4. Black-Scholes Models and PDEs - YorkU Math and...

Math6911 S08, HM Zhu

4. Black-Scholes Models and PDEs

2Math6911, S08, HM ZHU

References

1. Chapter 13, J. Hull

2. Section 2.6, P. Brandimarte


Outline

• Derivation of Black-Scholes equation

• Black-Scholes models for options

• Implied volatility


4.1 Derivation of the Black-Scholes-Mertion differential equation



Assumptions for Black-SholesEquation

• Asset price follows the lognormal random walk• The risk-free interest rate r and the asset volatility σ are

known functions of time over the life of an option• No transaction cost or taxes• No arbitrage possibilities• No dividends during the life of an option • Trading of the underlying asset is continuous• Short selling is permitted and the assets are divisible


Concepts underlying the Black-Scholes equation

• The option price & the stock price depend on the same underlying source of uncertainty

• We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty

• The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

• This leads to the Black-Scholes differential equation

Math6911, S08, HM ZHU

1 of 3: The Derivation of theBlack-Scholes Differential Equation

22 2

2 4 2

Model of the asset price: (4.1)

Option value V using Ito's lemma:

V V 1 V VV 2

We set up a portfolio consisting of short one option

dS S dt S dzˆ

d S S dt S dz ( . )S t S S

µ σ

∂ ∂ ∂ ∂µ σ σ∂ ∂ ∂ ∂

= +

⎛ ⎞= + + +⎜ ⎟⎝ ⎠

V long number of underlying assetS

∂∂


The value of the portfolio is given byV V+ (4.3)

The change in its value in time is given byV V+ (4.4)

Substituting Eq's (4,1) and (4

SS

dt

d d dSS

∂∂

∂∂

Π

Π = −

Π = −

22 2

2

12

.2) to Eq (4.4) gives

V t (4.5) Vd S dt S

∂σ∂

⎛ ⎞∂Π = − +⎜ ⎟∂⎝ ⎠



The return of the riskless portfolio would seea growth of t (4.6) over a time dt, where r is risk-free interest rate. Substituting Eq's (4.6) and (4.1

d r d

Π

Π = Π

22 2

2

1 02

) to Eq (4.5) leads tothe Black-Scholes equation:

V (4.7) V VS rS rVt S S

∂σ∂

∂ ∂+ + − =

∂ ∂



The Black-Scholes Differential Equation

• Any derivative security whose price is dependent only on the current stock price and t, which is paid for up-front, must satisfies the Black-Scholes differential equation or its variations

• Other options, for example, American options that depend on both the history and present values of the asset, can also fit into the Black-Scholes framework


4.2 More on the Black-Scholes-Mertion differential equation



• The delta given by is a measure of the

correlation/sensitivity between the movements of the option and the underlying asset

• The linear differential operator

is a measure of the difference between the return on a hedged option portfolio and the return on a bank deposit

The Derivative terms

SV∂∂

=∆

22 2

2

12

return on a bank depositreturn on hedged option portfolio

S rS rt S S

∂σ∂

∂ ∂+ + −

∂ ∂


Risk-Neutral Valuation

• The variable µ does not appear in the Black-Scholesequation

• The only parameter from the stochastic differential equation involved is the volatility σ

• It is independent of risk preference• We can assume that all investors are risk-neutral• Therefore, the expected return on any securities is

the risk-free interest rate r


Boundary and final conditions

• The Black-Scholes equation is a common type of PDEs, called backward parabolic equation

• For such an equation to have a unique solution, we will need the boundary and final conditions of V(S,t)

• For example, a typical boundary conditions are

• A typical final condition is the value of V(S, t) at t = T

( ) ( )( ) ( )⎩

⎨⎧

==

tVtbVtVtaV

b

a

,,

( ) ( )SVTSV T=,


4.3 Black-Scholes analysis for European options



Boundary and final conditions for European call options

• The value of a European call option satisfies the Black-Scholesequation

( )( )⎩

⎨⎧

∞→==StSC

tC as ?,

?,0:conditionsBoundary

( ) ?,:condition Final =TSC

22 2

2

1 02

s.t.

C C CS rS rCt S S

∂σ∂

∂ ∂+ + − =

∂ ∂


The Black-Scholes Formulas for C(S,t)(See p 48-49, [4]; proof of solving PDE, p76-79, [4])

( ) ( )

( )

( )

( )

2

1 2

2

1

2 1

12

2 2

2 2

where

r T t

yx

C S ,t S N( d ) K e N( d )

N x e dy

ln( S / K ) ( r / ) T td

T t

ln( S / K ) ( r / ) T td d T t

T t

π

σσ

σσ

σ

− −

−

−∞

= −

=

+ + −=

−

+ − −= = − −

−

∫


Approximate the cumulative normal distribution function N(x) (See p 297, Hull)

( ) dyexNx y

∫ ∞−

−= 2

2

21 π

( ) ( )( )( )

( ) 2/54

321

55

44

33

221

2

21,330274429.1,821255978.1

781477937.1,35656378.0,319381530.0

2316419.0x1

1k

where0 xif10 xif1

:accuracyplacedecimal 6givesion that approximat polynomialA

xexNaa

aaa

xNkakakakakaxN

xN

−=′=−=

=−==

=+

=

⎩⎨⎧

≤−−≥++++′−

=

π

γγ


The Black-Scholes Formulas for C(S,t)(Proof p 295, Hull; using risk-neutral evaluation)

( ) ( ) ( )( ) ( )

1 2

2

0

Another way to derive the Black-Scholes equation is to use risk-neutral valuation

where probability that the option will be e

r T tT

r T t r T t

ˆC S ,t e E max S K ,

e Se N( d ) K N( d )

N( d ) :

− −

− − −

= −⎡ ⎤⎣ ⎦⎡ ⎤= −⎣ ⎦

( )2

1 T T

xercised in a risk-neutral worldK probability that the strike price will be paid

expected value of a variable which equals S if S and is zero oth

r T t

N( d ) :

Se N( d ) : K− >erwise in a risk-neutral world


European call value C(S,t) as a function of S and t (r = 0.1, σ=0.4, K=1)


Boundary and final conditions for European put options

• The value of a European call option satisfies the Black-Scholesequation

( )( )

0Boundary conditions:

as P ,t ?

P S ,t ? S⎧ =⎪⎨ → →∞⎪⎩

( )Final condition: P S ,T ?=

22 2

2

1 02

s.t.

P P PS rS rPt S S

∂σ∂

∂ ∂+ + − =

∂ ∂


The Black-Scholes Formulas for P(S,t) (See pages 48-49, [4])

( )2 1

Using the put-call parity we can obtain:

r T tP K e N( d ) S N( d )− −= − − −

( )Put-call parity: r T tS P C Ke− −+ − =


European put value P(S,t) as a function of S and t (r = 0.1, σ=0.4, K=1)


4.4 Implied Volatility



Implied Volatility

• Although volatility can be estimated from a history of the stock price, traders often work with implied volatility

• The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price

• There is a one-to-one correspondence between prices and implied volatilities; therefore, an iterative search procedure can be used to find the implied volatility

• Often used to monitor the market’s opinion about the volatility of a particular stock

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