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Finance and Derivatives: Theory and PracticeFinance and Derivatives: Theory and PracticeSébastien Bossu and Philippe Henrotte

Chapter 10The Black-Scholes model

2Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice.Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

Chapter 10 The Black-Scholes model

1. The Black-Scholes Partial Differential Equation Like the binomial model, Black-Scholes assumes that

the underlying asset is subject to random variations with respect to its initial price level St between two infinitesimally close times t and t + dt.

However, Black-Scholes considers an infinity of final levels St+dt rather than only two outcomes: St(1 + u) and St(1 + d). The final levels are distributed according to a log-normal distribution, i.e.:St+dt = St(1 + X), where X is the normally distributed return.



Figure p.168: Binomial vs. Black-Scholes

St

St (1+u)

St (1+d)

St St+dtSt

St (1+u)

St (1+d)

St St+dt



1. The Black-Scholes Partial Differential Equation (2) The assumptions of the Black-Scholes model are:

The price (St) of the underlying asset follows a geometric Brownian motion:

The yield curve is flat and constant throughout time, with r being the continuous interest rate.

The underlying asset pays no income and has no cost of carry.

Together with the usual economic assumptions: There are no arbitrage opportunities on the markets. Transactions take place in continuous time, have no

cost, and assets are infinitely liquid. Short-selling is allowed.

t t t tdS S dt S dW



1. The Black-Scholes Partial Differential Equation (3) Let Dt be the value of the derivative at time t,

and assume that it is only a function of time and S:

Dt = f (t, St) for all t 0. If f is ‘sufficiently smooth’ then the Ito-Doeblin

theorem yields: 21 ²²2 ²t t t t t

f f f fdD df S S dt S dWt S S S



1. The Black-Scholes Partial Differential Equation (4) This equation means that between times t and

t + dt, the value of the derivative is exposed to a drift:

and to a volatility risk This risk is proportional to the volatility risk of

the underlying itself , and the coefficient of proportionality is

21 ²²2 ²t t

f f fS S dtt S S

t tfS dWS

fS

t tS dW


Chapter 10 The Black-Scholes modelFigure p.169: Drift & distribution of the underlying and the derivative

25

50

100

125

200

250

5

10

20

25

one-to-onecorrespondence

40

50

Distribution of the underlyingDistribution of the derivative

Drift of theunderlying

Drift of thederivative

Initial Spot

100

Initial Option Value

28

Final Spot Final Option Value

25

50

100

125

200

250

5

10

20

25

one-to-onecorrespondence

40

50

Distribution of the underlyingDistribution of the derivative

Drift of theunderlying

Drift of thederivative

Initial Spot

100

Initial Option Value

28

Final Spot Final Option Value



1. The Black-Scholes Partial Differential Equation (5) As with the binomial model, the holder of one

unit of derivative can eliminate at any time t the volatility risk on horizon t + dt by selling units of underlying.

Define Pt as the value at time t of a portfolio long one single unit of derivative and short units of underlying:

Pt = f(t, St) – St

At time t + dt, the portfolio value is:Pt + dt = f(t + dt, St + dt) – St + dt



1. The Black-Scholes Partial Differential Equation (6) After the Ito-Doeblin theorem and the price

model equation for S:

The portfolio is riskless between t and t + dt if and only if the change in value dPt has no random component:

21 ²²2 ²

t t dt t

t

t t t t t t

dP P Pdf dS

f f f fS S S dt S S dWt S S S

0t tfS SS



1. The Black-Scholes Partial Differential Equation (7)

Dividing both sides by σSt:

This result is expected since we already identified that the volatility risk of the derivative is exactly proportional to that of the underlying, with coefficient (or hedge ratio)

fS

fS




Substituting with its value in our equation for dPt yields:

This portfolio is riskless and there is no arbitrage. Its value must grow at the risk-free continuous interest rate r:

dPt = rPtdt

21 ²²2 ²t t

f fdP S dtt S




Since Pt = f (t, St) – St, we obtain another equation for dPt:

Equations (1) and (2) both characterize in deterministic terms the change in value of the portfolio between times t and t + dt, therefore their drift coefficients must be equal.

t t

t

dP r f S dt

frf r S dtS




This yields the Black-Scholes partial differential equation:

This partial differential equation has an infinite number of solutions, which define the admissible (or tradable) derivatives of the underlying asset S.

To determine an individual solution, additional constraints must be specified. In the case of European options, this condition is the payoff at maturity, for instance:

for a vanilla call : f(T, ST) = max(0, ST – K); for a vanilla put : f(T, ST) = max(0, K – ST).

21 ²²2 ²t t

f f frf rS St S S




The hedge ratio continuously changes over time: the Black-Scholes model relies on a dynamic strategy called delta-hedging, which we introduced earlier.

In theory, delta-hedging allows an option trader to perfectly replicate the payoff at a cost equal to the option value without incurring any risk.



2. Black-Scholes Formulas For vanilla European calls and puts, there are

closed-form solutions to the Black-Scholes partial differential equation:

where:

0 0 1 2

0 2 0 1

( ) ( )

( ) ( )

rT

rT

c S N d Ke N d

p Ke N d S N d

0 0

1 2 1

² ²ln ln2 2,

S Sr T r TK Kd d d T

T T



2. Black-Scholes Formulas (2) These formulas are identical to those obtained

in the log-normal model: e-rT is the discount factor S0erT is the forward price F0 of the underlying for

maturity T.

What is added value of Black-Scholes over the log-normal model?

Arbitrage argument: should the price of the derivative differ from its theoretical value, Black-Scholes assures us that it is possible to implement a delta-hedging strategy and make riskless profits.



3. Volatility Recall the model of asset prices for (St):

The drift coefficient is the mean continuous rate of return of the underlying.

This parameter does not appear in the Black-Scholes formulas: the value of an option does not depend on the expected return of the underlying.

This is because asset prices already include expectations of future growth. When increases, St will usually increase as well, and this change will be reflected in the option price.

tt

t

dSdt dW

S



3. Volatility (2) The volatility coefficient is the standard

deviation of the return of the underlying: Option prices are extremely sensitive to volatility. The non-linearity of call and put payoffs results in a

higher option value when volatility goes up.

There are two commonly used techniques to determine the volatility parameter :1. historical volatility2. implied volatility



3.1. Historical Volatility Historical volatility is determined by

calculating the annualized standard deviation of asset returns.

The historical approach leaves several issues unresolved: How far back in time should one go? Should prices be observed every second, hour, day,

or month? Is past volatility a good estimate for future volatility?

Because these questions do not have definite answers, historical volatility is only used as a very rough estimate to determine the Black-Scholes value of an option.



3.2. Implied Volatility The implied approach consists in finding the

value of the parameter which matches the Black-Scholes value of an option with its present market price,

Such value for is called implied volatility.



Figure p.174: Black-Scholes pricing

BlackScholes

SKTr

Parameters

TheoreticalValue



Figure p.174: Implied volatility

BlackScholes

SKTr

Parameters

TheoreticalValue

imp



3.2. Implied Volatility (2) Implied volatility can then be used to compute

the theoretical value of other options with similar characteristics on the same underlying.

On most markets, all the other parameters are known with certainty and options prices are actually quoted in implied volatility rather than price.

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