The Black-Scholes Model - City University of New...

The Black-Scholes Model

Liuren Wu

Zicklin School of Business, Baruch College

Options Markets

Liuren Wu (Baruch) The Black-Scholes Model Options Markets 1 / 55

Outline

1 Brownian motion

2 Ito’s lemma

3 BSM

4 Risk-neutral valuation

5 Measure change

6 Dynamic hedgingDeltaVegaGamma

7 Static hedging


The Black-Scholes-Merton (BSM) model

Black and Scholes (1973) and Merton (1973) derive option prices under thefollowing assumption on the stock price dynamics,

dSt = µStdt + σStdWt

The binomial model: Discrete states and discrete time (The number ofpossible stock prices and time steps are both finite).

The BSM model: Continuous states (stock price can be anything between 0and ∞) and continuous time (time goes continuously).

Scholes and Merton won Nobel price. Black passed away.

BSM proposed the model for stock option pricing. Later, the model hasbeen extended/twisted to price currency options (Garman&Kohlhagen) andoptions on futures (Black).

I treat all these variations as the same concept and call themindiscriminately the BSM model.


Primer on continuous time process


The driver of the process is Wt , a Brownian motion, or a Wiener process.

Wt generates random “continuous movements:” Movements arrivecontinuously, & the sizes of the movements are random and center aroundzero.

Contrast: Merton (1976) proposes a stock price process with bothcontinuous movements (Wt) and compound Poisson jumps: The jumparrives randomly over time (say on average once every two years). Given onejump occurring, the jump size is randomly distributed.

It is appropriate to use Brownian motion to capture daily random pricemovements, and use the compound Poisson jumps to rare but extremeevents (such as market crashes, credit events, etc).


Properties of a Brownian motion


The process Wt generates a random variable that is normally distributedwith mean 0 and variance t, φ(0, t).

The process is composed of independent normal increments dWt ∼ φ(0, dt).

I “d” is the continuous time-time limit of the discrete time difference(∆).

I ∆t denotes a finite time step (say, 3 months), dt denotes an extremelythin slice of time (smaller than 1 milisecond).

I It is so thin that it is often referred to as instantaneous.I Similarly, dWt = Wt+dt −Wt denotes the instantaneous increment

(change) of a Brownian motion.

By extension, increments over non-overlapping time periods areindependent: For (t1 > t2 > t3), (Wt3 −Wt2 ) ∼ φ(0, t3 − t2) is independentof (Wt2 −Wt1 ) ∼ φ(0, t2 − t1).


Properties of a normally distributed random variable


If X ∼ φ(0, 1), then a + bX ∼ φ(a, b2).

If y ∼ φ(m,V ), then a + by ∼ φ(a + bm, b2V ).

Since dWt ∼ φ(0, dt), the instantaneous price changedSt = µStdt + σStdWt ∼ φ(µStdt, σ2S2

t dt).

The instantaneous return dSS = µdt + σdWt ∼ φ(µdt, σ2dt).

The annualized instantaneous return 1dt

[dSS

]∼ φ(µ, σ2).

I Under the BSM model, µ is the annualized mean of the instantaneousreturn — instantaneous mean return.

I σ2 is the annualized variance of the instantaneous return —instantaneous return variance.

I σ is the annualized standard deviation of the instantaneous return —instantaneous return volatility.


Geometric Brownian motion

dSt/St = µdt + σdWt

The stock price is said to follow a geometric Brownian motion.

µ is often referred to as the drift, and σ the diffusion of the process.

Instantaneously, the stock price change is normally distributed,φ(µStdt, σ2S2

t dt).

Over longer horizons, the price change is lognormally distributed.

The log return (continuous compounded return) is normally distributed overall horizons:d ln St =

(µ− 1

2σ2)dt + σdWt . (By Ito’s lemma).

I d ln St ∼ φ(µdt − 12σ

2dt, σ2dt).I ln St ∼ φ(ln S0 + µt − 1

2σ2t, σ2t).

I ln ST/St ∼ φ((µ− 1

2σ2)

(T − t), σ2(T − t)).

Integral form: St = S0eµt− 1

2σ2t+σWt , ln St = ln S0 + µt − 1

2σ2t + σWt


Aggregating normally distributed random variables

If yi is independent of one another and yi ∼ φ(m,V ) for all i , then∑Ni=1 yi ∼ φ(Nm,NV ).

Mean and variance increase with time periods N, volatility increases withsquare root of time periods,

√N.

Application: Instantaneous return d ln St ∼ φ(µdt − 12σ

2dt, σ2dt) hasidentical and independent normal distribution (iid normal), the aggregatedreturn over finite time period (say, ln ST/St) remains normal with mean andvariance proportional to the time period (T − t).

If yi is independent but not identical: yi ∼ φ(mi ,Vi ) with (mi ,Vi ), the

distribution of∑N

i=1 yi is unknown (no longer normal).

I Example: The instantaneous price change dSt ∼ φ(µStdt, σ2S2t dt),

independent but not identical (it varies with St level). The aggregatedprice change is no longer normally distributed.


Examples on time aggregation

Suppose daily returns are iid normal with mean 0.05% and standarddeviation is 1.5%. What is the distribution of the aggregated return over 1year (252 business days)?

I The aggregated 1-year return is also normally distributed, and has amean of 0.05%× 252 = 12.6%.

I The annual return has a variance of (1.5%)2 × 252 = 0.0567.

I The annual return has a volatility of 1.5%×√

252 = 23.81%.

Pay attention to the aggregation on volatility.

I Another example: Suppose the annual return volatility for a stock is20%, under the iid normal assumption, what’s the monthly returnvolatility?

Remember: Wt has a volatility of√

t.

I Which of the following pairs are identical in distribution?(A) (2Wt ,W2t), (B) (4Wt ,W2t), (C) (4 + 4Wt , 2 + W2t), (D)(4 + 9Wt , 4 + W3t).


Normal versus lognormal distribution

dSt = µStdt + σStdWt , µ = 10%, σ = 20%,S0 = 100, t = 1.

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

PD

F of

ln(S

t/S0)

ln(St/S

0)

50 100 150 200 2500

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

PD

F o

f St

St

S0=100

The earliest application of Brownian motion to finance is Louis Bachelier in hisdissertation (1900) “Theory of Speculation.” He specified the stock price asfollowing a Brownian motion with drift:

dSt = µdt + σdWt


Simulate 100 stock price sample paths

dSt = µStdt + σStdWt , µ = 10%, σ = 20%,S0 = 100, t = 1.

0 50 100 150 200 250 300−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Daily

retu

rns

Days0 50 100 150 200 250 300

60

80

100

120

140

160

180

200

Stoc

k pr

ice

days

Stock with the return process: d ln St = (µ− 12σ

2)dt + σdWt .

Discretize to daily intervals dt ≈ ∆t = 1/252.

Draw standard normal random variables ε(100× 252) ∼ φ(0, 1).

Convert them into daily log returns: Rd = (µ− 12σ

2)∆t + σ√

∆tε.

Convert returns into stock price sample paths: St = S0e∑252

d=1 Rd .


µ versus µ− 12σ

2

Suppose we have daily data for a period of several months

Arithmetic mean: If you compute daily percentage return ∆S/S , and takethe simple sample average on the return. After annualization, it should beclose to µ.

µ =252

1

[1

N

N∑d=1

∆Sd

Sd

]

Geometric mean: If you start at day 1 and compound continuously (or daily,it should be close), the return per year is close to µ− 1

2σ.

I Daily: 2521

{[(1 + ∆S1

S1

)(1 + ∆S2

S2

)· · ·(

1 + ∆SN

SN

)]1/N

− 1

}I Continuous: 252

1

{1N ln

[S1

S0

S2

S1· · · SN

SN−1

]}= 252

N

{ln[

SN

S0

]}


Example

Suppose that returns in successive years are:15%, 20%, 30%, -20% and 25%.

Arithmetic mean: (0.15+0.20+0.30−0.20+0.25)5 = 0.14.

Geometric mean:[(1 + 0.15) (1 + 0.20) (1 + 0.30) (1− 0.20) (1 + 0.25)]1/5 − 1 = 0.124.

Comments:

I Geometric mean is normally lower than arithmetic mean due tosomething called “concavity (convexity) adjustment.”

I If Rt ∼ φ(m,V ), then E [Rt ] = m, but

E [eRt ] = em+ 12 V , ln E [eRt ] = m +

1

2V


Ito’s lemma on continuous processes

For a generic continuous process xt ,

dxt = µxdt + σxdWt ,

the transformed variable y = f (x , t) follows,

dyt =

(ft + fxµx +

1

2fxxσ

2x

)dt + fxσxdWt

Example 1: dSt = µStdt + σStdWt and y = ln St . We haved ln St =

(µ− 1

2σ2)dt + σdWt

Example 2: dvt = κ (θ − vt) dt + ω√

vtdWt , and σt =√

vt .

dσt =

(1

2σ−1

t κθ − 1

2κσt −

1

8ω2σ−1

t

)dt +

1

2ωdWt

Example 3: dxt = −κxtdt + σdWt and vt = a + bx2t . ⇒ dvt =?.


Ito’s lemma on jumpsFor a generic process xt with jumps,

dxt = µxdt + σxdWt +∫

g(z) (µ(dz , dt)− ν(z , t)dzdt) ,= (µx − Et [g ]) dt + σxdWt +

∫g(z)µ(dz , dt),

The random counting measure µ(dz , dt) realizes to a nonzero value for agiven z if and only if x jumps from xt− to xt = xt− + g(z) at time t.

The process ν(z , t)dzdt compensates the jump process so that the last termis the increment of a pure jump martingale.

The integral is over all possible jump sizes, R0, which means “the whole realline excluding zero.” — By definition, z = 0 is not a jump.

The transformed variable y = f (x , t) follows,

dyt =(ft + fxµx + 1

2 fxxσ2x

)dt + fxσxdWt

+∫

(f (xt− + g(z), t)− f (xt−, t))µ(dz , dt)−∫

fxg(z)ν(z , t)dzdt,=

(ft + fx (µx − Et [g ]) + 1

2 fxxσ2x

)dt + fxσxdWt

+∫

(f (xt− + g(z), t)− f (xt−, t))µ(dz , dt)


Ito’s lemma on jumps: Merton (1976) example

dyt =(ft + fxµx + 1

2 fxxσ2x

)dt + fxσxdWt

+∫

(f (xt− + g(z), t)− f (xt−, t))µ(dz , dt)−∫

fxg(z)ν(z , t)dzdt,=

(ft + fx (µx − Et [g(z)]) + 1

2 fxxσ2x

)dt + fxσxdWt

+∫

(f (xt− + g(z), t)− f (xt−, t))µ(dz , dt)

Merton (1976)’s jump-diffusion process:

dSt = µSt−dt + σSt−dWt +∫

St− (ez − 1) (µ(z , t)− ν(z , t)dzdt) , andy = ln St . We have

d ln St =

(µ− 1

2σ2

)dt + σdWt +

∫zµ(dz , dt)−

∫(ez − 1) ν(z , t)dzdt

f (xt− + g(z), t)− f (xt−, t) = ln St − ln St− = ln(St−ez)− ln St− = z .

The specification (ez − 1) on price guarantees that the maximum downsidejump is no larger than the pre-jump price level St−.

In Merton, ν(z , t) = λ 1σJ

√2π

e− (z−µJ )2

2σ2J .

Read his article for better understanding of jumps.


The key idea behind BSM

The option price and the stock price depend on the same underlying sourceof uncertainty.

The Brownian motion dynamics implies that if we slice the time thin enough(dt), it behaves like a binominal tree.

Reversely, if we cut ∆t small enough and add enough nodes, the binomialtree converges to the distribution behavior of the geometric Brownianmotion.

I Under this thin slice of time interval, we can combine the option withthe stock to form a riskfree portfolio.

I Recall our hedging argument: Choose ∆ such that f −∆S is riskfree.I The portfolio is riskless (under this thin slice of time interval) and must

earn the riskfree rate.I Magic: µ does not matter for this portfolio and hence does not matter

for the option valuation. Only σ matters.


The hedging proof

The stock price dynamics: dS = µSdt + σSdWt .

Let f (S , t) denote the value of a derivative on the stock, by Ito’s lemma:dft =

(ft + fSµS + 1

2 fSSσ2S2)dt + fSσSdWt .

At time t, form a portfolio that is long 1 unit of the derivative contract f andshort ∆ = fS unit of the stock. The value of the portfolio is P = f − fSS .

The instantaneous uncertainty of the portfolio is fSσSdWt − fSσSdWt = 0.Hence, instantaneously the delta-hedged portfolio is riskfree.

Then, the portfolio must earn the riskfree rate: dP = rPdt.

dP = df − fSdS =(ft + fSµS + 1

2 fSSσ2S2 − fSµS

)dt = r(f − fSS)dt

Hence, the fundamental partial differential equation:ft + rSfS + 1

2 fSSσ2S2 = rf

No where do we see the drift of the price dynamics (µ).I If you intend to predict stock return, the focus is µ.I If you intend to price options while performing delta hedge, regardless

of whether the stock price is fair or not (hence with no intension ofpredicting whether stock will go up or down), focus on σ.


Partial differential equation

The hedging argument leads to the following partial differential equation:

∂f

∂t+ (r − q)S

∂f

∂S+

1

2σ2S2 ∂

2f

∂S2= rf

I The only free parameter is σ (as in the binominal model).

Solving this PDE, subject to the terminal payoff condition of the derivative(e.g., fT = (ST − K )+ for a European call option), BSM derive analyticalformulas for call and put option value.

I Similar formula had been derived before based on distributional(normal return) argument, but µ was still in.

I The realization that option valuation does not depend on µ is big.Plus, it provides a way to hedge the option position.

The PDE is generic for any derivative securities, as long as S followsgeometric Brownian motion.

I Given boundary conditions, derivative values can be solved numericallyfrom the PDE.


Explicit finite difference method

One way to solve the PDE numerically is to discretize across time using Ntime steps (0,∆t, 2∆t, · · · ,T ) and discretize across states using M grids(0,∆S , · · · ,Smax).

We can approximate the partial derivatives at time i and state j , (i , j), bythe differences:fS ≈ fi,j+1−fi,j−1

2∆S , fSS ≈ fi,j+1+fi,j−1−2fi,j∆S2 , and ft =

fi,j−fi−1,j

∆t

The PDE becomes:fi,j−fi−1,j

∆t + (r − q)j∆Sfi,j+1−fi,j−1

2∆S + 12σ

2j2 (∆S)2 = rfi,j

Collecting terms: fi−1,j = aj fi,j−1 + bj fi,j + cj fi,j+1

I Essentially a trinominal tree: The time-i value at j state of fi,j is aweighted average of the time-i + 1 values at the three adjacent states(j − 1, j , j + 1).


Finite difference methods: variations

At (i , j), we can define the difference (ft , fS) in three different ways:

I Backward: ft = (fi,j − fi−1,j)/∆t.I Forward: ft = (fi+1,j − fi,j)/∆t.I Centered: ft = (fi+1,j − fi−1,j)/(2∆t).

Different definitions results in different numerical schemes:

I Explicit: fi−1,j = aj fi,j−1 + bj fi,j + cj fi,j+1.I Implicit: aj fi,j−1 + bj fi,j + cj fi,j+1 = fi+1,j .I Crank-Nicolson:−αj fi−1,j−1 +(1−βj)fi−1,j−γj fi−1,j+1 = αj fi,j−1 +(1+βj)fi,j +γj fi,j+1.


The BSM formula for European options

ct = Ste−q(T−t)N(d1)− Ke−r(T−t)N(d2),

pt = −Ste−q(T−t)N(−d1) + Ke−r(T−t)N(−d2),

where

d1 =ln(St/K)+(r−q)(T−t)+ 1

2σ2(T−t)

σ√

T−t,

d2 =ln(St/K)+(r−q)(T−t)− 1

2σ2(T−t)

σ√

T−t= d1 − σ

√T − t.

Black derived a variant of the formula for futures (which I like better):

ct = e−r(T−t) [FtN(d1)− KN(d2)],

with d1,2 =ln(Ft/K)± 1

2σ2(T−t)

σ√

T−t.

Recall: Ft = Ste(r−q)(T−t).

Once I know call value, I can obtain put value via put-call parity:ct − pt = e−r(T−t) [Ft − Kt ].


Cumulative normal distribution

ct = e−r(T−t) [FtN(d1)− KN(d2)] , d1,2 =ln(Ft/K )± 1

2σ2(T − t)

σ√

T − t

N(x) denotes the cumulative normal distribution, which measures theprobability that a normally distributed variable with a mean of zero and astandard deviation of 1 (φ(0, 1)) is less than x.

Most software packages (including excel) has efficient ways to computingthis function.

Properties of the BSM formula:

I As St becomes very large or K becomes very small, ln(Ft/K ) ↑ ∞,N(d1) = N(d2) = 1. ct = e−r(T−t) [Ft − K ] .

I Similarly, as St becomes very small or K becomes very large,ln(Ft/K ) ↑ −∞, N(−d1) = N(−d2) = 1. pt = e−r(T−t) [−Ft + K ].


Implied volatility

ct = e−r(T−t) [FtN(d1)− KN(d2)] , d1,2 =ln(Ft/K )± 1

2σ2(T − t)

σ√

T − t

Since Ft (or St) is observable from the underlying stock or futures market,(K , t,T ) are specified in the contract. The only unknown (and hence free)parameter is σ.

We can estimate σ from time series return. (standard deviation calculation).

Alternatively, we can choose σ to match the observed option price —implied volatility (IV).

There is a one-to-one correspondence between prices and implied volatilities.

Traders and brokers often quote implied volatilities rather than dollar prices.

The BSM model says that IV = σ. In reality, the implied volatility calculatedfrom different options (across strikes, maturities, dates) are usually different.


Options on what?

As long as we assume that the underlying security price follows a geometricBrownian motion, we can use (some versions) of the BSM formula to priceEuropean options.

Dividends, foreign interest rates, and other types of carrying costs maycomplicate the pricing formula a little bit.

A simpler approach: Assume that the underlying futures/forwards price (ofthe same maturity of course) process follows a geometric Brownian motion.

Then, as long as we observe the forward price (or we can derive the forwardprice), we do not need to worry about dividends or foreign interest rates —They are all accounted for in the forward pricing.

Numerical trips: I like to normalize option price in terms of the forwardoption value in percentage of the underlying forward.

ct =er(T−t)ct

Ft= N(d1)− K

FtN(d2).

Also normalize strike k = ln K/Ft (percentage return over forward),ct = N(d1)− ekN(d2).— always between (0, 1) like a probability.


Risk-neutral valuation

Recall: Under the binomial model, we derive a set of risk-neutralprobabilities such that we can calculate the expected payoff from the optionand discount them using the riskfree rate.

I Risk premiums are hidden in the risk-neutral probabilities.I If in the real world, people are indeed risk-neutral, the risk-neutral

probabilities are the same as the real-world probabilities. Otherwise,they are different.

Under the BSM model, we can also assume that there exists such anartificial risk-neutral world, in which the expected returns on all assets earnrisk-free rate.

The stock price dynamics under the risk-neutral world becomes,dSt/St = (r − q)dt + σdWt .

Simply replace the actual expected return (µ) with the return from arisk-neutral world (r − q) [ex-dividend return].

We label the true probability measure by P andthe risk-neutral measure by Q.


The risk-neutral return on spots

dSt/St = (r − q)dt + σdWt , under risk-neutral probabilities.

In the risk-neutral world, investing in all securities make the riskfree rate asthe total return.

If a stock pays a dividend yield of q, then the risk-neutral expected returnfrom stock price appreciation is (r − q), such as the total expected return is:dividend yield+ price appreciation =r .

Investing in a currency earns the foreign interest rate rf similar to dividendyield. Hence, the risk-neutral expected currency appreciation is (r − rf ) sothat the total expected return is still r .

Regard q as rf and value options as if they are the same.


The risk-neutral return on forwards/futures

If we sign a forward contract, we do not pay anything upfront and we do notreceive anything in the middle (no dividends or foreign interest rates). AnyP&L at expiry is excess return.

Under the risk-neutral world, we do not make any excess return. Hence, theforward price dynamics has zero mean (driftless) under the risk-neutralprobabilities: dFt/Ft = σdWt .

The carrying costs are all hidden under the forward price, making the pricingequations simpler.


Return predictability and option pricingConsider the following stock price dynamics,

dSt/St = (a + bXt) dt + σdWt ,dXt = κ (θ − Xt) + σxdW2t , ρdt = E[dWtdW2t ].

Stock returns are predictable by a vector of mean-reverting predictors Xt . Howshould we price options on this predictable stock?

Option pricing is based on replication/hedging — a cross-sectional focus,not based on prediction — a time series behavior.

Whether we can predict the stock price is irrelevant. The key is whether wecan replicate or hedge the option risk using the underlying stock.

Consider a derivative f (S , t), whose terminal payoff depends on S but not onX , then it remains true that dft =

(ft + fSµS + 1

2 fSSσ2S2)dt + fSσSdWt .

The delta-hedged option portfolio (f − fSS) remains riskfree.

The same PDE holds for f :ft + rSfS + 1

2 fSSσ2S2 = rf


Return predictability and risk-neutral valuation

Consider the following stock price dynamics (under P),

dSt/St = (a + bXt) dt + σdWt ,dXt = κ (θ − Xt) + σxdW2t , ρdt = E[dWtdW2t ].

Under the risk-neutral measure (Q), stocks earn the riskfree rate, regardlessof the predictability: dSt/St = (r − q) dt + σdWt

Analogously, the futures price is a martingale under Q, regardless of whetheryou can predict the futures movements or not: dFt/Ft = σdWt

Measure change from P to Q does not change the diffusion componentσdW .

The BSM formula still applies.


Stochastic volatility and option pricing

Consider the following stock price dynamics,

dSt/St = µdt +√

vtdWt ,dvt = κ (θ − vt) + ω

√vtdW2t , ρdt = E[dWtdW2t ].

How should we price options on this stock with stochastic volatility?

Consider a call option on S . Even though the terminal value does not dependon vt , the value of the option at other periods have to depend on vt .

I Assume f (S , t) does not depend on vt , we havedft =

(ft + fSµS + 1

2 fSSvtS2)dt + fSσS

√vtdWt , which obviously

depend on vt .

For f (S , v , t), we have (bivariate version of Ito):dft =

(ft + fSµS + 1

2 fSSvtS2 + fvκ (θ − vt) + 1

2 fvvω2vt + fSvωvtρ

)dt +

fSσS√

vtdWt + fvω√

vtdW2t .

Since there are two sources of risk (Wt ,W2t), delta-hedging is not going tolead to a riskfree portfolio.


Pricing kernel

In the absence of arbitrage, in an economy, there exists at least one strictlypositive process, Mt , the state-price deflator, such that the deflated gainsprocess associated with any admissible strategy is a martingale. The ratio ofMt at two horizons, Mt,T , is called a stochastic discount factor, or moreinformally, a pricing kernel.

I For an asset that has a terminal payoff ΠT , its time-t value ispt = EP

t [Mt,T ΠT ].I The time-t price of a $1 par riskfree zero-coupon bond that expires at

T is pt = EPt [Mt,T ].

I In a discrete-time representative agent economy with an additive CRRAutility, the pricing kernel is equal to the ratio of the marginal utilities ofconsumption,

Mt,t+1 = βu′(ct+1)

u′(ct)= β

(ct+1

ct

)−γ


From pricing kernel to exchange rates

Let Mht,T denote the pricing kernel in economy h that prices all securities in

that economy with its currency denomination.

The h-currency price of currency-f (h is home currency) is linked to thepricing kernels of the two economies by,

S fhT

S fht

=M f

t,T

Mht,T

The log currency return over period [t,T ], ln S fhT /S

fht equals the difference

between the log pricing kernels of the two economies.

Let S denote the dollar price of pound (e.g. St = $2.06), then

ln ST/St = ln Mpoundt,T − ln Mdollar

t,T .

If markets are completed by primary securities (e.g., bonds and stocks),there is one unique pricing kernel per economy. The exchange ratemovement is uniquely determined by the ratio of the pricing kernels.

If the markets are not completed by primary securities, exchange rates (andcurrency options) help complete the markets by requiring that the ratio ofany two viable pricing kernels go through the exchange rate.


From pricing kernel to measure change

In continuous time, it is convenient to define the pricing kernel via thefollowing multiplicative decomposition:

Mt,T = exp

(−∫ T

t

rsds

)E

(−∫ T

t

γ>s dXs

)

r is the instantaneous riskfree rate, E is the stochastic exponentialmartingale operator, X denotes the risk sources in the economy, and γ is themarket price of the risk X .

I If Xt = Wt , E(−∫ T

tγsdWs

)= exp

(−∫ T

tγsdWs − 1

2

∫ T

tγ2

s ds)

.

I In a continuous time version of the representative agent example,dXs = d ln ct and γ is relative risk aversion.

I r is usually a function of X .

The exponential martingale E(·) defines the measure change from P to Q.


Measure change defined by exponential martingales

Consider the following exponential martingale that defines the measure changefrom P to Q:dQdP∣∣t

= E(−∫ t

0γsdWs

),

If the P dynamics is: dS it = µiS i

tdt + σiS itdW i

t , with ρidt = E[dWtdW it ],

then the dynamics of S it under Q is: dS i

t =µiS i

tdt + σiS itdW i

t + E[−γtdWt , σiS i

tdW it ] =

(µi

t − γσiρi)S i

tdt + σiS itdW i

t

If S it is the price of a traded security, we need r = µi

t − γσiρi . The riskpremium on the security is µi

t − r = γσiρi .

How do things change if the pricing kernel is given by:

Mt,T = exp(−∫ T

trsds

)E(−∫ T

tγ>s dWs

)E(−∫ T

tγ>s dZs

),

where Zt is another Brownian motion independent of Wt or W it .


Bond pricing under P and Q

The time-t price of a $1 par riskfree zero-coupon bond that expires at T is:

pt = EPt [Mt,T ]

= EPt

[exp

(−∫ T

trsds

)E(−∫ T

tγ>s dXs

)]= EQ

t

[exp

(−∫ T

trsds

)].

We need to know the short rate dynamics under Q for bond pricing.

Suppose the pricing kernel is:

Mt,T = exp(−∫ T

trsds

)E(−∫ T

tγ>s dXs

)E(−∫ T

tη>s dZs

)with X ,Z

orthogonal, and r = r(X ).Bond price will only depend on X , not Z .


Measure change defined by exponential martingales: Jumps

Let X denote a pure jump process with its compensator being ν(x , t) underP.

Consider a measure change defined by the exponential martingale:dQdP∣∣t

= E (−γXt),

The compensator of the jump process under Q becomes:ν(x , t)Q = e−γxν(x , t).

Example: Merton (176)’s compound Poisson jump process,

ν(x , t) = λ 1σJ

√2π

e− (x−µJ )2

2σ2J . Under Q, it becomes

ν(x , t)Q = λ 1σJ

√2π

e−γx− (x−µJ )2

2σ2J = λQ 1

σJ

√2π

e−

(x−µQJ

)2

2σ2J with µQ = µJ − γσ2

J

and λQ = λe12γ(γσ2

J−2µJ ).

Reference: Kuchler & Sørensen, Exponential Families of Stochastic Processes, Springer, 1997.


The BSM DeltaThe BSM delta of European options (Can you derive them?):

∆c ≡∂ct

∂St= e−qτN(d1), ∆p ≡

∂pt

∂St= −e−qτN(−d1)

(St = 100,T − t = 1, σ = 20%)

60 80 100 120 140 160 180−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Delta

Strike

BSM delta

call deltaput delta

60 80 100 120 140 160 1800

10

20

30

40

50

60

70

80

90

100

Delta

Strike

Industry delta quotes

call deltaput delta

Industry quotes the delta in absolute percentage terms (right panel).

Which of the following is out-of-the-money? (i) 25-delta call, (ii) 25-deltaput, (iii) 75-delta call, (iv) 75-delta put.

The strike of a 25-delta call is close to the strike of: (i) 25-delta put, (ii)50-delta put, (iii) 75-delta put.


Delta as a moneyness measureDifferent ways of measuring moneyness:

K (relative to S or F ): Raw measure, not comparable across different stocks.

K/F : better scaling than K − F .

ln K/F : more symmetric under BSM.

ln K/F

σ√

(T−t): standardized by volatility and option maturity, comparable across

stocks. Need to decide what σ to use (ATMV, IV, 1).

d1: a standardized variable.

d2: Under BSM, this variable is the truly standardized normal variable withφ(0, 1) under the risk-neutral measure.

delta: Used frequently in the industry

I Measures moneyness: Approximately the percentage chance the optionwill be in the money at expiry.

I Reveals your underlying exposure (how many shares needed to achievedelta-neutral).


OTC quoting and trading conventions for currency options

Options are quoted at fixed time-to-maturity (not fixed expiry date).

Options at each maturity are not quoted in invoice prices (dollars), but inthe following format:

I Delta-neutral straddle implied volatility (ATMV):A straddle is a portfolio of a call & a put at the same strike. The strikehere is set to make the portfolio delta-neutral ⇒ d1 = 0.

I 25-delta risk reversal: IV (∆c = 25)− IV (∆p = 25).I 25-delta butterfly spreads: (IV (∆c = 25) + IV (∆p = 25))/2− ATMV .I Risk reversals and butterfly spreads at other deltas, e.g., 10-delta.

When trading, invoice prices and strikes are calculated based on the BSMformula.

The two parties exchange both the option and the underlying delta.

I The trades are delta-neutral.


The BSM vega

Vega (ν) is the rate of change of the value of a derivatives portfolio withrespect to volatility — it is a measure of the volatility exposure.

BSM vega: the same for call and put options of the same maturity

ν ≡ ∂ct

∂σ=∂pt

∂σ= Ste

−q(T−t)√

T − tn(d1)

n(d1) is the standard normal probability density: n(x) = 1√2π

e−x2

2 .

(St = 100,T − t = 1, σ = 20%)

60 80 100 120 140 160 1800

5

10

15

20

25

30

35

40

Vega

Strike−3 −2 −1 0 1 2 30

5

10

15

20

25

30

35

40

Vega

d2

Volatility exposure (vega) is higher for at-the-money options.


Vega hedging

Delta can be changed by taking a position in the underlying.

To adjust the volatility exposure (vega), it is necessary to take a position inan option or other derivatives.

Hedging in practice:

I Traders usually ensure that their portfolios are delta-neutral at leastonce a day.

I Whenever the opportunity arises, they improve/manage their vegaexposure — options trading is more expensive.

I As portfolio becomes larger, hedging becomes less expensive.

Under the assumption of BSM, vega hedging is not necessary: σ does notchange. But in reality, it does.

I Vega hedge is outside the BSM model.


Example: Delta and vega hedging

Consider an option portfolio that is delta-neutral but with a vega of −8, 000. Weplan to make the portfolio both delta and vega neutral using two instruments:

The underlying stock

A traded option with delta 0.6 and vega 2.0.

How many shares of the underlying stock and the traded option contracts do weneed?

To achieve vega neutral, we need long 8000/2=4,000 contracts of thetraded option.

With the traded option added to the portfolio, the delta of the portfolioincreases from 0 to 0.6× 4, 000 = 2, 400.

We hence also need to short 2,400 shares of the underlying stock ⇒ eachshare of the stock has a delta of one.


Another example: Delta and vega hedging

Consider an option portfolio with a delta of 2,000 and vega of 60,000. We plan tomake the portfolio both delta and vega neutral using:

The underlying stock

A traded option with delta 0.5 and vega 10.

How many shares of the underlying stock and the traded option contracts do weneed?

As before, it is easier to take care of the vega first and then worry about thedelta using stocks.

To achieve vega neutral, we need short/write 60000/10 = 6000 contracts ofthe traded option.

With the traded option position added to the portfolio, the delta of theportfolio becomes 2000− 0.5× 6000 = −1000.

We hence also need to long 1000 shares of the underlying stock.


A more formal setupLet (∆p,∆1,∆2) denote the delta of the existing portfolio and the two hedginginstruments. Let(νp, ν1, ν2) denote their vega. Let (n1, n2) denote the shares ofthe two instruments needed to achieve the target delta and vega exposure(∆T , νT ). We have

∆T = ∆p + n1∆1 + n2∆2

νT = νp + n1ν1 + n2ν2

We can solve the two unknowns (n1, n2) from the two equations.

Example 1: The stock has delta of 1 and zero vega.

0 = 0 + n10.6 + n2

0 = −8000 + n12 + 0

n1 = 4000, n2 = −0.6× 4000 = −2400.

Example 2: The stock has delta of 1 and zero vega.

0 = 2000 + n10.5 + n2, 0 = 60000 + n110 + 0

n1 = −6000, n2 = 1000.

When do you want to have non-zero target exposures?


BSM gamma

Gamma (Γ) is the rate of change of delta (∆) with respect to the price ofthe underlying asset.

The BSM gamma is the same for calls and puts:

Γ ≡ ∂2ct

∂S2t

=∂∆t

∂St=

e−q(T−t)n(d1)

Stσ√

T − t

(St = 100,T − t = 1, σ = 20%)

60 80 100 120 140 160 1800

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Vega

Strike−3 −2 −1 0 1 2 30

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Vega

d2

Gamma is high for near-the-money options.High gamma implies high variation in delta, and hence more frequent rebalancingto maintain low delta exposure.


Gamma hedging

High gamma implies high variation in delta, and hence more frequentrebalancing to maintain low delta exposure.

Delta hedging is based on small moves during a very short time period.

I assuming that the relation between option and the stock is linearlocally.

When gamma is high,

I The relation is more curved (convex) than linear,I The P&L (hedging error) is more likely to be large in the presence of

large moves.

The gamma of a stock is zero.

We can use traded options to adjust the gamma of a portfolio, similar towhat we have done to vega.

But if we are really concerned about large moves, we may want to trysomething else.


Dynamic hedging with greeks

The idea of delta and vega hedging isbased on a locally linear approximation(partial derivative) of the relationbetween the derivative portfolio valueand the underlying stock price andvolatility.

60 80 100 120 140 160 180−20

−10

0

10

20

30

40

50

60

70

80

Cal

l

ST

Since the relation is not linear, thehedging ratios change as theenvironment change.

Dynamic hedging, hedging basedon partial derivatives, often asks forfrequent rebalancing.

Dynamic hedging works well if

I The overall relation is close tolinear. Hence, the hedgingratio is stable over time.

I The underlying variable (stockprice, volatility) variessmoothly and only changes alittle within a certain timeinterval.


Dynamic versus static hedging

Dynamic hedging can generate large hedging errors when the underlyingvariable (stock price) can jump randomly.

I A large move size per se is not an issue, as long as we know how muchit moves — a binomial tree can be very large moves, but delta hedgeworks perfectly.

I As long as we know the magnitude, hedging is relatively easy.

I The key problem comes from large moves of random size.

An alternative is to devise static hedging strategies: The position of thehedging instruments does not vary over time.

I Conceptually not as easy. Different derivative products ask for differentstatic strategies.

I It involves more option positions. Cost per transaction is high.

I Monitoring cost is low. Fewer transactions.


Example: Static hedging long-term option using short-termoptions

“Static hedging of standard options,” Carr and Wu, wp.

A European call option maturing at a fixed time T can be replicated by astatic position in a continuum of European call options at a shorter maturityu ≤ T :

C (S , t; K ,T ) =

∞∫0

w(K)C (S , t;K, u)dK,

with the weight of the portfolio given by w(K) = ∂2

∂K2 C (K, u; K ,T ), thegamma that the target call option will have at time u, should the underlyingprice level be at K then.

Assumption: The stock price can jump randomly. The only requirement isthat the stock price is Markovian in itself: St captures all the informationabout the stock up to time t.


Practical implementation and performance

Target Mat 2 4 12 12 12 2 4 12

Strategy Static with Five Options Daily DeltaInstruments 1 1 1 2 4 Underlying Futures

Mean -0.00 -0.10 -0.01 -0.07 0.01 0.16 0.03 -0.01Std Err 0.23 0.50 0.87 0.64 0.42 0.57 0.65 0.88RMSE 0.23 0.51 0.86 0.64 0.42 0.59 0.65 0.87Min -0.62 -1.55 -2.32 -1.97 -1.58 -1.28 -2.23 -2.51Max 0.50 1.09 1.56 1.10 0.86 1.45 1.42 1.76Target Call 3.68 6.16 10.94 10.76 11.52 3.65 6.14 11.58


Semi-static hedge of barriers

Option being hedged: a one-month one-touch barrier option with a lowerbarrier and payment at expiry

Hedging instruments: vanilla and binary put options, binary call.

Procedure:

I At time t, sell the barrier, and put on a hedging position with vanillaoptions with the terminal payoff1(ST < L)

(1 + ST

L

)= 2(S < L)− (L− S)+/L

I If the barrier L never hits before expiry, both the barrier and thehedging portfolio generate zero payoff.

I If the barrier is hit before expiry, then:

F Sell a European option with the terminal payoff

1(ST < L)(

STL

)= (S < L)− (L− S)+/L.

F Buy a European option that pays 1(ST > L).F The operation is self-financing under the BSM when r = q.


Hedging barriers: Dynamic and static hedging performance

JPYUSD:Strategy Mean Std Max Min Skew Kurt

Value 55.40 3.71 67.39 37.77 0.55 2.77No Hedge -49.46 50.01 0.00 -100.00 -0.02 1.00Delta -44.24 20.88 76.67 -138.26 0.79 7.86Vega -44.45 17.50 67.47 -130.67 1.40 10.14Vanna -45.38 18.01 67.47 -137.32 1.32 10.08

Static -50.86 10.68 48.94 -68.02 6.66 62.00


Hedging barriers: Dynamic and static hedging performance

GBPUSD:Strategy Mean Std Max Min Skew Kurt

Value 49.35 1.66 54.98 42.86 0.21 4.09No Hedge -32.21 46.73 0.00 -100.00 -0.76 1.58Delta -36.39 19.81 70.02 -127.86 1.20 6.34Vega -38.24 16.58 55.74 -124.94 1.85 8.82Vanna -39.07 16.33 54.92 -119.33 1.85 8.56

Static -46.55 10.80 49.19 -52.62 5.07 39.81


Hedging using nearby contracts

To hedge against risk exposure (such as greeks), one needs to know the riskexposures first. To the extent that the exposure is measured/modeledwrong, so is the hedge.

The vega of a 10-year option is not the same as the vega of 1-month option.

We propose a new methodology to hedge based on affinity of contractcharacteristics, not risk exposures...


The Black-Scholes Model - City University of New...

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