and other products of the bank.

� Structurers� design new products that are attractive to customers.

� Customers choose them if they offer low risk, high profit and small premium


Inside the dealing room: Traders

� Traders� Hedge the position that the structurers open. � They buy sell/options to minimise the sensitivity of the bank’s portfolio to movements of the underlying.

� “Prop-traders”� Take position based on their expectation about the market’s next move.


Introduction: « Quants »

� Who: � Develop and implement mathematical models to price the products of structurers and calculate the risk for the bank.

� Where: � Investment banks, hedge funds and more generally in anyfinancial institution dealing with derivatives and market risk.

� Background: � Mathematics, � Physics, � Engineering, � Economy.


How a bank makes money

� Buying low & sell high� Selling financial products to customers� “Bid-offer” spread (buy price: bid, sell price: offer)

� Banks compete to offer best spread to customer

� Spread cannot go too high� The customer will go to someone else

� Spread cannot go too low� The bank will not have enough money to buy the hedge


Derivative products: a reminder

� Main idea behind Options: pay now a small premium to have a choice in the future

� Example: exchange 1ml EUR for 1,3ml USD in one year� What is this option worth today ?

� Can be used as insurance, for example:� If we don’t want to risk receiving less than 1,3m USD(We need the money to fund a US company)

� Can be used for speculation, for example� If we believe that the USD will weaken



� Underlying asset:� Any asset sold/bought on a stock market or trading room� Example:Stocks Bonds Metals Grains ElectricityInterest-rates Indices Currencies Gas Oil

� "Spot" Transaction:� We buy or sell an underlying

� Example: Microsoft shares, USD

� Market price is fixed through supply and demand.



� Derivative product� Its price fluctuates as a function of the value of the underlying.� Requires either no or small initial investment

� Its settlement is made at a future date

� Derivative market growing rapidly since 1980s� Requires numerical and heavy mathematical methods� Requires strong computational power & IT infrastructure � Need to process market data & produce option premium and risk

� Now present in the bulk of financial activity� Derivative pricing

� Requires maths and IT


Derivative products: discounting (1)� Today we put in a bank account N euro� Bank pays interest rate r once a year

� After 1 year we receive N·(1+r)� After 2 years we receive N·(1+r) ·(1+r)� …� After n years we receive N·(1+r)n


Derivative products: discounting (2)� Today we put in a bank account N euro� Bank pays interest rate r two times in a year

� After 1 year we receive N·(1+r/2)·(1+r/2)� After 2 years we receive N·(1+r/2)4

� …� After n years we receive N·(1+r/2)2n

� If bank pays interest rate m times in a year for n years:� Return after n years: N·(1+r/m)mn

� If payment is very often : “continuously compounded interest rate”� In this case, the return after n years is: N·(1+r/m)nm= N·e nr


Derivative products: discounting (3)

� This implies that an amount N today is equivalent to N·erT in T years

� Similarly, an amount N in T years is equivalent to N·e-rT today

� Exercise: Show that in the limit where m is very large we have (1+r/m)mn=er·n

� Remark: This is the same



� What is the “fair” value of an option?� Some intuition:

� More risk for the issuer, more expensive� Longer maturity, more expensive� More volatile market, more expensive


Derivatives: finding the fair price� In the horse races there are two horses

� Horse A, wins 75% of races� Horse B, wins 25% of races

� The booker pays� 100€ if horse A wins� 200€ if horse B wins

� You want to buy the right to choose your horse after the end of the race

� How much is this option worth ?


Derivatives: finding the fair price

� Fair price = average profit

� Average profit = 100 € · ¾ + 200 € · ¼ = 75 € + 50 € = 125 €

� Option’s fair price = 125 €

A (75%)

B (25%)

100 €

200 €

Horse race


Derivatives: finding the fair price in stock options

� Central idea is similar:� Fair price ~ Average payoff� Simulate stock many times

� Record final value� Calculate payoff for that path� Average over all paths

� Discounting� This “average” price is valid at maturity� To calculate the equivalent price

today:� N € in a bank account today=

N · e rT € after T years� Inversely,

P at maturity = P · e-rT today

Option price =Discounted Average Payoff

Average taken over probabilities that eliminate all risk: Risk-neutral measure



� History� 6th century BC: Greek philosopher Thales of Miletus reports that options were used to secure a low price of olives in advance of harvest.

� Middle Ages: futures contracts to fix in advance the price ofimports of goods from Asia

� Holland 1637: The "Tulip Mania" one of the first speculativebubbles.


Derivative products: a reminder� Two most simple and popular:

� Call = right to buyat an agreed future datea certain amount of the underlying assetat a price fixed today.

� Put = right to sellat an agreed future datea certain amount of the underlying assetat a price fixed today.

� Terminology� “Agreed future date” = Maturity of the option� “Amount of underlying” = Notional� “Price fixed today” = Strike


Derivative products: a reminder� The payout of an option

� what the option would bring to its owner at maturity (T), � depends on price of the underlying at that time (ST).

� Long Call payout = max(0, ST- K)� Go « Long » a Call if you think the underlying will increase

K ST

Call

Long (the case of a buyer of a call)

Short (the case of a seller of a call)

payout = ST-K


Derivative products: a reminder� Long Put payout = max(0, K- ST)

� Go « long » a Put if you think the underlying will go lower

� Calls and Puts are called vanillas� Vanilla flavour = simple.

K

ST

Put Long (the case for an owener of a Put)

Short (the case for a seller of a Put)

payout = K- ST


Derivative products: a reminder� Barrier options

� Advantage: Cheaper than vanilla options� Disadvantage: More risky

K ST

At maturity (T)Regularbarrier

Reverse barrier

•Knock-In = the option is activated if the spot hits the barrier

•Knock-Out = the option is disactivated if the spot hits the barrier



� Price of an option

K ST

Call

payout = ST-K

At maturity (T)

Today (t<T)

StTime value



How option parameters affect the price. Examples:

� If spot goes up, call price goes up� The right to buy cheap shares is more expensive because underlying became more expensive

� If vol goes up, call price goes up� The right to buy cheap shares is more expensive because the underlying is more risky


Derivative products: a reminder� How the option parameters affect the option price:

-+r

++

+-K

-+S

PutCall


Derivative products: terminology� European Payout: payout is uniquely determined by the value of the underlying at maturity

� American Payout: payout is function of the evolution of the underlying during the lifetime of the option

� European exercise: the owner can only exercise the option atmaturity

� American exercise: the owner can exercise the option any time during the lifetime of the option

� European barrier: the barrier is active only at maturity� American barrier: the barrier is active continuously during the lifetime of the option


Some derivative strategies

� Call spread(K1, K2) = Call(K1)- Call(K2)

=

� Cheaper than a simple call� Profit is limited to K2-K1 for spots>K2

+Call(K1)

-Call(K2)K1

K2

K1 K2


Some derivative strategies

� Straddle(K) = Call(K) + Put(K)

� Expensive� If ST>K: gives the right to buy cheap� If ST<K: gives the right to sell expensive

K

CallPut


Mathematical reminder


The exponential function

-2 -1 1 2

1234567

ex = Exp(x)

ex is always positive



• e=2.71828182845904523536028747135…• ex = Exp(x), e0 = 1, e1 = e

∑∞

=

=++++++=0

5432

!...

!5!4!321

i

ix

ixxxxxxe

ex ~2.716ex=1+x+ x/2! + x/3! + x/4! + x/5!

ex ~2.708ex=1+x+ x/2! + x/3! + x/4!

ex ~2.66ex=1+x+ x/2! + x/3!

ex ~2.5ex=1+x+ x/2!

ex ~2ex=1+x

ex ~1ex=1



� LN(e)=1� eln(x) = x, or ln(ex) = x� Logarithm in base e � Defined only for x>0

...5432

)1(5432

++−+−=+yyyyyyLn

• The function LN (Neperian logarithm):


Mathematical reminder� The derivative of a function: slope of a function at 1 point

Numerical approximation:

or∆

−∆+≈

)()()(' ooo

xfxfxf∆

∆ ∆

∆′−∆+′

≈′′ )()()( ooo

xfxfxf

∆∆−−∆+

≈2

)()()(' ooo

xfxfxf

2

)(2)()()(∆

−∆−+∆+≈′′

⇔

oooo

xfxfxfxf

• The 2nd derivative: curvature of a function in 1 pointNumerical approximation:


Numerical differentiation� Expand the function around the neighbourhood of x

� Non-central difference derivative has error O(h)

� Central difference derivative has error O(h2)

� Numerical error: central difference < non-central difference

( ) ( ) ( ) ( ) ( ) L+′′′+′′⋅+′⋅+=+ hfhhfhhfhxfhxf 32

!31

21

( ) ( ) ( ) ( ) ( ) L+′′′−′′⋅+′⋅−=− hfhhfhhfhxfhxf 32

!31

21

44 344 21L

error

)(21)()()(

+′′⋅+′=−+ xfhxf

hxfhxf

44 344 21L

error

2 )(!31)(

2)()(

+′′′+′=−−+ xfhxf

hhxfhxf


Some analytical derivatives


Mathematical reminder� Integral of a function


Mathematical reminder� Primitives of some commonly used functions


Mathematical reminder� Numerical integration of a function

Method of lowerrectangles

Method of upperrectangles

Trapezoidal method



� Taylor series: approximating a function around a point x0

� Converts a complex function into a simple power-series

� Examples� exp(x) around x0=0:

� cos(x) around x0=0:

� around x0=0:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0)(00

200000 !

121 xfxx

nxfxxxfxxxfxxf nn−++′′⋅−+′⋅−+=+ L

L+++= 2

211 xxex

L+−= 2

211)cos( xx

x−11

L+++=−

2111 xxx


Taylor-series approximation to ATM options

� Black-Scholes formula for call options at the money (where S=Ke-rT)

� Approximate cumulative normal N(x) for small x=σ√T:

� This gives the approximate call price:

T

TrKSd

σ

σ )221()/ln( 0 ±+

=±)()( −+ ⋅−⋅= dNSdNSC

( ) ( ) ( ) ( )

)(21

21

)(02100

2

32

xOx

xONxNxNxN

++=

+′′⋅+′⋅+=

π

TSC σπ

⋅=21


Root finding: bisection method� Consider portfolio of options

� Find Vol that makes portfolio value equal to 0� Choose two points xmin, xmax� Take the middle point: xmid� If f(xmid)<0

� xmin=xmid� xmax=xmax� Repeat

� If f(xmid)>0� xmax=xmid� xmin=xmin� Repeat

� Here: function increases with variable

� Similar when function is decreasing


Interpolation methods: Linear

� The line connecting two points (x1,y1) and (x2,y2) is

� Check:� If x=x1 then y=y1� If x=x2 then y=y2

( ) 1121

21 yxxxxyyy +−⋅

−−

=


Interpolation methods: Cubic spline (1)

� n data points {(x1,y1),…, ( xn,yn)}

� Fit a piecewise function S(x) of the form

� where si(x) cubic polynomials

[ ][ ]

[ ]

∈

∈∈

=

−− nnn xxxxs

xxxxsxxxxs

xS

,if)(

,if)(,if)(

)(

11

322

211

iiiiiiii dxxcxxbxxaxs +−⋅+−⋅+−⋅= )()()()( 23


Interpolation methods: Cubic spline (2)

� Require:� For i=1,…,n (for all points)

� For i=2,…,n-1 (for all interior points)

� This gives: n+3(n-2) equations, 4(n-1) unknowns� Remaining 2 unknowns fixed by boundary conditions:

� “natural spline”� “clamped spline”� Other choices are possible

ii yxs =)(

)()()()()()(

1

1

1

iiii

iiii

iiii

xsxsxsxsxsxs

−

−

−

′′=′′′=′

=

0)()( 1 =′′=′′ nxSxSvxSuxS n =′=′ )()( 1


Differentiation of an integral

� If the differentiation variable lies in the integration limits then the following formula is useful

� Example: Second derivative of a market-quoted call option gives the market-implied terminal density of the spot

( )( )

( )

( ) ( )( )

( )

∫∫ ⋅∂

∂+

∂∂

⋅−∂∂

⋅=⋅∂∂ xa

xb

xa

xb

dsx

xsFxbxbsF

xaxasFdsxsF

x),()(,)(,,

[ ]

)(DF

)(DF

)()(DF)0,max(EDF

mkt2

mkt2

mktmkt

mktmkt

KPKC

SPdSK

C

KSSPdSKSC

KTT

KTTTT

⋅=∂

∂

⋅⋅−=∂

∂

−⋅⋅⋅=−⋅=

∫

∫∞

∞


Random variables and stochastic processes

Basic notions


Random variables and stochastic processes

� Random variable� a variable whose value is determined by the outcome of an experiment� We don’t know its value only how likely it is

� Discrete random variable: � Can take on only certain separated (discrete) values� Example: the result of throwing a dice.

The probability of every outcome is 1/6

� Continuous random variable: � Can take on any real value from a range� Example: the price of an stock. The probability that the price is within a

certain interval depends on the distribution of the random variable.

� Stochastic process� represents the evolution in time of a random variable


Properties of random variables

� Probability of an event: 0≤Prob(event) ≤1� Prob=0: certainty that event will not happen� Prob=1: certainty that event will happen

� Normalisation: Prob(ev1)+… +Prob(evN) =1

� Prob(ev1 OR ev2) = Prob(ev1) + Prob(ev2) � Example: probability that a dice is either “1” or “2” = 1/6 + 1/6

� If ev1 is independent of ev2 then: Prob(ev1 AND ev2) = Prob(ev1) · Prob(ev2)� Example: Prob that two dice are both “1” = 1/6 · 1/6


Uncorrelated vs independent r.v.

� Two sets of random variables xi, yi are uncorrelated if

� Two random variables x, y are independent if

� If xi, yi are independent then they are uncorrelated (converse not true!)

( )( )[ ] 0E =−− yyxx

( ) ( ) ( )yxx,y ProbProbProb ⋅=


Random variables� Characterised by:

� The probability density distribution function f(x)� Prob that event x will happen

� The cumulative distribution function� Prob that the outcome of the experiment will be less than x

� The mathematical expectation (mean)� The average by repeating the experiment many times

� The moments (order n) :� First moment is the mean� Second moment is related to the variance� Third moment is related to the skewness� ...

∫∞−

⋅=x

dxxfxF )()(

[ ] ∫∞

∞−

⋅⋅== dxxfxx )(Eµ

∫∞

∞−

⋅⋅= dxxfxXM nn )()(


a b

∫ ⋅=∈b

a

dxxfbaXP )(]),[(

Interpretation of distribution function� The surface under the curve between a and b is the probability that the value of the random variable is between aand b :


Central moments

� The central moments (of order n): remove the mean

� The variance (n=2), characterises the amplituded around the mean:

� Standard Deviation = √ variance � In finance, the standard deviation is the volatility

∫∞

∞−

⋅⋅−= dxxfxXCM nn )()()( µ

( )[ ] [ ] 22222 )()()( σµµµ =−=−=−= ∫∞

∞−

xExEdxxfxxV


�Skewness (n=3), describes the asymmetry:

�Kurtosis (n=4), describes the effects of «fat» tails:

<3 : distribution platykurtic>3 : distribution leptokurtic

3

3 )()()(

σ

µγ

∫∞

∞−

−=

dxxfxX

4

4 )()()(

σ

µδ

∫∞

∞−

−=

dxxfxX 3)lawnormal( =δ

Central moments


Skewness & kurtosisAsymmetry: skewness Fat tails: kurtosis


Meaning of “fat tails”� Represents a high probability of extreme events.

� Catastrophic market crashes (1927, 1987)� Money lost is more than ½ of all money lost in the next 20 years

� Catastrophic earthquakes (Chile 1960 9.5R, Sumatra 2004 9.1R)� Energy released is more than ½ of total energy released by crust

� Such events are characterised by � Very low probability � Very high impact


Examples of “fat tails”� Fat tails means that the extreme-event probability is low, but much higher than we expect !


Variance of a distribution

� Controls size of deviations away from the mean


Variance of a distribution

� Small variance = large certainty

� All distributions look the same when variance 0

� Graph opposite:� Lognormal vs Normal� variance=0.01

� Which is which ?


Distribution vs cumulative� Some important properties

� Definition or and

� Distribution function is normalized:

� Cumulative is between 0 and 1, always increasing

dXXdFXf )()( =

0)( =−∞F

1)( =+∞F

∫∞−

=X

dUUfXF )()(

f(x) F(x)

]1,0[)( ∈XF

∫∞

∞−

=⋅ 1)(xfdx


Some important properties� Integral of the distribution: probability that the randomvariable will be less than a certain value

� Probability that the random variable is between two values:

∫∞−

=x

dssfxF )()(

)()()()( BXAPdxxfAFBFB

A

≤≤=⋅=− ∫


Sampling from a distribution

� This is an important application of cumulative functions� Problem: generate random variables from specific distribution� Matlab, Excel,… provide the uniform random number generator� This selects uniformly a number between 0 and 1� We use the inverse cumulative function of the distribution

Pseudo code• Draw a uniform random number Z in [0,1]• Pass it through the InvCum of the required distribution (e.g. normal)•Result is a number sampled from the required distribution•Example: + ·Z ~ N( , 2)


Use of distributions in finance

� Financial derivatives require us to calculate theexpectation of a function of a random variable

� Example: a Call option

where (ST) is the distribution function of the final spot

∫ ⋅⋅== dxxXgXgE )()()]([derivative ϕ

( ) ( )∫+∞

⋅⋅−=−=0

)(0,max]0,[maxCall TTTT dSSKSKSE ϕ


Normal Distribution� Normal Distribution N(µ,σ)

� Special case: µ = 0 and σ = 1 denoted N(0,1)

µ = mean

σ= standard deviation


Normal Distribution

Exercise :

� What are (i) the mean and (ii) the standard deviation of the index EUROSTOXX50, if we suppose that it follows a law a+bX whereX follows a centered normal distribution (a and b are 2 constants) ?

� Calculate the mathematical expectation of eαX where X follows a centered normal distribution

� Calculate the expectation of S=S0e(r-q-σ²/2)T+xσ √T where X follows a centered normal distribution


Log-normal Distribution

� Very important in finance

� If X follows a normal law X~N(µ, σ),

� Then Y=eX is distributed log-normally.

� Relations between the function of X and Y, related by X = f(Y):

YYfYffYf

YXXfYfXXfdYYf

xy

xyxy

∂∂

=

∂∂

=⇔=

)())(()(

)()(d)()(Exercise: recover the Log-Normal distribution law


Log-normal Distribution

e21),;( ²2

)²-(x-σµ

πσσµ =Xf

x

Starting from a normal distribution for X

We find the log-normal law for Y=eX

e21),;( ²2

)²-(ln(Y)-σ

µ

πσσµ

YYf

y=

Exercise: Calculate the mean and variance of a log-normal function with parameters ,


Central Limit Theorem

� This theorem is the reason why normal distributions are present so often!

� The sum of N independent, identically distributed random numbers is normally distributed

� The N numbers do not have to be normally distributed! � N numbers, x1,…, xN each with mean m, variance s� The random variable x1+ x2 …+ xN follows

( )( )

2

2

2N1 2

1xxyProb NsmNy

esN

⋅−−

⋅=++=π

L


Central Limit Theorem at work� For N = 5, 20, 100

� Sample N random variables from some distribution (here lognormal) and sum them: x1+…+ xN

� For each N, repeat many times and plot histogram� Observations:

� For small N, only central region looks normally distributed ! � For large N, the sum resembles the normal distribution very well


Sum of lognormal variables

� Because of the Central Limit Theorem� A sum of normal variables is normal� A sum of lognormal variables is not lognormal

� In finance however we often approximate a sum of lognormal variables by a lognormal


Commutation of integration & differentiation

� The order of integration and differentiation can be interchanged

� Example: the derivative of a call with respect to strike

� since the expectation is simply an integral

( ) ( )∫∫ ∂∂

⋅=⋅∂∂ zxf

xdzzxfdz

x,,

( )[ ] ( )

−∂∂

=−∂∂ 0,maxE0,maxE KS

KKS

K TT

( )[ ] ( )0,max)(0,maxE0

KSSdSKS TTTT −⋅⋅=− ∫∞

ϕ


Commutation of integration & differentiation

� We can use this trick to compute moments of a distribution� Example, 2nd moment of a central normal distribution:

( )

( ) ( )

2

21

1

22

1

2

2

2

1

1

221

1

221

lim22lim-

21

2lim

22

lim2

22

22

22

σ

λλ

σλσ

λ

σλ

λσπλ

σσπλσπ

λλ

λσ

λ

λσ

λ

σ

=

∂∂

−=

∂∂

=

−⋅⋅⋅∂∂

=

−⋅

∂∂

⋅=⋅⋅

−

→→

∞

∞−

⋅

−

→

∞

∞−

⋅−

→

∞

∞−

−

∫

∫∫

x

xx

edx

edxxedx


Jensen’s inequality (1)� Which is bigger?

� Denote and Taylor expand f(x) around x0

� Apply the expectation

� If then

� If then

[ ]( ) ( )[ ]xfxf EE?<

[ ]xx E0 =

L+′′−+′−+= )()(21)()()()( 0

20000 xfxxxfxxxfxf

[ ][ ]( )

[ ][ ]

[ ] L43421321

+′′−+′−+==−==

)()(E21)()(E)()(E 0

200

0E

0

)E

0

0

xfxxxfxxxfxfxxxf

0)( >′′ xf ( )[ ] [ ]( )xfxf EE >

0)( <′′ xf ( )[ ] [ ]( )xfxf EE <


Jensen’s inequality (2)

� Example: Is an “Asian” option cheaper than Vanilla?� Asian: spot of payoff is average over various spots

� Since f(x)=max(x,0) is a convex function (with f’’(x)>0) then

� Asian < sum of vanillas

[ ]( )0,Emax0,1maxAsian spots1

KSKSN

N

ii −=

−= ∑

=

( ) ( )[ ]0,maxE0,max1 vanillasofsum spots1

KSKSN

N

ii −=−= ∑

=


Jensen’s inequality (3)

� Exercise: Using Jensen’s inequality show that the call price E[max(S-K,0)] cannot be less than S-K


Relation between mean and variance

� Variance in terms of simple expectations Var[x] = E[x2]-E2[x]

� Derivation:

( ) [ ]( )[ ][ ] [ ][ ]

[ ] [ ] [ ] [ ][ ] [ ]XX

XXXXXXXX

XXX

22

22

22

2

EEEEE2E

EE2EEEVar

−=

+⋅⋅−=

+⋅⋅−=

−=


Basic notions of VBA Excel


Basic notions of VBA Excel� Enter the VBA environment : Alt+F11� Write a function in a MODULE



� HeaderOption ExplicitOption Base 1

� Create a VBA functionFunction GetDelta(ByVal a As Integer, ByVal b As Integer, ByVal c As Integer)Dim delta As Longdelta = b * b - 4 * a * cGetDelta = delta

End Function

� Declare a variable Dim nom_variable As type_variable (double, long, string, Range…)



� Create a VBA macroSub SommeDeuxValeurs()'declarationDim nb1 As IntegerDim nb2 As IntegerDim somme As Long'Lecturenb1 = InputBox("nbre 1")nb2 = InputBox("nbre 2")'Traitementsomme = nb1 + nb2'AffichageMsgBox "La somme est " & somme

End Sub



� Loops “For ... To ... Next”Function GetFactoriel(ByVal a As Integer)Dim fact As LongDim i As Integerfact = 1For i = 1 To afact = fact * i

Next iGetFactoriel = fact

End Function



� Tests “If ... Then ... Else”Function EstPositif(ByVal a As Double)If a > 0 ThenEstPositif = 1

ElseIf a < 0 ThenEstPositif = -1

ElseEstPositif = 0

End IfEnd Function



� Some useful functions

� Histogram of a distribution:� Function « frequence » in Excel

In Excel

•ALEA()

•LOI.NORMALE.STANDARD( x )

•LOI.NORMALE.INVERSE(x ;0;1)

In VBA Excel

• Rnd

•Application.WorksheetFunction.NormSDist( x )

•Application.WorksheetFunction.NormSInv( x )


Numerical methods in finance:some background history


Brownian Motion

� Robert Brown (botanist)� Observed motion of pollen

particles suspended in water (1827).


Stochastic methods in finance

� Louis Bachelier (1870 – 1946)� Considered as the founding fatherof financial mathematics.

� Was the first to have appliedmathematical models to the analysis of financial markets

� Stock prices evolve according to Brownian motion


Models for Brownian Motion

� Thorvald N. Thiele (1880), was the first to propose a mathematical theory to explainBrownian motion� Danish astronomer� Founder of an insurance company

� Louis Bachelier (1900) used Brownian motion in his thesis « La théorie de la spéculation » to describe stock prices

� Albert Einstein (1905) makes a statisticaltheory that explains Brownian motion and allows predictions

Random Walk


Why Brownian motion in finance?

� Paths resemble stock market indices � Problem: Brownian motion can turn negative !


Brownian motion (Wiener process)

� Brownian motion is stochastic process (=sequence of r.v.)� W(0), W(1), W(2), ...

� Main properties:� W(0) = 0� The increments W(2)-W(1), W(3)-W(2),...

are independent of each other� The increments W(t)-W(s) are normally distributed N(0,√(t-s) )

� Property 3 implies

� Property 2 implies

[ ] [ ] stWWWW stst −=−=− Varand0E

( )( )[ ] 0E =−− uvst WWWW


Useful properties

� A normal random variable X with mean m and variance v can be written as

� If x,y are constants

( )1,0~ with NWWvmX ⋅+=

[ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]ZWWZW,Z

W,ZxyZyWxyZxWWxxWyxW

yWxyxWWxxW

EEECovCov2VarVarVar

VarVarVarEE

EE

22

2

−=⋅++=+

==+

+=+=


Brownian motion: an example� Bob finishes his job at 5pm and before going home he makes a stop at the bar

� There he drinks a bit more than he should

� He leaves the bar at 8pm and usually (after some zig-zags) arrives home at midnight

� His home is just 500m away

� This means he proceeds towards home with an average speed of 0.5/4 = 0.125 km/hr

This is an example of a random walk


Brownian motion: an example

� Notation: Xt position at time tT=24hr t0=21hr Xt0=X0=0

� Random-walk model:� Position at next step Xt+1 given

position at previous step Xt

� Randomness comes through the increment Wt ~N(0,t)

� What is the meaning of and ?

� Bob takes first step:

�

� in this model is average speed� = 0.125 km/hr

�

� Small : random walk is confined� Large : random walk can make big jumps

ttt WtXX ∆⋅+∆⋅+=+ σµ1

ttt WtXX ⋅+⋅+= σµ0

[ ] tX t ⋅= µE

[ ] [ ]

[ ] tXtt

WtWtX

t

ttt

⋅+=

⋅+=

⋅++=

22

222

22222

E

2EE

σ

σµ

σµσµ

[ ]tXtVar12 =σ


Brownian motion: an example

� After several steps Bob arrives home

� The model describes his random walk as

� In the limit t 0:

� We are facing a problem:

� What is the meaning of an integral over a stochastic differential ?

� Stochastic calculus

∑∑∑

∑∑∑

=+

==

=+

=+

=

∆⋅+∆⋅+=

−⋅+−⋅+=

N

ii

N

ii

N

it

N

iii

N

iii

N

itT

WtX

WWttXX

i

i

01

00

01

01

0)()(

σµ

σµ

∫∫∑ ⋅+⋅+==

T

tt

T

t

N

itT dWdtXXi

000

σµ


� Kiyoshi Itô (1940s) develops stochastic calculus

� Itô integral :

with stochastic differential dW

� Itô’s lemma: differentiation of stochastic functions

� Robert Merton (1969) introduces stochastic calculusin finance to explain the price of financial products

� S ~ eW(t) >0 : The value of an underlying staysalways positive!

Stochastic calculus in mathematical finance

∫t

sdWsH0

)()(


� Robert Merton, Fisher Black & Myron Scholespublished the famous work on option pricing (1973)

� The model allows to derive analytic expression for the fairprice of call and put options

� A significant contribution to the growth of derivatives� Merton and Scholes receive the Nobel price of economics1997 (F. Black had died in 1995)

Option pricing with stochastic calculus


Stochastic integral (1)

� Definition:

� A useful property: The mean of a stochastic integral is zero

� Derivation

( )∫ ∑=

+∞→−⋅=⋅

b

a

N

itttNtt WWWgdWWg

01)(lim)(

( ) ( )[ ]

00)(Elim

E)(Elim)(limE

0

100

1

=⋅

=

−⋅

=

−⋅

∑

∑∑

=∞→

+=

∞→=

+∞→

N

itN

tt

N

itN

N

itttN

Wg

WWWgWWWg Independents increments

Mean of N(0,1)=0


Stochastic integral (2)

� Property

� Derivation

∫∫ =

T

t

T

tt dtWgdWWg0

2

2

0

)()(E

( ) ( )

( )

=

=

−⋅=

=

−⋅−⋅⋅=

∫

∑

∑∑∫

=+∞→

= =++∞→

T

t

n

iiiin

n

i

n

jjjiijin

T

tt

dtWg

ttWg

WWWWWgWgdWWg

0

2

11

2

1 111

2

0

)(E

)(Elim

)()(Elim)(ETerms with i not equal to j have zero expectation

dW2=dt


Exercises

� Show that E[(Wt-Ws)2] = t-s

� Show that E[Wt·Ws] = min(t,s)


How to model Brownian motion? (1)

� Choose a starting value of the random walk: e.g. at t=0 we have W0

For i=1 To NWt+ t = Wt + n · √∆t

Next i

� N is the number of steps in the random walk� t = Total time of walk / N

� n =normal random variable N(0,1)� VBA: n=Application.NormSInv(rnd), rnd=uniform random variable in [0,1]� n·√ t= normal random variable N(0, t)



� What will happen if we modify the previous code to:

For i=1 To N Wt+ t = Wt + v · n · √∆t

Next i

� What is the effect of the variable v?� It multiplies the random variable n

� If v is large, the random walk will show big fluctuations� If v is small, the random walk will show small fluctuations

� For this reason we interpret v as the volatility: it controls the size of fluctuations in the random walk



� What will happen if we modify the previous code to:

For i=1 To N Wt+ t = Wt + v · n · √∆t + d · t

Next i

� The new term is independent of the random variables � It always adds d · t to the previous position

� If d>0 the random walk will drift (on average) to positive values� If d<0 the random walk will drift (on average) to negative values

� Therefore we interpret d as the drift of the random walk


Variance of Brownian motion� We defined a Brownian motion as described by Wt which is a random variable normally distributed of zero mean and variance t

� Why is the variance of Wt equal to t ?

� At t=0: W(0)=W0� At t=1: W(1)=W0 + dW1� At t=2: W(2)=W0 + dW1 + dW2� ....� At t=N: W(N)=W0 + dW1 + dW2+…+ dWN

� At time step N, the position of the random walker is a sum of N independent normally distributed random variables

� Central limit theorem: variance of W(N) is proportional to N


The Black & Scholes model


The Black-Scholes model

Cartoon by S Harris


The Black & Scholes model� Simple brownian motion

� dS = · dW

� Black & Scholes model� dS = S · · dt + S · · dW

� S : value of underlying� stock, foreign exchange rate, etc

� µ : drift� the price of risk-free interest rate – annualised dividend: r-q (Equity) � Domestic minus foreign interest risk-free rates: rdom-rfor (Forex)

� : volatility (annualised)

� t : time (expressed in years)

� W: Wiener process (Brownian)


How to model the B&S stochastic eqn?

Black-Scholes model the underlying asdS = S · · dt + S · · dW

This means St+ t - St = S t · · t + St · · n · √∆t

Here is a pseudo-code:

For i=1 To N Si+1 = Si · (1 + · t + ·n·√(ti+1-ti) )

Next i

where n·√ t= normal random variable N(0, t)


What is the difference with Brownian motion?� Brownian motion

� St can be negative

� Geometric Brownian motion (Black & Scholes)

� St is strictly positive

WdSdtSrSd σ+=

WddtrSd σ+=


Exercise

� Can we model:

� Interest rates with Brownian motion?

� Bond prices with Brownian motion?


Itô’s Lemma� Itô’s process:

� x solution of dx=a(x,t) dt + b(x,t) dW� Consider a function G(x,t):

�

� dx² = [a(x,t) dt + b(x,t) dW]2= ??� Some properties in differential stochastic calculus:

� dt . dt = 0� dW . dt = 0� dW . dW=dt

22

2

21),( dx

xGdt

tGdx

xGtxdG

∂∂

+∂∂

+∂∂

=Additional term fromstochastic calculus

dWbxGdtb

xG

tGa

xGtxdG ⋅

∂∂

+

∂∂

+∂∂

+∂∂

= 22

2

21),(


Itô’s Lemma in two dimensions

dxdyyx

GdyyGdx

xGdy

xGdx

xGdt

tGtyxdG

∂∂∂

+∂∂

+∂∂

+∂∂

+∂∂

+∂∂

=2

22

22

2

2

21

21),,(

� Ito’s lemma in two dimensions (x and y are stochastic)

� Example in finance:� Two coupled stochastic processes used to describe the underlying� Spot and volatility are both stochastic� Spot and interest-rates are both stochastic

� Exercise: Calculate the stochastic differential of Z=X·Y and Z=X/Y


Itô’s Lemma� Exercise:Black-ScholesWhat is the differential of ln(S) ?

What is the value of S(T) ?

WdSdtSSd σµ +=

?)(ln =Sd


Itô’s Lemma

� Exercise:

� If the spot St follows a lognormal process what is the process followed by Xt=1/St ?

� Application: St is the EURUSD exchange rate while Xt is the USDEUR rate.



WddtSSd σµ +=

dt

µ

SSd

dtµ

dWσ

Differential equation of Black & Scholes

Random variable, distributed according to a normal distribution of 0 mean & variance t

Solution of the differential equation of Black & Scholes

)(2²

)0()(tWt

eStSσσµ +

−

=

Itô calculus

}}


Numerical solution of BS stoch. eqn.

� Stochastic differential equation

� For i=1… N

� Next i

� normal random variable

� Result: sequence of spots for all times

� Solution of stochastic differential equation

� For i=1…N

� Next i

� normal random variable

� Result: sequence of spot for all times

tttt WdSdtSSd σµ +=

( )iiii tttt xtSSS ⋅+∆⋅⋅+=

+σµ

1

,...,,210 ttt SSS

( ) ( ) iiiii

ii

xtttttt eSS ⋅−+−⋅− ++

+⋅= 11

221

1

σσµ

,...,,210 ttt SSS

ix ix

SAME


Absence of arbitrage� Absence of Arbitrage (AOA)

� Normally there can be no profit without taking a risk. � However, if an opportunity for riskless profit arises, the market reacts immediately, and soon the opportunity disappears.

� It is the basis of the Black-Scholes model � ...and of most other derivative models.

� This condition allows us to determine the expectation of the underlying

� An example …


No-arbitrage (1)� EURUSD = 1.3 = So (1 EUR equals 1.3 USD)

� 1 EUR = underlying, USD payment currency� I start with no money� I borrow 1 EUR from a European bank, with 1 year maturity, interest rate q. In one year I must pay back eqT (=1 + q T + …)

� I convert today my EUR to USD, I receive So USD� I enter into a Forward contract (for free), allowing me to change USD into EUR within a year, at a fixed rate Fo.

� I deposit So USD into an american bank with interest rate r. After 1 year I receive: So erT

� After 1 year, I will have gained (without taking any risk):- eqT (money to pay back in european bank) + So erT / Fo (money I receive from american bank in EUR)

� AOA implies that the forward contract has value Fo = So e(r-q)T


No-arbitrage (2)� This allows us to fix the variable that appeared in

� Since we found that the forward rate F must be

� And because it represents the expected spot

� Therefore

µtttt WdSdtSSd σµ +=

( )TqrT eSF −= 0

[ ] ( ) [ ] TxTTTT eSeeSSF ⋅− ⋅=⋅== µσσµ

00 EE2

21

qr −=µ


Derivation of the Black-Scholes PDE� Composition of portfolio:

� 1 option of value V(S,t)� An amount of the underlying

� We adjust the amount ∆ such that the portfolio is not sensitive to risk (suchas small random movements of the underlying)

� Putting it together, the portfolio P consists of:� P = V + ∆ S

� The variation of the portfolio after an very small amount of time is� dP = dV + ∆ dSWith� dS = (r – q) S dt + σ S dw (differential equation of B&S)

� ( )22

2

21 dS

SVdS

SVdt

tVdV

∂∂

⋅+∂∂

+∂∂

=

Classic differentialcalculus

Additional term in stochasticdifferential calculus


Derivation of the Black-Scholes PDE� Some useful rules of the stochastic differential calculus

� dt · dt = 0� dW · dt = 0� dW · dW=dt

� (dS)² = ?� dS · dS = [µ S dt + σ S dw] · [µ S dt + σ S dw]

= σ² S² dt• We arrive at the variation of our portfolio P:

� dSdtSSVdS

SVdt

tVdP ∆+

∂∂

⋅+∂∂

+∂∂

= 222

2

21 σ


Derivation of the Black-Scholes PDE

• We suppress all sources of risk (risk=randomness) of the underlying (dS):� � « delta » of an option

• We arrive at the variation of the portfolio P� The remaining portfolio contains more sources of risk: it must evolve as money placed into a "safe" savings account with interest rate r

dSdtSSVdS

SVdt

tVdP ∆+

∂∂

⋅+∂∂

+∂∂

= 222

2

21 σ

SV

SV

∂∂

−=∆⇔=∆+∂∂ 0

dtSSVVrdtPrdtS

SVdt

tVdP ⋅

∂∂

−⋅=⋅⋅=∂∂

⋅+∂∂

= )(21 22

2

2

σ

PDE of Black-Scholes


The three forms of the B&S model

� Stochastic differential equation

� Solution of the stochastic differential equation

� Partial differential equation governing the evolution of the price

of a derivative (pricing equation)

WddtrSSd σ+=

)(2²

)0()(tWtr

eStSσσ

+

−

=

021

2

222 =−

∂∂

+∂∂

+∂∂ rV

SVrS

SVS

tV σ


� Call and Put options

Solution of the Black & Scholes model

TdT

TqrKSd

TTqrKSd

dNeSdNeKpdNeKdNeSc

qTrT

rTqT

σσ

σ

σσ

−=−−+

=

+−+=

−−−=

−=−−

−−

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(where

)()(

)()(


Derivation of the Call price for the Black-Scholes model� At maturity, the call value is g(ST) = max(0,ST-K) (ST-K)+� Call price: expectation of the payoff, discounted to the value of today

� S(ST): Distribution function of the random variable ST� The assumed process for the random variable ST

has solution

� where X a normal random variable (mean 0, variance 1)

[ ] TTTSrT

TrT dSSgSeSgEe ⋅⋅== ∫

+∞

∞−

−− )()()(Call ϕ

WdSdtSSd σµ +=

2

21

21)(

X

XeX

−=

πϕ

[ ] dXXSgXXSgECalleTXT

rT ⋅⋅==⋅ ∫+∞

∞−

))(()())(( ϕ

ST: spot

K: strikee-rT: Discount factor

( ) XTTT eSS σσµ +−⋅=

221

0


[ ] ( )[ ] dXKeSeXSgECalle XTTX

TrT ⋅−⋅⋅==⋅

+

+⋅−+∞

∞−

−

∫ σσµ

π2

25.0

021

21))((

( )( )

T

TSK

XKeS XTT

σ

σµσσµ

2

05.00

5.0ln

02

⋅−−

>⇔>−⋅ +⋅−

κ( )[ ]

( )dXKedXe

eS

dXKeSeCalle

XXTXT

XTTXrT

.21

2

21

222

22

21

215.0

0

5.00

21

∫∫

∫∞+

−∞+

+−⋅−

+

+⋅−+∞

−

⋅−⋅⋅⋅

=

⋅−⋅⋅=⋅

κκ

σσµ

σσµ

κ

ππ

π

A B

Derivation of Black-Scholes call price


[ ]

[ ] )()(1

)()(.21 2

21

κκ

κϕπ κκ

−⋅=−⋅=

−+∞⋅=⋅⋅=⋅= ∫∫+∞+∞

−

NKNK

NNKdXKdXKeBX

X

The easy part:

The more difficult part:( )

∫∞+

+−⋅−

⋅⋅⋅

=κ

σσµ

πdXe

eSA

XTXT 2

2

215.0

0

2

We would like to bring this to an integral of the form2

21

dUz

eU−

∞−∫

« Complete the square » TUTTXXTX

U

221

212

212

212

21 2

2

σσσσ +−=+−−=+−

4434421

[ ][ ] [ ])()(1

)()(2

00

021

02

κσσκ

σκπ

µµ

µ

σκ

µ

−⋅⋅=−−⋅⋅=

−−+∞⋅⋅=⋅⋅⋅

= ∫+∞

−

−

TNeSTNeS

TNNeSdUeeS

A

TT

T

T

UT

Most common way to do this is:


Finaly the value of the Call:

[ ] )()(0

κκσ −⋅−−⋅⋅=−⋅= ⋅−⋅−− NKeTNeSBAeCall TrTqrT

Equivalently, in the standard notation:

)()(210

dNKedNeSCall TrTq ⋅−⋅⋅= ⋅−⋅−

TdT

TqrKSd

TTqrKSTd

σσ

σκ

σσκσ

−=−−+

=−=

+−+=−=

10

2

01

)2/2()/ln(

)2/2()/ln(where

Exercise: calculate the price of a « digital » option (it pays at maturity 1 unit of underlying if ST>K)


Interpretation of the Black-Scholes formula

� N(d2): probability that spot finishes in the money

� N(d1): measures how far in the money the spot is expected to be if it finishes in the money

� Call price: value of receiving the stock in the event of exerciseminus cost of paying the strike price

)()( 21 dNKedNSeC TrTq ⋅⋅−⋅⋅= ⋅−⋅−


Interpretation of Black-Scholes formula

� What does N(d2) mean?� To answer this: calculate probability that spot finishes in the money:

[ ] ( )

)()()(

Indicator)(

Indicator)(Prob

2

21

0

2

2

2

dNxdxxdx

KeSxdx

KSxdxKS

d

d

xTTr

TT

=⋅=⋅=

>⋅⋅=

>⋅⋅=>

∫∫

∫

∫

∞−

∞

−

∞

∞−

+

−

∞

∞−

ϕϕ

ϕ

ϕ

σσ

≤>

=0 xif00 xif1

x)Indicator( on distributiNormal21)(

2

21

==− x

exπ

ϕwhere


Digital: An important market quote

� A digital option pays at maturity

� This means the value today is (see previous computation)

� Digital gives probability spot finishes in the money

≤>

=>KSKS

KST

TT if0

if1)Indicator(

K

1

( )[ ] ( )2IndicatorEDigital dNeKSe rTT

rT −− =>=


Black-Scholes and risk-neutrality

� The Black-Scholes formula

depends on the Spot, Volatility, Interest-rates and time.� None of these parameters involves the risk-preference of the investor.

� Therefore, the B&S formula does not depend on any assumption about the risk-preferences of the investors

)()( 21 dNKedNSeC TrTq ⋅⋅−⋅⋅= ⋅−⋅−


Assumptions of the B&S model

� More Important� Underlying evolves according to a lognormal process� Volatility ( size of fluctuations) is constant and known� No arbitrage opportunities exist

� Less important� No dividends� No transaction costs� Risk-free rates are constant


How realistic are the assumptions of the B&S model ?

� In real markets the size of the fluctuations is not constant� The underlying can make big jumps on some economic news� Calculating the volatility is not trivial� The process of the underlying is typically not lognormal� Interest rates are not constant

� All assumptions are wrong in reality !� They are made only to simplify the calculations


Comparison with a earlier model: Bachelier

� Black-Scholes (1973)� Assumptions

� Result:

ttt WdSSd σ=

� Bachelier (1900)� Assumptions

� Result:

tt WdSSd 0σ=tWTT

t eSS ⋅⋅+−⋅= σσ 221

0( )tt WSS ⋅+⋅= σ10

( )( ) ( )

+⋅−

+⋅=

−⋅⋅= ∫ +−

TTNK

TTNS

KeSxdx

kS

kS

xTT

σσ

σσ

ϕ σσ

2212

21

0

0

00

221

lnln

0,max)(Call ( )

−⋅⋅−

−⋅−=

−⋅+⋅⋅⋅= ∫

TSKSTS

TSKSNKS

KxSxdx

σϕσ

σ

σϕ

0

00

0

00

0

)(

0,)1(max)(Call

N(x): cumulative normal, (x): normal density


Exercise

� Derive the call price for Bachelier’s model:

� Derive the price today of a quadratic call payoff for a lognormal model:

( )

−⋅⋅−

−⋅−=

−⋅+⋅⋅⋅= ∫

TSKSTS

TSKSNKS

KxSxdx

σϕσ

σ

σϕ

0

00

0

00

0

)(

0,)1(max)(Call

( )∫ −⋅⋅= 0,max)(Call 22 KSxdx Tϕ


How wrong was Bachelier?

� Call option with� T=0.75� vol = 0.25� K=90

� In practice, indistinguishable!


Option pricing after the 2008 crisis (1)

� The crisis showed that even top institutions can fail� Merrill Lynch, AIG, Lehman Brothers,…� Northern Rock, Bear Sterns, Fortis

� Even countries are at risk � Greece, Portugal, Hungary, Ireland

� Is there then a truly “risk-free” rate ?� risk-free = guaranteed 100%?� If this doesn’t exist our analysis is not correct !



� The crisis changed completely the way transactions are made in practice

� Because of the fear of a counterparty failure, now both counterparties agree to put into a separate account an amount of money that can be used as a guarantee in the case of failure.

� This amount of money is called collateral



� Standard theory

today maturity

Party A buys

Party B sells

premium

Party A

Party B

Cashflow of option



� How it works in practise

today maturity

Party A buys

Party B sells

Party A

Party B

Cashflow of option

Party A buys

Party B sells

premium

Party A

Party B

collateralInterest on

collateral



� What are the implications of the collateral exchange?� The equation

for the evolution of an amount B in the savings account is not true. It is not guaranteed 100% to grow always.

� The collateral amount of money needs to be funded (obtained from another source). Therefore we need to borrow this collateral by paying a new interest rate.

� Option pricing theory needs to include this adjustment

dtrBdB ⋅⋅=


Call-Put parity relation

� Call-Put = = S·e-qT-K·e-rT =(F-K)·e-rT

� The price of a call is linked to the price of a put through the forward



� Solution of the Black-Scholes model for the price of a call/put with barrier� Barrier « in » : the option is activated only if the barrier is touched

� Barrier « out » : the option is dead if the barrier istouched



� Solution of the Black-Scholes model for the price of a call/put with barrier� Barrier « up » : the barrier must be touched while the spot rises

� Barrier « down » : the barrier must be touched whilethe spot declines

Call / Put, in / out, up / down � 8 possible combinations



� Parity relations:

c = cui + cuoc = cdi + cdop = pui + puop = pdi + pdo



� Price of barrier options



� Price of « touch » options

� One-Touch Up with So<H

� One-Touch Down with So>H

( )T

TqrSH

yσ

σ )2/(ln 2

1

+−+=

TTqrKSd

σσ )2/2()/ln( 0

2−−+

=

( )

( ))( 22

1

22

1

dNterm

TyNSHterm

o

=

+−⋅

=

−

σλ

( )

( ))( 22

1

22

1

dNterm

TyNSHterm

o

−=

−⋅

=

−

σλ

( )21Pr termtermeice rT += −


Important identities in the B&S model (1)

� and

� Derivation:

σσ12 dd

−=∂∂

σσ21 dd

−=∂∂

( )

( )

( )( )

σσ

σ

σ

σ

σσσ

σ

σ

σσ

2

221

2

221

221

1

ln1

ln)()(

ln

dT

TqrKS

T

TqrKSTTT

T

TqrKS

d

−=

−−+

−=

+−+

−⋅

=

+−+

∂∂

=∂∂



� and and similarly and

� Derivation

σT

rd

=∂∂ 2

σT

rd

=∂∂ 1

( )

σσ

σ

σ

TT

T

T

TqrKS

rrd

==

+−+

∂∂

=∂∂

ln 221

1

σT

qd

−=∂∂ 1

σT

qd

−=∂∂ 2



� where and

� Derivation

� We will show that

� Start from right-hand side

)()( 210 dneKdneS rTqT ⋅⋅=⋅⋅ −−2

21

21)(

dedn

−=

π)()( dNdn ′=

( )2122

0

1

20

21)(ln

)()( ddTqr

KS

dndn

eKeS

rT

qT

−−=−+

⇒=

⋅⋅

−

−

( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )TqrKS

TTqrTqr

TTqrTqr

dddddd

KS

KS

KS

KS

−+

=

+−−−−−+⋅

⋅

+−++−−+−=

−+−=−−

0

2212

21

2212

21

121221

22

ln

lnln

lnln2121

21

00

00

σσσ

σσσ


The Greek Letters

� Delta :

� Gamma :

� Vega :

� Theta :

SP

∂∂

=∆

SSP

∂∆∂

=∂∂

=Γ 2

2

συ

∂∂

=P

tP∂∂

=Θ

The mostimportant quantity for the dailymanagement of the tradingbooks


The Greek Letters

� They represent sensitivities of the portfolio with respect to market parameters

� They allow us to monitor the risk of the portfolio

� They can be applied to a single derivative or to a portfolio of derivatives


GreeksAnalytic expressions for the Greeks (here for a Call):�

�

�

�

TSedN

o

qT

σ

−′=Γ

)( 1

qTo edNTS −′= )( 1υ

rTqTo

qTo edNrKedNqS

TedNS −−

−

−+′

−=Θ )()(2)(

211 σ

N’(x) =ϕ(x)

probability density of a normal random variable

)( 1dNe qT−=∆


Demonstration: Delta

� and

� Derivation:

� Now use the fact thatand and

� And also the identity we proved:

� to eliminate the two right-most terms and obtain the result

)(Call1

0

dNeS

qT−=∂

∂=∆ )(Put

10

dNeS

rT −−=∂∂

=∆ −

( ) ( )( )

( ) ( ) ( )0

2

0

101

21000

Call

SdNeK

SdNeSdNe

dNeKdNeSSS

rTqTqT

rTqT

∂∂

⋅−∂

∂⋅−⋅=

⋅⋅−⋅⋅∂∂

=∂∂

−−−

−−

0

2,1

2,1

2,1

0

2,1 )()(Sd

ddN

SdN

∂∂

⋅∂

∂=

∂∂

)()(

2,12,1

2,1 dnddN

=∂

∂ 2

21

21)(

dedn

−=

π0

2

0

1

Sd

Sd

∂∂

=∂∂

)()( 210 dneKdneS rTqT ⋅⋅=⋅⋅ −−


Example� A bank has sold

� European call option for $300,000 � on 100,000 shares � of a non-dividend paying stock

� Market parameters areS0 = 49 σ = 20%, K = 50 T = 20 weeksr = 5%

� The Black-Scholes value of the option is $240,000� How does the bank hedge its risk to lock in a $60,000 profit?

example


Naked & Covered Positions

� Naked positionTake no action

� Covered positionBuy 100,000 shares today

� Both strategies leave the bank exposed to significant risk


Delta� Delta (∆) is the rate of change of the option price with respect to the underlying

� Delta small option price does not move when spot moves� Delta large option price moves when spot moves

Optionprice

A

B Slope = ∆

Stock price


Delta Hedging� This involves maintaining a delta neutral portfolio

� Delta neutral: ∆=0� This means that if the spot makes a small change the value of the portfolio does not change

� Eliminates spot risk

� Delta hedging is done by buying/selling the underlying (e.g. cash or stocks)

� Black-Scholes theory shows� that a Delta-neutral portfolio is possible� what is the correct amount of the underlying to short


Delta: an example

� Call option with: � Premium 400€� Delta 50%� Spot today is at S0=100

� This means that � If spot moves to S0=110� The premium will move to 405€ (=400 + 0.5 * 10) � (with all other market parameters unchanged)


Theta

� Theta (Θ) is the change in value of the derivative with respect to the passage of time

� The theta of a call or put is usually negative. � meaning: as time passes the value of the option decreases

� Practically, change in time is 1 day.

Exemple2


Theta: an example

� Call option which today is worth: � Premium 20€� Theta -0.5

� This means that � tomorrow the premium goes to 19.5€� (with all other market parameters unchanged)


Gamma

� Gamma (Γ) is the rate of change of delta (∆) with respect to the price of the underlying asset� Gamma is small Delta is stable under spot movements� Gamma is large Delta is not stable under spot movements

� Gamma neutral hedge:� portfolio and Delta are stable under spot movements.� better hedge than simple Delta-neutral (but more expensive!)

� Gamma is the second derivative of the derivative value withrespect to the underlying price


Interpretation of Gamma� Gamma Addresses Delta Hedging Errors Caused By Curvature

S

CStock price

S'

Callprice

C''C'


Relationship Between Delta, Gamma, and Theta

For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q

Θ ∆ Γ Π+ − + =( )r q S S r12

2 2σ


Vega

� Vega (ν) represents the change in value of a derivative with if market volatility moves by 1%

� Vega tends to be greatest for options that are close to the at-the-money� Risk that volatility can move the spot out of the money


Vega: an example

� Call option with � Premium 20€� Vega 0.5� Market Vol 20%

� This means that � If market Vol goes to 21%� Premium goes to 20.5€


Managing Delta, Gamma, & Vega

� ∆ can be changed by taking a position in the underlying

� To adjust Γ & ν it is necessary to take a position in an option or other derivative


00.000010.000020.000030.000040.000050.00006

0.8 1 1.2 1.4 1.6 1.8

spot

vega

0.6y1y

� Call option, strike 1.25

Price Delta Gamma

Vega� Option price becomes linear for large spots� Delta ~ cumulative function� Convexity risk (Gamma) highest at-the-money� Vol risk (vega) is highest at-the-money

Spotladders: vanilla

00.51

1.52

2.53

3.5

0.8 1 1.2 1.4 1.6 1.8

spot

gamma

0.6y1y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.8 1 1.2 1.4 1.6 1.8

spot

price

0.6y1y

0

0.2

0.4

0.6

0.8

1

1.2

0.8 1 1.2 1.4 1.6 1.8

spot

delta

0.6y1y


Spotladders: barrier option

� Knock-out option, strike 1.25, barrier 1.35

Price Delta Gamma

Vega� Option price: 0 at barrier and out-of-the-money � Delta, Gamma, Vega can be negative unlike vanilla!

00.00050.0010.00150.0020.00250.0030.00350.0040.0045

0.8 0.9 1 1.1 1.2 1.3 1.4

spot

price

0.6y1y

-0.06-0.05-0.04-0.03-0.02-0.01

00.010.020.03

0.8 0.9 1 1.1 1.2 1.3spot

delta

0.6y1y

-0.6

-0.4

-0.2

0

0.2

0.4

0.8 1 1.2spot

gamma

0.6y1y

-0.000008

-0.000006

-0.000004

-0.000002

0

0.000002

0.000004

0.8 1 1.2

spot

vega 0.6y

1y


Rho

� Rho is the rate of change of the value of a derivative with respect to the interest rate


Example of risk hedging� The forward (=the expected spot) is set by the market� Imagine you are a trader in EURUSD � What are the risks in each of the positions?

-5 mlGAMMA-5 mlGAMMA+5 mlGAMMA+5 mlGAMMA

-1 mlDELTA+1 mlDELTA-1 mlDELTA+1 mlDELTA

1.431M FORWARD1.431M FORWARD1.431M FORWARD1.431M FORWARD

1.41SPOT1.41SPOT1.41SPOT1.41SPOT

EURUSD positionEURUSD positionEURUSD positionEURUSD position

Spot is expected to go up.

Delta is >0, Gamma>0

Probably will make money

If Spot increases (as expected by mkt) Delta<0 will lose money but Gamma>0 will soon turn Delta from <0 to >0.

If Spot increases Delta>0 will make money but Gamma<0 will soon turn Delta from >0 to to <0. Then will lose money.

If Spot increases Delta<0 will lose money while Gamma<0 make Delta even more negative. Losses will mount!


Volatility « smile »

A practitioner’s introduction


Implied volatility

� Traders often quote vols instead of prices

� This means:

� Implied vol: the vol to put in the BS pricer to get the price

� It is not equivalent to historical vol: � measure of historical fluctuations

� It does not give information about the dynamics

BS pricervol price

BS pricervol price


Numerical computation of the implied vol

� Aim: Compute the implied vol corresponding to a call price C� Use the bisection method:

1. Take two extreme points volmin and volmax2. Compute the middle point: volmid=0.5 (volmin+volmax)3. Compute the BS Call price using volmid: BSC4. If the BSC > C then

set volmax=volmidElse set volmin=volmid

5. Repeat 2-5 till BSC comes very close to C


Historical vs implied volatility

� Historical Volatility (Annualised):

� Represents the size of fluctuations in the process S� A measure for the vol of the past

� Implied Volatility:

� Represents the price of a vanilla option today� A measure for the vol of the future

( )11

2hist ln with 11

−=

=

−= ∑

i

ii

N

ii S

SrrrNT

σ


Measuring historical volatility

� EURUSD� 6-month data, closing of day� Historical vol = 5.2%� Implied vol in Apr2010 = 17%

� Measuring historical vol is not easy� Which data set do we take?� min, hourly, daily intervals?� How do we account for low/high?

� Black-Scholes assumption on volis wrong:� Apr-Jun: high volatility� Oct-Nov: low volatility


Interpretation of implied volatility (1)

� If instead of the Black-Scholes assumption

� we consider

� where (t) a deterministic “instantaneous” vol� Then we have a solution for the spot process

� And therefore for the vanilla price and implied vol we can write

tttt dWSdtSdS ⋅⋅+⋅⋅= σµ

tttt dWtSdtSdS ⋅⋅+⋅⋅= )(σµ

⋅+

−⋅= TWTTrSTS σσ 2

21exp)0()( ∫ ⋅=

T

dttT 0

22 )(1 σσ

( ) ( )σσ CallCall impl = → σσ =impl


Exercise (instantaneous vol = no smile)� Show that� leads to with

� Solution:� From Ito’s Lemma we have� with mean and variance:

� This means lnST is normal with mean and variance as above, so:

tttt dWtSdtSdS ⋅⋅+⋅⋅= )(σµ

⋅+

−⋅= TWTTrSTS σσ 2

21exp)0()( ∫ ⋅=

T

dttT 0

22 )(1 σσ

( ) ( )∫+−+=T

tT dWtTSS0

221

0lnln σσµ

[ ] ( )TSST2

21

0lnlnE σµ −+=

( )[ ] [ ] ( ) ( )∫∫ =

=−

TT

tTT dttWdtSS0

2

2

0

22 ElnElnE σσ

( ) ( ) T

T

T WdttTSS ~lnln0

2221

0 ⋅+−+= ∫σσµ


Interpretation of implied volatility (2)

� This relation implies that the implied variance can be seen as an average over instantaneous variances

∫ ⋅=T

dttT 0

22impl )(1 σσ

(1) (2) (3)

Spot

timet1 t2 t3 t4


Black-Scholes vs market� Black&Scholes-price < market-price, for very low / very high strikes� Plug market-price in BS formula to calculate implied volatility� Black-Scholes theory assumes that the spot fluctuations (vol) do not

depend on the strike of the option� Here we observe a parabolic-shape looking like a smile

Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 1.5500

strike

USD

cas

h

Black-Scholes

Market

Smile

13.00%

13.50%

14.00%

14.50%

15.00%

1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 1.5500 1.6000

StrikeVo

latil

ity

Black-Scholes

Market


Reasons for the smile? (1)

Supply and demand:

� Traders buy out-of- the-money puts as protection for market crashes

� As option price rises, so do the implied vols

� Traders look at support/resistance levels which set the strike where they buy


Reasons for the smile? (2)� Black-Scholes assumption for log-normality is wrong� Plotting the market-implied terminal spot density reveals fat tails

( ) 2

mkt2

0Call

2

KeSSP Tr

T ∂∂

⋅= ⋅

Fat tails:

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes:

•Spot dynamics is not lognormal

•Spot fluctuations (vol) are not constant


Reasons for the smile? (3)

� In real trading there are jumps in the spot that are not foreseen in the Black-Scholes model

� Extreme events appear more often than predicted by the lognormal distribution

� The volatility we observe is not constant

� Jumps are observed in the evolution of prices


Why use Black-Scholes model

� Despite its shortcomings the Black-Scholes model is accepted due to its simplicity

� Today it is used as a means of communication: to make sure all parties talk about the same option

� Risk-sensitivities (Delta, Vega, etc) are quoted in terms of their implied Black-Scholes value


Vanilla as function of vol

� A simple and useful property of the vanilla price is the monotonicity of the price with respect to the volatility

� This will be used extensively in subsequent proofs


At the money� Various definitions across asset classes

� A common one:

� The at-the-money point is the point where the delta of the call equals the delta of the put (in absolute value)

� Solving this relation for Katm gives

� F: the forward, T: maturity

),(),( atmatmputatmatmcall σσ KK ∆−=∆

TeFK

⋅⋅=

2atm2

1

atm

σ


Smile Stickiness

� Two types of market smiles:

� Sticky-strikeThe smiles changes when the spot changesNatural definition of market smile

� Sticky-deltaThe smiles changes when the “moneyness” changesEfficient when the market spot changes rapidly, e.g. in Forex

� Moneyness: number of standard deviations that strike is away from forward

TFK

σ

ln


Sticky delta

� Market quotes are given in terms of the Black-Scholes Delta

� For example, “vol at 25-Delta-Call =10%”

� This equation provides the strike K25

41)1.0,( 2525call ==∆ ∆∆ σK


Market quotes (1)

� Market quotes the smile in an implicit way� Quotes KATM and two other strikes K1, K2� 3 vol quotes are given

� ATM� Butterfly: implies the smile convexity� Risk-Reversal: implies the smile skew

� If RR<0 2< 1left-side higher than right-side

� If BF is large: the distance between ATMand the average vol at K1 and K2 is large

( )12RR

ATM21BF

ATM

21

σσσ

σσσσ

σ

−=

−+=

1

2ATM

K1 KATM K2


Market quotes (2)

� Special case: RR=0 and BF=0 implies that

� 1= 2= ATM

� The smile is a flat line

� We return to Black-Scholes world


Market quotes (3)

� RR: controls skew� BF: controls convexity


Stylised facts (1): ATM vs smile-min

� The ATM point of Katm=Fexp(½ atmT) is close to the minimum of the smile curve


Stylised facts (2): convexity vs maturity

� Smiles of short-maturities are more pronounced


Stylised facts (3): spot vs vol� F. Black: “I have noticed that when spot increases voldecreases and when spot decreases vol increases”

� Similarly for spot vs realised vols


Contruction of the implied vol surface

� Market quotes few points on along the strike axis� 3 points in illiquid underlyings� 5 points in liquid underlyings

� Practitioners use interpolation/extrapolation methods to obtain the non-quoted part of the curve

� In general:� Time-interpolation: linear-in-variance� Strike-interpolation: natural cubic spline

� Caution to no-arbitrage conditions!


No-arbitrage conditions: spot PDF� The market quoted smile must lead to a non-negative spot PDF

( ) 2

mkt2

0Call

2

KeSSP Tr

T ∂∂

⋅= ⋅


Market data: vols, BFs, RRs, spread


Market data interpolation: rates� The market is quoting the rates

� r1 between t0 (today) and t1� r2 between t0 (today) and t2

� What is the rate between t1 and t2 ?

� The interpolation is linear due to the continuity required by discounting:

� DF(t0, t2) = DF(t0, t1) · DF(t1, t2)

� This implies:

t0 t1 t2

Mkt: r1 Mkt: r2

interpolation: r

12

011022)()()( )()(12011022

ttttrttrreee ttrttrttr

−−⋅−−⋅

=⇒⋅= −⋅−−⋅−−⋅−


Market data interpolation: vol (in time)

� The market is quoting the implied vols� 1 between t0 (today) and t1� 2 between t0 (today) and t2

� What is the vol between t1 and t2 ?� The interpolation is “linear-in-variance”:

� This implies

t0 t1 t2

Mkt: 1Mkt: 2

interpolation:

( ) ( )( )12

012102

222

tttttt

−−⋅−−⋅

=σσσ

( ) ( ) ( )

( ) ( ) ( ) ( )[ ]01121022

22

12

22

12

2

12

2

1

11 1

0

2

0

2

1

tttttttt

dssdsstt

dsstt

t

tinst

t

tinst

t

tinst

−⋅−−⋅−

=

−

−=

−= ∫∫∫

σσ

σσσσ


Other possibilities to get to the smile� If Black-Scholes assumption is wrong then let us consider that vol is not constant

� Calculations are not as elegant and simple anymore� Two of the mainstream models:

WdStSdtSSd ),(σµ +=

[ ]

=+−=

+=

∞

dtWdWdEWdVdtVVdV

WdSVdtSSd

ρεκ

µ

21

2

1

)(

� Local Volatility model

� Volatility is deterministic: depends on the time and spot

� This model can reproduce the smile

� Stochastic Volatility model

� Spot: Geometric Brownian motion� Variance is stochastic:by design returns to a long-term mean value V∞


Local-vol vs Stochastic-vol� Local- and Stochastic-vol models can reproduce the vanilla-

smile perfectly � But can differ dramatically when pricing exotics!


Hedging with smile

� Without correct fitting of the true forward smiles, hedging is a problem

Example: Delta hedging

� Price depends on the implied-vol, which is constructed for a given spot

� Bumping the spot, means that the implied-vol surface will change

� “Model Risk”: risk induced by difference in models

� Different models will give a different and

( )( ) ( )( )S

SSSSSSSS∆⋅

∆−∆−−∆+∆+=∆

2;Price;Price implimpl σσ

( )SS ∆+implσ ( )SS ∆−implσ


Mainstream models

Three main categories

� Volatility is local � Non-parametric, e.g. Dupire� Parametric, e.g. Quadratic Local Vol

� Volatility is stochastic� Mean-reverting, e.g. Heston

� Spot process allows for jumps� Merton jump-diffusion

All combinations are possible:

� Stochastic Local Vol model

� Heston + Jumps = Bates model

� …


Mainstream models: Black-Scholes (1973)

� Constant vol (size of spot fluctuations)� Lognormal process for the spot

dWSdtSdS tttt ⋅⋅+⋅⋅= σµ


Mainstream models: Bachelier (1900)

� Simpler than Black-Scholes� Allows for smile (in the Black-Scholes world)

� Can be seen as a primitive “local vol” model

� Has analytic solution

� But can lead to negative values for the underlying

dWS

SdtS

SdSt

tt

tt ⋅

⋅+⋅

⋅=

σµ

dWdtdSt ⋅+⋅= σµ


Mainstream models: Merton (1976)

� Lognormal process for spot � Poisson frequency of jumps

� Jumps is a realistic ingredient� Model allows full analytic solution for vanillas� Improves Black-Scholes in the fat tails

( ) tttttt dqdWSdtSkdS +⋅⋅+⋅⋅−= σλµ


Mainstream models: Dupire (1994)

� Developed by Dupire and Derman & Kani� Reproduces market vanillas� Non-parametric, non-arbitrageable

� But historical data does support the idea that spot-volatility is a deterministic function

� This can lead to wrong hedging

dWStSdtSdS ttttt ⋅⋅+⋅⋅= ),(σµ


Mainstream models: Quadratic Local Vol

� Similar to Dupire’s Local Vol

� But in parametric form

� Proxy parameters: 0~ATM, ~ skew, ~convexity� Allows analytic solution and fits smile dynamics

dWSSdtSdS ttttt ⋅⋅+⋅⋅= )(σµ

( )2

000 11

−⋅+

−⋅+=

FF

FFS tt

t βασσ


Mainstream models: Heston (1993)

� Stochastic Vol model

� Spot: lognormal process� variance: mean-reverting process � Has analytic solution for vanillas

+−=

+=

∞ 2

1

)( WdVdtVVdV

WdSVdtSSd

εκ

µ


Mainstream models: SLV (2007)

� Stochastic Local Vol

� Uses Dupire Local Vol in the calibration process� Fits the smile � Hard to implement

Sttttt dWStZtSdtSqrdS ⋅⋅⋅+⋅⋅−= )(),()( σ

( ) Ztttt dWdtZZd ⋅+⋅−= λθκ lnln


Model selection

� Large number of alternative models:

� Volatility is stochastic� Spot process is not lognormal � Random variables are not Gaussian � Random path has memory (“non-markovian”)� The time increment is a random variable (Levy processes)� Correlation between vol and spot is stochastic� And many many more…

� A successful model must allow quick and exact pricing of vanillas to reproduce smile

� Wilmott: “maths is like the equipment in mountain climbing: too much of it and you will be pulled down by its weight, too few and you won’t make it to the top”


Model selection: Calibration

� Calibrationminimize (model output – market observable)2

Example (model ATM vol – market ATM vol)2

� Parameter space should not be� too small: model cannot reproduce all market-quotes across tenors� too large: more than one solution exists to calibration


How to simulate a stochastic process

� Consider the stochastic differential equation

� With solution

� To solve it numerically:� Euler discretization: freeze integrals at their lower limit value

( ) ( ) tttt dWXbdtXadX ⋅+⋅=

( ) ( )∫∫ ⋅+⋅+=t

uu

t

ut dWXbduXaXXττ

τ

( ) ( ) ( ) ( )ττττ τ WWXbtXaXX tt −⋅+−⋅+=


Model selection: One-touch tables� OT tables measure model success vs market price� OT price ≈ probability of touching barrier (discounted)� Collect mkt prices for TV in the range:

0%-100% (away-close to barrier)� Calculate model price – market price� The better model gives model-mkt≈0

OT table

-7.00%

-6.00%

-5.00%

-4.00%

-3.00%

-2.00%

-1.00%

0.00%

1.00%

2.00%

3.00%

0 0.2 0.4 0.6 0.8 1

TV price

mkt - mod

el

VannaVolgaLocalVolHeston

OT tables depend on

� nbr barriers

� Type of underlying

� Maturity

� mkt conditions


Numerical methods


Models, numerical methods and payouts

� Payout � describes the derivative product, the rights and obligations of the owner and of the issuer (no maths!).

� Model � Assumptions concerning the evolution of the underlying in the market

� Numerical method � The way of calculating the price of the payout, depending on the chosen model


Models, numerical methods and payouts

Models :Black-ScholesStochastic VolLocal Vol

Jump Diffusion…Numerical methods:analytic solutionStatic replicationBinomial treeMonte Carlo

Finite differences…

Payout :Call, Put, barriers, european, americanCallable, touch …

A model associated with a numerical method allows us to give the price of a payout(derivative product)


Numerical methods� Analytic solution:

� Very fast� « Exact » result� Very easy to implement� Exists only for a few payouts , with some models

� Monte Carlo� Relatively easy to implement� Can be applied practically on all payouts, with all models� Can be applied on payouts with several underlyings� Easy to parallelize computations� Slow� More difficult to implement on options with American exercise� Calculation of greeks is not easy


Numerical methods

� Binomial Tree (or trinomial):� Relatively easy to implement� Exists for many payouts (barriers), with only some models

� Partial differential equation (PDE) grid� Can be applied on many payouts, with most models

� limited to 2-3 underlyings� Very stable for the calculation of the greeks� Fast� Difficult to parallelise computations� Relatively difficult to implement


Binomial Trees

� Binomial trees are frequently used to approximate the movements of an underlying

� In each small interval of time the stock price can� move up by a proportional amount u � move down by a proportional amount d


Binomial Trees

� We discretise time in small steps� At each time step the underlying can only have twopossibilities :� Increase by a factor « u » (>1)� Decrease by a factor « d » (<1)


Movements in Time ∆t

Su

SdS

p

1 – p

∆t

p = probability that underlying increases

1-p = probability that underlying decreasesp, u, d ?

uSS

o

=

dSS

o

=


Risk Neutral Pricing

� If we know the value of the underlying today S(t)=So� The expected value at a future time t+∆t is E[S(t+∆t)] = S(t) e(r-q)∆t

� r is the interest rate of the currency of the underlying� q is the divident rate (for stocks), or, the interest rate of currency 1 (for Forex)

� on average an underlying evolves according to the risk-free interest rate (=savings account) of the currency on which it is expressed


Parameters p, u, and d are chosen so that the tree gives correct values for the mean & variance of the stock price changes in a risk-neutral world

Mean: E[St+∆t /St]=e(r-q)∆t = pu + (1– p )d

dudep

tqr

−−

=⇔∆− )(

Eq.1

Tree Parameters (1)


Tree Parameters (2)

Var σ2∆t = pu2 + (1– p )d 2 – e2(r-q)∆t

A further condition often imposed is u = 1/ d

Var[St+∆t /St]=?

)()(.)( tttt

ttt WWtqrS

SSapproxWddtqrSSd

−+∆−=−

+−= ∆+∆+ σσ

)()(1 tttt

tt WWtqrS

S−+∆−+= ∆+

∆+ σ22

var

−

=

∆+∆+∆+

t

tt

t

tt

t

tt

SSE

SSE

SS

Eq.2

Eq.3


When ∆t is small a solution to the equations is

tqr

t

t

eadudap

ed

eu

∆−

∆σ−

∆σ

=−−

=

=

=

)(

Solution to Binomial Tree


Exercise

� Prove the previous statement� Start with eq.2

� Now try the solution

� For small t it gives (Taylor expansion):

( ) ( )( )( )

( ) uddueddude

ddupdppu

t

t

−−=

=++−=

+−=−+

∆⋅

∆⋅

µ

µ

1

2eq.1

22222

tt edeu ∆−∆ == σσ ,

( ) ( ) ( ) ( )t

ttttteuddue tt

∆=

∆−−∆−⋅∆+−∆−+∆+=−+ ∆∆

2

211111σ

µσσσσµµ

tdtu ∆−≈∆+≈ σσ 1,1


The Complete Tree

S0u 2

S0u 4

S0d 2

S0d 4

S0

S0u

S0d S0 S0

S0u 2

S0d 2

S0u 3

S0u

S0d

S0d 3

Today

Maturity


Backwards Induction

� We know the value of the option at the final nodes

� We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate


Example: Put Option

S0 = 50; K = 50; r =10%; σ = 40%; T = 5 months = 0.4167; ∆t = 1 month = 0.0833

The parameters imply u = 1.1224; d = 0.8909; a = 1.0084; p = 0.5073


Example (continued)89.070.00

79.350.00

70.70 70.700.00 0.00

62.99 62.990.64 0.00

56.12 56.12 56.122.16 1.30 0.00

50.00 50.00 50.004.49 3.77 2.66

44.55 44.55 44.556.96 6.38 5.45

39.69 39.6910.36 10.31

35.36 35.3614.64 14.64

31.5018.50

28.0721.93

So

Stage 1 : complete the values of the underlying(top box)

Put�Max(0,K

-S)

Stage 2 : Determine the value of the option at the end nodes


Example (continued)

� Step 3: Go through the whole tree from right to left by completing the boxes on the bottom of each cell (option value)

2.66=(p x 0 + (1-p) x 5.45 ) x e-r∆t


Calculation of Delta

Delta is calculated from the nodes at time ∆t

Delta =−−

= −2 16 6 965612 44 55

0 41. .. .

.


Calculation of Gamma

Gamma is calculated from the nodes at time 2∆t

03.065.11

=Gamma

64.069.395036.1077.3;24.0

5099.6277.364.0

21

21

=∆−∆

−=−−

=∆−=−

−=∆

11.65=½(62.99-50)+½(50-39.69)


Calculation of Theta

Theta is calculated from the central nodes at times 0 and 2∆t

Theta= per year

or - . per calendar day

377 44901667

43

0012

. ..

.−= −


Calculation of Vega

� We can proceed as follows� Construct a new tree with a volatility of 41% instead of 40%.

� Value of option is 4.62� Vega is

4 62 4 49 013. . .− =per 1% change in volatility


Options on Indices, Currencies, Futures

As with Black-Scholes:� For options on stock indices, q equals the dividend yield on the index

� For options on a foreign currency, q equals the foreign risk-free rate

� For options on futures contracts q = r


Alternative Binomial Tree

Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 and

ttqr

ttqr

ed

eu∆σ−∆σ−−

∆σ+∆σ−−

=

=)2/(

)2/(

2

2


Trinomial Tree

61

212

32

61

212

/1

2

2

2

2

3

+

σ−

σ∆

−=

=

+

σ−

σ∆

=

== ∆σ

rtp

p

rtp

udeu

d

m

u

t

S S

Sd

Su

pu

pm

pd


Time Dependent Parameters in a Binomial Tree

� Making r or q a function of time does not affect the geometry of the tree. The probabilities on the tree become functions of time.

� We can make σ a function of time by making the lengths of the time steps inversely proportional to the variance rate.


Pricing an american put with a binomial tree

� « American » = the owner of the option has the right to exercise at any moment before expiry (or, atexpiry).

� Begin in the same way as for the european option


Example American Put 89.070.00

79.350.00

70.70 70.700.00 0.00

62.99 62.990.64 0.00

56.12 56.12 56.122.16 1.30 0.00

50.00 50.00 50.004.49 3.77 2.66

44.55 44.55 44.556.96 6.38 5.45

39.69 39.6910.36 10.31

35.36 35.3614.64 14.64

31.5018.50

28.0721.93

So

Stage 1 : complete the values of the underlying(top box)

Put�Max(0,K

-S)

Stage 2 : Determine the value of the option at the end nodesassuming that the option was not exercised before


Example American Call

If immediate exercise:

If no immediate exerciseValue =(p x 0.69 + (1-p) x 0.43 ) x e-r∆t =0.55

Call : Max(0,(2.06-1.5)) = 0.56

2,19

0,69

1,930,43

2,060,55

S 1,5call Put cstrike 1,5volatility 20%r1 5%r2 3%T 1

American n

Nsteps 10

dt 0,1u 1,06528839d 0,93871294a 0,998002p 0,46840882phi 1

0.56


Example American CallS 1,5 dt 0,1 2,823340166

call Put c u 1,06528839 1,323340166strike 1,5 d 0,93871294 2,65030595volatility 20% a 0,998002 1,15030595r1 5% p 0,46840882 2,4878765 2,487876496r2 3% phi 1 0,9878765 0,987876496T 1 2,33540186 2,33540186

American y 0,83540186 0,83540186

Nsteps 10 2,19227195 2,19227195 2,19227195

0,69227195 0,69227195 0,692271952,05791405 2,05791405 2,05791410,55791405 0,55791405 0,5579141

1,93179055 1,93179055 1,93179055 1,93179060,43179055 0,43179055 0,43179055 0,4317906

1,81339679 1,81339679 1,81339679 1,810,31660783 0,31339679 0,31339679 0,31

1,70225904 1,70225904 1,70225904 1,70225904 1,7022590370,2244334 0,21690461 0,20926664 0,20225904 0,202259037

1,59793259 1,59793259 1,59793259 1,59793259 1,597932590,1541648 0,14448285 0,13310696 0,11869566 0,09793259

1,5 1,5 1,5 1,5 1,5 1,50,10291859 0,09311931 0,08148539 0,06675155 0,04573508 0

1,40806941 1,40806941 1,40806941 1,40806941 1,408069410,05834495 0,04838689 0,03645986 0,02135854 0

1,32177298 1,32177298 1,32177298 1,32177298 1,321772980,02803347 0,01949579 0,00997456 0 0

1,2407654 1,2407654 1,2407654 1,24076540,0102576 0,00465818 0 0

1,16472254 1,16472254 1,16472254 1,164722540,0021754 0 0 0

1,09334012 1,09334012 1,093340120 0 0

1,02633252 1,02633252 1,0263325210 0 0

0,96343162 0,963431620 0

0,90438573 0,904385730 0

0,8489585900,796928414

0

Cells in red:

Immediate exercise more interestingthan keeping the option

Can occur for a call if r1 (q) >0

Can occur for a put if r2 (r) >0


Demo binomial tree (american)

American Exercise


Pricing of a KO put with binomial tree

KO Barrierlevel = 1.5


Demo binomial tree (Barrier)

Tree Barrier


Monte Carlo Method


Monte Carlo method

Cartoon by S Harris


Monte Carlo� In most cases analytic formula is too hard to find� An practical alternative is pricing via simulations

� We simulate the evolution of the underlying a large number of times (~10000).

� For every simulation we calculate the expected gain for the owner of the option

� Option price = (average of gains) x (disc-fact)

e-rT


Monte Carlo

� Each simulation describes a randomly chosen path of the underlying

� The name “Monte Carlo” comes from the resemblance to casino games


Random numbers

� Simulations require sampling random numbers� Typical simulation: 105 paths & 102 steps� Deviations away from required statistics produce unwanted bias� Main problem: Random numbers are not truly random

� there is a formula behind taking as input the computer clock

� After a while “random numbers” will repeat themselves� Good random numbers have a long period before repetition occurs� This effect is more pronounced as the number of dimensions (=number of steps * number of paths) increases


Mersenne uniform random numbers

� Have a period that is a Mersenne number, i.e. can be written as 2n-1, for example n=19937

� Mersenne numbers are popular due to � They are quickly generated� Sequences are uncorrelated � Eventually (after many draws) they fill the space uniformly


Box-Muller normal random numbers

� Transform a pair of uniform random numbers u,v into normal random numbers x,y

� Transformation

� with

� x,y: Normally-distributed � Advantages: One transformation, two random numbers N(0,1)� Disadvantages: ln(…), cos(…), sin(…) are CPU-consuming

)2sin(ln2

)2cos(ln2

vuy

vux

π

π

⋅⋅−=

⋅⋅−=

( )( )

22

21

21

21

21

,,Jacobian

yxee

yxvu −−

⋅=∂∂

=ππ


Sobol’ numbers

� Sobol’ numbers are not random: � They are “low-discrepancy”: meant to split equidistantly the space� Quality depends on nbr of dimensions = nbr Paths x nbr Steps� Uniformity is good in low dimensions� Uniformity is bad in high dimensions� Are convenient because … they are not random !

� Calculating the Greeks with finite difference requires the same sequence of random numbers, for example the Delta

� The calculation of the Greeks should differ only in the “bumped” parameter (e.g. the spot) not the path of the spot process

( ) ( )S

SSSS∆⋅

∆−−∆+=∆

2PricePrice


Random numbers comparison (1)

Dimension:

65532 x 2


Random number comparison (2)� Mersenne vs Sobol’ (uniform)� Box-Muller vs inverse cumulative method (generate normal variates)� Check convergence to N(0,1)


Sobol’ number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

0.5 0.5 0.5 0.5 0.5 0.5 0.50.25 0.75 0.25 0.75 0.25 0.75 0.250.75 0.25 0.75 0.25 0.75 0.25 0.750.875 0.875 0.125 0.625 0.375 0.375 0.6250.375 0.375 0.625 0.125 0.875 0.875 0.1250.125 0.625 0.875 0.875 0.625 0.125 0.3750.625 0.125 0.375 0.375 0.125 0.625 0.8750.6875 0.8125 0.8125 0.1875 0.0625 0.6875 0.56250.1875 0.3125 0.3125 0.6875 0.5625 0.1875 0.06250.4375 0.5625 0.0625 0.4375 0.8125 0.9375 0.31250.9375 0.0625 0.5625 0.9375 0.3125 0.4375 0.81250.8125 0.6875 0.4375 0.0625 0.9375 0.3125 0.68750.3125 0.1875 0.9375 0.5625 0.4375 0.8125 0.18750.0625 0.9375 0.6875 0.3125 0.1875 0.0625 0.43750.5625 0.4375 0.1875 0.8125 0.6875 0.5625 0.93750.59375 0.96875 0.34375 0.90625 0.78125 0.84375 0.031250.09375 0.46875 0.84375 0.40625 0.28125 0.34375 0.531250.34375 0.71875 0.59375 0.65625 0.03125 0.59375 0.781250.84375 0.21875 0.09375 0.15625 0.53125 0.09375 0.281250.96875 0.59375 0.96875 0.78125 0.15625 0.21875 0.156250.46875 0.09375 0.46875 0.28125 0.65625 0.71875 0.656250.21875 0.84375 0.21875 0.53125 0.90625 0.46875 0.906250.71875 0.34375 0.71875 0.03125 0.40625 0.96875 0.406250.65625 0.65625 0.03125 0.34375 0.34375 0.90625 0.093750.15625 0.15625 0.53125 0.84375 0.84375 0.40625 0.593750.40625 0.90625 0.78125 0.09375 0.59375 0.65625 0.843750.90625 0.40625 0.28125 0.59375 0.09375 0.15625 0.34375

Draw (n x m) table of Sobol’ numbers

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0.000 0.200 0.400 0.600 0.800 1.000

, 2 )

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0.000 0.200 0.400 0.600 0.800 1.000

Pair( 10 , 20 )

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0.000 0.200 0.400 0.600 0.800 1.000

Pair( 13 , 40 )

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0.000 0.200 0.400 0.600 0.800 1.000

Pair( 20 , 881 )

Plot pairs of columns(1,2) (10,20)

Non-uniform filling for large dimensions!

(13,40) (20,881)

Nbr Steps Nbr Paths


Sobol’ in high dimensions� 8191x32 dimensions� Convert Sobol’ to normal variates� Patterns form. Non-uniform filling of the space!


Monte Carlo method� It is a method for finding the average of a functiong of a random variable X:� We are interested in calculating integrals of the form:

� where f(x) is the probability density of x in the interval [a,b]

� Example

� where (ST) is the spot terminal density in the interval [0,∞]� call(ST) = max(ST-K,0)

[ ] ( ) dxxfxgxgGb

a

⋅⋅== ∫ )()(E

[ ] ( ) TTTT dSSSSG ⋅⋅== ∫∞

0

)(call)(callE ϕ


Monte Carlo method� Obtain estimator of G� by producing large number of realisations of x: (x1,x2…,xN).

� Estimator

� Theoretical mean

� The larger the N, the more accurate the estimator

[ ] ( ) dxxfxgxgb

a

⋅⋅= ∫ )()(E

∑=

=N

iiN xg

Ng

1

)(1~


Monte Carlo method: an example� Calculate the mean of N lognormal

variables� Sample N lognormal variables� Sum them up� Repeat for various values of N� Small N: fluctuations� Large N: convergence to mean

� How to sample at random a lognormally-distributed variable in Excel:� X = RAND()� Y = LOGINV(X,mean,std)

� where mean=mean of Lognormal distrib.where std=standard dev of Lognormal distrib.


Monte Carlo Simulation and π

� Calculate π by randomly sampling points in the square?

Exercice


Monte Carlo Simulation and Options

When used to value European stock options, Monte Carlo simulation involves the following steps:

1. Simulate one path for the stock price in a risk neutral world2. Calculate the payoff from the stock option3. Repeat steps 1 and 2 many times to get many sample payoffs

4. Calculate mean payoff5. Discount mean payoff at risk free rate to get an estimate of the value of the option


Sampling Stock Price Movements

� In a risk neutral world the process for a stock price is

� We can simulate a path by choosing time steps of length ∆t and using the discrete version of this

where ε is a random sample from Ν(0,1)tStSS ∆εσ+∆µ=∆ ˆ

dS S dt S dz= +$µ σ

=LOI.NORMALE.INVERSE(ALEA();0;1)


An alternative approach

( ) ttetSttS ∆+∆−=∆+ 2/2)()( εσσµ

•More accurate in most cases

•The options with a european payout require only one time step

=LOI.NORMALE.INVERSE(ALEA();0;1)

Often instead of using the BS stochastic differential equation, we use its solution:


Extensions to several underlyings

When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world to

calculate the values for the derivative


Sampling from Normal Distribution

� The simplest way to sample from N(0,1) :

� Generate 12 random numbers between 0.0 & 1.0� use the Excel function alea() (=random())

� Sum them up and subtract 6.0

� Exercise: calculate the mean and the variance of V=U1 + U2 … +U12 - 6

� In Excel =LOI.NORMALE.INVERSE(ALEA();0;1)gives a random sample from N(0,1)


Example: pricing a call option

� for i=1…N� Generate standard normal variable Ui� Set Si(T) = S(0) exp[ (r-½σ2)T+ √T Ui]� Set Calli = e-rT max(Si(T)-K,0)

� Call = (Call1+…+ CallN)/N

� Exercise: show that this converges to the result given by the Black-Scholes formula


Confidence interval

� Calculate the standard deviation of the Monte Carlo result

� For a 95% confidence interval find z /2=Ninv(1- /2) with =5%� Ninv is the inverse cumulative normal function� 95% confidence interval means =5% and z /2=1.96

� The confidence interval is within the values� Average - zδ/2 · SD/√n� Average + z /2 · SD/√n

( )∑=

=N

iiN 1

2Average-Result1SD


Obtain two correlated Normal Samples

� Obtain independent normal samples x1 and x2 and set

� A procedure known as Cholesky’s decomposition� =[-1…1] measures correlation:

� =1 then 1= 2 : perfect correlation� =0 then 1= x1 and 2 =x1 : no correlation� =-1 then 1=-ε2 : perfect anti-correlation

� Used when samples are required from two (or more) normal variables

� Exercise: show that the correlation between 1 and 2 is

2212

11

1 ρ−+ρ=ε=ε

xxx


Cholesky exercise

� x,y,z are three correlated random variables. Find the correct coefficients in

� so that

� Considering that are uncorrelated

321

21

1

εεεεε

ε

⋅+⋅+⋅=⋅+⋅=

=

fdczbay

x

[ ] [ ] [ ][ ] [ ] [ ] 1E1E1E

EEE222 ===

=⋅=⋅=⋅

zyx

zyzxyx yzxzxy ρρρ

( )1,0~,, 321 Nεεε


Application of Monte Carlo Simulation

� Monte Carlo simulation can deal with � path dependent options (e.g. Asians, barriers,…)� options dependent on several underlying state variables (e.g. Forex & interest rates)

� options with complex payoffs

� It cannot easily deal with American-style options


Example: pricing an Asian call option

� An Asian option averages the payoff spot over several intermediate dates T1,… ,TN

� This is a path-dependent option

� for i=1… nbrPaths� for j=1… N

� Generate standard normal variable Ui,j� Set Si(Tj) = S(Tj-1) exp[(r-½σ2)(Tj-Tj-1)+ √(Tj-Tj-1) Ui,j]

� Set meanSpoti =(Si(T1)+…+Si(TN))/N� Set Calli = e-rT max(meanSpoti-K,0)

� Call = (Call1+…+ CallN)/N

−= ∑

=

N

iT KS

N i1

0,1maxAsian


Monte Carlo and barrier options� If the barrier monitored continuously, it requires a simulation withmany points:

� What happens between ti and ti+1 is unknown. Was the barriertouched ?� Put more points (CPU time increases!), or � Smarter : Compute the pobability of touching the barrier between ti and ti+1

( ) )t-t()t-t(2/ˆ i1ii1i2

)()1( ++ +−=+ ieiSiS εσσµ


Monte Carlo and barrier options� Estimating probability of not touching barrier:

� Total survival probability:

� Knock-out option = DF · Payoff(S) · Psurv

NN ttsurv

ttsurv

ttsurv PPPP →→→ −⋅⋅⋅= 13221

surv L


Monte Carlo and barrier options

� For knock-in options we use the decomposition

� KI = Vanilla – KO

� and we price the two right-hand side instruments based again on the survival probability formula


Determining Greek Letters

� For ∆� Make a small change to asset price� Carry out the simulation again using the same random numbers

� Estimate ∆ as the change in the option price divided by the change in the asset price

� Proceed in a similar manner for other Greek lettersdS

dSSdSS⋅

−−+=∆

2)(Price)(Price 00

2000

)()(Price)(Price2)(Price

dSdSSSdSS −+⋅−+

=Γ( ) ( )

σσσσ

∆−∆+

=PricePriceVega


Demonstration XL

Monte-Carlo


Finite Difference Methods

� Finite difference methods represent the differential equation as a difference equation

� Practically speaking, we transform

� into

� and we solve for P(t): the price at the previous time step� is the risk-neutral drift

PrSPS

SPS

tP

⋅=∂∂

⋅⋅⋅+∂∂

⋅⋅+∂∂

2

222

21 σµ

PrS

SSPSPSSPSS

SSPSSPSt

tPttP⋅=

∆∆−+−∆+

⋅⋅+∆⋅

∆−−∆+⋅⋅+

∆−∆+

222 )()(2)(

21

2)()()()( σµ


Finite Difference Methods: the main idea

� We form a grid with equally spaced time-values and stock-price values

� Define ƒi,j as the value of ƒ at time i∆t when the stock price is j∆S� Knowing the payoff at maturity we solve PDE backwards till T=today

time

Spot

today maturity

strike

Call payoff: f

fi,j

i

j


Finite Difference Methods: an example (1)

� Consider the simplified equation

� It implies that

� Isolate C(S,t) on the left-hand side:

2

222

21

SCS

tC

∂∂

⋅⋅⋅−=∂∂ σ

⇒∆

∆+∆−+∆+−∆+∆+⋅⋅−=

∆−∆+

222 ),(),(2),(

21),(),(

SttSSCttSCttSSCS

ttSCttSC σ

∆+∆−+∆+⋅

∆∆

⋅−−∆+∆+⋅

∆∆

⋅⋅⋅= ),(),(22),(21),(

2

22222 ttSSCttSC

tS

SttSSC

StStSC

σσ

Black-Scholes equation for r=q=0


Finite Difference Methods: an example (2)

� This means that if we know the value of P at time t+ t and for spot values S-∆ S, S, S+ S we can immediately obtain P(S,t)

� Example: Call option with maturity 1y, strike 95, vol 10%� We solve backwards starting from the (known) payoff at maturity

Spot 80 90 100 110 120

+⋅

∆∆

⋅−−⋅

∆∆

⋅⋅⋅= ∆−∆+∆+

∆+∆+

SStt

Stt

SStt

St CC

tS

SC

StSC

2

22222 22

21

σσ

Value 0 0 5 15 25


Finite Difference Methods� Explicit method� Spot derivatives are calculated at t=(i+1)· t

� Implicit method� Spot derivatives are calculated at t=i· t

21,1,11,1

2

2

1,11,1

,,1

)(2

2

Sfff

Sf

Sff

Sf

tff

tf

jijiji

jiji

jiji

∆

+−=

∂∂

∆⋅

−=

∂∂

∆

−=

∂∂

−++++

−+++

+

21,,1,

2

2

1,1,

,,1

)(2

2

Sfff

Sf

Sff

Sf

tff

tf

jijiji

jiji

jiji

∆

+−=

∂∂

∆⋅

−=

∂∂

∆

−=

∂∂

−+

−+

+


Explicit method

� The difference equation becomes

� and after some re-arrangement:

� more compactly:

� For i+1 =Tmat the function fi+1,j is fully known� Solve above equation iteratively for fi,j in every (i,j) until i=today

jijijijijijijiji fr

Sfff

SjSff

Sjt

ff,12

1,1,11,1221,11,1,,1

)(2

)(21

2 +−++++−++++ ⋅=

∆

+⋅−∆⋅+

∆⋅

−⋅∆⋅+

∆

−σµ

( )

∆+∆+∆−∆−+

∆−∆= +++−+ tjtjftrtjftjtjff jijijiji µσσµσ

21

211

21

21 22

1,122

,122

1,1,

jjijjijjiji CfBfAff ⋅+⋅+⋅= +++−+ 1,1,11,1,


Explicit method schematically � To calculate the option value at

the boundary spots � Smin (with j=1)� Smax (with j=nbrSpots)we need extra equations, the boundary conditions

� We obtain these by requiring that at very low and very high spots the option has no convexity:

� This implies:

time=i t time=(i+1) t

Spot = (j+1) S

Spot = j S

Spot = (j-1) S

0)1()(2)1(02

2

=−+−+⇒=∂∂ jCjCjCSC

)2()1(2)()3()2(2)1(−−−=

−=NCNCNC

CCC


Explicit method at work

� PDE solution with� 100 time steps� 100 spots � t = 0.005� S = 0.025

� converges to the correct Black-Scholes solution


Explicit method (not) at work

� Unstable if number of time-steps is not big enough

� Oscillations are produced and propagate to all spots


Implicit method� More complex but avoids instabilities of explicit method

� The difference equation becomes

� and after some re-arrangement:

� more compactly:

� For i+1 =Tmat the function fi+1,j is fully known� Solve above equation iteratively for fi,j in every (i,j) until i=today

jijijijijijijiji fr

Sfff

SjSff

Sjt

ff,2

1,,1,221,1,,,1

)(2

)(21

2⋅=

∆

+⋅−∆⋅+

∆⋅

−⋅∆⋅+

∆

− −+−++ σµ

( ) jijijiji ftjtjftrtjftjtjf ,122

1,22

,22

1, 21

211

21

21

++− =

∆−∆−+∆+∆++

∆+∆− µσσµσ

jijjijjijji fCfBfAf ,11,,1, ++− =⋅−⋅+⋅


Implicit method schematically

� 1 equation, 3 unknowns !� We have to solve the entire system of equations for each time step

� Linear algebra methods� LU decomposition

� Boundary conditions remain as before

time=i t time=(i+1) t

Spot = (j+1) S

Spot = j S

Spot = (j-1) S


Explicit vs Implicit methods

� In practise we use a combination of the two methods� Crank-Nicolson method� Combines efficiency and stability


Risk management and calculation of VAR


VAR (Value At Risk)� VAR is a measure of market risk on a group of assets.

� Def: Maximum loss that can be reached in x days such thatthere is a small probability p that the realised loss is bigger.

� It can be calculated at different levels: single portfolios, smallgroup of portfolios, bank portfolios,…

� It is not additive (diversification effect)

� It computes the amount of capital the bank must hold to cover its risks� Bassel accord: p=1%, x=10 days.


VAR: historical approach

� identify the parameters of the market that influence the value of the portfolio:� V=f(S1, S2, …..)� Si: Forex spots, swap rates, market vols, etc

� on a large sample of historical data (two or more years), calculate the daily returns of these market parameters:

( ) ( )( )1

1)(−

−−=

tStStSt

i

iiiα


� Apply these returns from the past to today’smarket data and recalculate the value of the portfolio� Vj=f(S1·α1(t0-j), S2·α2(t0-j), …..)� j=1→N (number of daily observations)

� For each scenario replayed, calculate the profit or loss:� PLj= Vj- V0

� Order the PnL from the smaller (great loss) to the larger (great gain)

VAR: historical approach


p=5%

Var is the largest value such that at least (1-p) of observations are above it


Temporal extrapolation

� The VAR obtained in this way corresponds to a horizon of « 1-day »

� Assuming the daily increments are i.i.d.� Independent� Identically distributed

nVARnVAR )day 1()days( =


Quantile extrapolation

� The VAR previously obtained are for p=5%� Assuming the observations of PnL are normallydistributed

� N-1(p): inverse cumulative normal function

)()(

)()(1

12

1

12 pNpN

pVARpVAR −

−

⋅=


Example

� The 5% VAR of 1-day is 42,000$, what is the value of 10-day 1% VAR?


Disadvantages of historical VAR� It is based on historical data. Implicitly assumes that the markets will behave in the

future as they behaved in the past.

� It reduces the measure of risk to a single digit. This does not necessarily representthe potential damage

The two distributions have the same VAR!


Conditional VAR (CVAR)� Measurement of the average loss exceeding VAR

The two distributions do not have the same CVAR !


VAR: different possible implementations� Historical simulation

� Advantages� Easy to calculate� Matches data distributions

� Disadvantages� Depends on limited experience (past data)� not enough extreme events

� Monte-Carlo simulation: daily returns are randomly sampled based on a model� Advantages

� Can generate lots of data & scenarios� Disadvantages

� Introduces «model risk»: dependence on the assumed distibution of daily returns


� VAR for a continuous distribution

∫

∫

∫

−

∞−

−

∞−

+∞

−

=

=

VAR

VARVAR

dxxf

dxxxfCVAR

dxxfp

)(

)(

)(

VAR: some useful identities


Introduction to Credit Risk


Merton’s « Default Risk Model » (1)

� A company has asset value At

� This company has debt D� The company has equity value Et

� The company� Can pay its debt if At>D. Then Et=At-D� Cannot pay its debt if At<D . Then Et=0

� The company’s equity value is Et=max(At-D,0)

� This is equivalent to the payoff of a call !



� Similar analysis to Black-Scholes model� Interested in: asset value and the probability of default

� Assume that assets follow geometric brownian motion:

� Equivalent analysis as in Black-Scholes gives the equity value today:

dWAdtArdA A ⋅⋅+⋅⋅= σ

( ) ( )2100 dNeDdNAE rT−⋅−⋅= TddTT

DeA

d AAA

rT

σσσ

−=+

⋅

= 12

0

1 21ln

Risk-free rate

Asset volatility



� The amount of debt D today is worth D*=De-rT. Define L=D*/A as a measure of leverage. Then from the previous equation

� This equation gives us the value of the company’s assets today A0� But how do we calculate the volatility of the assets?� (assets = value of company’s buildings, of people working there, etc) � This is not directly observable !

( ) ( )( )2100 dNLdNAE ⋅−=

TddTTLd AA

A

σσσ

−=+−

= 121 21ln



� Since the equity value is a function of the asset value E=E(A), we use Ito’s Lemma:

� With

� Equalising the left- and right-hand side (the dW terms) gives

� We have expressed in terms of which is observable !

( )22

2

21 dA

AEdA

AEdt

tEdE

∂∂

+∂∂

+∂∂

=

dWAdtArdA A ⋅⋅+⋅⋅= σ dWEdtErdE E ⋅⋅+⋅⋅= σ

AAEE AE ⋅

∂∂

=⋅ σσ

Aσ Eσ

Vol of equity



� From this equation and the Black-Scholes equity price formula

� we obtain

� From these equations we know and we solve them to obtain

� Then the probability of default can be computed: it is the probability that the “call” will be out of the money, i.e.

( )( ) ( )21

1

dNLdNdNA

E ⋅−⋅

=σσ

AA σ,0

( ) ( )( )2100 dNLdNAE ⋅−=

TLE E ,,,0 σ

( )2PD dN −=


Two possible measures of the default probability:

� Actuarial: we measure the credit risk on statisticalbasis of default of payment. Data produced by rating agencies.

� Implicit: deducing the default risk of certain marketprices.


Actuarial measure of the default risk (1)



� Marginal default rate during a period T: Probability of default during the year T, given that no default has occurred inprevious years dT

� Cumulative rate of default between 0 and T:probability that at least one default occurs between 0 and T: CT� Link between CT and dT……

� Survival rate between 0 and T :� St=(1-d1)⋅(1-d2)…⋅(1-dT)



� The measurement of default rates over a long period of time may be problematic (small sample)

� A more robust approach: Transition probabilityfrom one state to another:

Example : a company with a rating « B » has a probability of

12% to be upgraded to « A » within a year.


� What is the cumulative probability that a companycurrently rated as « A » faces default in the next 3 years?

Exercise


Trading in the real world


Classical theory of financial markets

� Efficient market hypothesis� Assumes: All information concerning a financial asset is already incorporated into the current price

� Implies: risk-free profit is impossible, traders are completely rational

� Asset increments are

� Independent from one tick to the next

� Identically distributed

� Normally distributed

( ) ( )( ) )1(

)(log11)(

−≈

−−−

=∆tStS

tStStStS


Market empirical (stylized) facts

� Fat tails� The market-realised distribution of log-returns is not Normal

� Opposite graph� S&P500 density of log-returns� Normal density with same mean and variance

� Y-axis in log-scale

� Example: � Probability of a daily move of -6%� Market: 0.02%� Normal: 0.000005%



� Volatility clustering� Periods of high volatility� Periods of low volatility

� Not reproduced by a time series of normal N(0,1) increments



� Decaying autocorrelations

� Dependence of market-returns between different times

� Graph opposite

as function of

where1

1

−

−−=

t

ttt S

SSx

[ ]τ+⋅ tt xxE


A simple trading strategy: Pairs trading

� Find two stocks that are consistently correlated

� Wait till one of them breaks the pattern

� Then buy the cheap one, sell the expensive one

� Wait till the trend reverses to the normal pattern

� Then close the position


Pairs trading at work� Several implementations exist. A possible one:

� Measure distances between stocks, Sa and S b, across timeseries

� When the distance is too far away from the mean: trade

� Backtest the algorithm and optimise through modifying� Distance threshold (based on e.g. multiple of the standard deviation)� Size of data� Asset classes of stocks� The measure of distance (alternative to above can be correlation)� …

( )∑=

−=N

iibiaba tStSd

0

2, )()(


Pairs trading at work: an example

� Algorithm gives signals for distances higher than 1.5·standard deviation of the mean


Kelly’s criterion

� You are a gambler

� You know your game and you win with probability 55%

� How much of your capital should you bet each time ?

Historical background� J L Kelly (1956)� Bells’ labs USA� Develops analysis for maximizing expected capital

� Mathematician Ed Thorp uses the analysis at Las Vegas casinos

� Reportedly made fortune� Author of best-seller book “Beat the Dealer” 1962700,000 copies sold

� Founder of hedge fund


Kelly’s criterion for coin-tossing

Notation� You played N times

� Number of times you won: W� Number of times you lost: L

� Win probability p=W/N� Lose probability q=1-p

� Initial capital X0

Strategy� Each time you bet a fraction of your

remaining capital fExample:

� 1st time:� Capital to bet: f ·X0� Capital that remains: (1- f) ·X0� This time you lose

� 2nd time: � Capital to bet: f ·(1- f )·X0� Capital that remains: (1- f) ·(1- f) ·X0� …

� After n rounds� Capital that remains: (1- f)L ·(1+ f)W ·X0



� Remaining capital after n rounds Xn=(1- f)L ·(1+ f)W ·X0

� Ratio (in logarithm):

� Take expectations:

)1log()1log(log)(

1

0

fnLf

nW

XXfG

nn

n −++=

=

)1log()1log(

)1log()1log(E)(

fqfp

fnLf

nWfg

−⋅++⋅=

−++=



Choose� fopt maximizes the Kelly function� This is the optimal fraction that leads to the maximal expected capital

Avoid� “Ruin” fraction fruin that leads to a negative capital: you lose all your money

� fopt =p-q


References

� Options, futures and otherderivativesJ. Hull (2008) Prentice Hall

� Monte Carlo Methods in FinanceP. Jäckel (2003) Wiley

� Stochastic Calculus for Finance II: Continuous-Time ModelsS. Shreve (2004) Springer Finance

� Pricing Financial Instruments: The finite-difference methodD. Tavella and C. Randal (2000) Wiley

� Monte Carlo methods in financialengineeringP. Glasserman (2000) Springer

� Paul Wilmott on Quantitative Finance 3 Vol SetPaul Wilmott (2000) Wiley

� The Concepts and Practice of Mathematical FinanceM. Joshi (2003) Cambridge Univ Press

� Financial Risk Manager HandbookP. Jorion (2009) Wiley Finance


Exercises:1. Decompose the following strategies into simple Call and Put positions (short or long).

Discuss advantages and disadvantages of each of the strategies

2. Integrate numerically the function exp(-x²/2) between –4 and +4, using an interval of dx=0.01. The answer must be close to √2π due to the normalisation property of the normal distribution function. Also integrate the function exp(-(x-1)²/2). The answershould be the same as before (explain why).

3. Differentiate numerically and analytically the function exp(-x²/2).

4. Write a program in VBA that calculates the functions min(a,b) and max(a,b) using the min / max of two numbers.

5. Write a program in VBA to generate a brownian motion W(t). The input parametersare: the number of time steps, the final time. As an output, the function should return the simulated trajectory.


Exercices:6. Use the function of exercise 4 to calculate the variance of the final value

of a brownian trajectory (10 time steps spaced by 3 sec), on the basis of 1000 realisations.

7. Using Excel or VBA show that the variance of random variable is givenby a difference in means: V(X) = E(X²)-(E(X))²

8. What are (i) the mean (ii) the standard deviation of returns of the index EUROSTOXX50, if we consider that it follows the law a+bX where X isa normal gaussian variable (a and b are 2 constants) ?

9. Using Excel or VBA calculate the mean and the variance of eaX where X is a guassian normal random variable. Compare against the exact analytic result.

10. Using Excel or VBA calculate the expectation of S=e(r-q-σ²/2)T+Xσ√T where X is a guassian normal random variable . Compare against the exact analytic result.

11. Write a programe in VBA to compute a Black-Scholes pricer (analyticformula) for a Call option: Call(S, K, σ, r, q, T).


Exercises:12. Compare the price of a simple call option to the price call with a barrier

where the barrier level H increases .

13. What is the value of a 3m call on EUR/USD, rEUR = 4%, rUSD = 5% vol=25%, K=1.3 for different values of the spot. For each point of the curvecalculate the Delta using finite differences and the analytic formula. If S=1.27, what is the cost of an option on 1,000,000 EUR notional? And on an option on 1,000,000 USD notional?

14. Show that for small ∆t, the relations

15. Derive the density function of a logNormal random variable.

16. Calculate the mean and the variance of a log-normal density withparameters µ, σ .

tqr

t

t

eadudap

ed

eu

∆−

∆σ−

∆σ

=−−

=

=

=

)(

are solutions of dudep

tqr

−−

=∆− )(

σ2∆t = pu2 + (1– p )d 2 – e2(r-q)∆t

u = 1/ d


Exercises:17. Calculate with Monte Carlo the value of an Asian put option and compare

with the value of the corresponding vanilla put. How do you explain the difference in the prices?

18. Calculate the number π using a Monte-Carlo method

19. Programm a VBA function allowing the pricing of a Call with Monte-Carlo: Call(S, K, s, r, q, T, Nsimu). Compare with the exact solution from Black-Scholes formula

20. Show that the variables 1, 2 obtained from Cholesky’s decomposition have a correlation equal to

21. Compute analytically the Delta, Gamma and Vega of a Put option

2212

11

1 ρρε

ε

−+=

=

xx

x


Exercises:22. Using Itô’s lemma, and starting from the differential equation of Black-Scholes

dS=µSdt+σSdW, calculate the differential of ln(S). Derive an expression for S(t).

23. Using Itô’s lemma compute the stochastic differential of the variable Z=X·Y where X and Y are stochastic variables. Repeat the same for the variable Z=X/Y.

24. Calculate the price of a digital option (at maturity it pays 1 unit of underlying if ST>K). Write a VBA program that calculates with Monte Carlo simulations.

25. Calculate the price of a knock-out option using Monte Carlo and the formula for the surviving probabilities

26. Price a put option using the explicit PDE method and compare the result to the Black-Scholes formula.

27. Bachelier vs Black-Scholes: Price a call option with the monte carlo methodusing (i) brownian motion (Bachelier model) and (ii) geometric brownianmotion (Black-Scholes model).


Exercises:28. Find the stochastic derivatives of the process: Xt=Wt

2-t and Xt=Wt2-Wt ·t

29. Write a Monte Carlo program in VBA that simulates a coin-tossing game and verify that the optimal fraction of capital fopt proven by Kelly leads to the maximum expected capital

30. Demonstrate that if Wt is a brownian motion then E[(Wt-Ws)2]=t-s

31. Write a VBA program that generates variables of a normal distribution of mean and variance using the VBA uniform random number generator. Calculate the mean and the variance of the samples.

32. Write an Excel method that calculates the cumulative function of a normal density function e-x*x/2/√(2 )

33. Using Excel calculate Black-Scholes spotladder (price of a call option for various spot levels) for different values of (i) volatility, (ii) maturity, (iii) rates. What is the impact of each of these on the price of the option?

34. Write a VBA function that calculates the root of f(x)=(x-2)2-1 from x=2 to x=4 using the bisection method

Computational methods in finance - KU Leuvennikos/papers/methodsinfinance.pdf · 2012-01-01 ·...

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Transcript of Computational methods in finance - KU Leuvennikos/papers/methodsinfinance.pdf · 2012-01-01 ·...