ch10 by Salim Kasap

Finance and Derivatives: Theory and Practice SĂŠ bastien Bossu and Philippe Henrotte

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.

Chapter 10 The Black-Scholes model

Figure p.168: Binomial vs. BlackScholes

St (1+u)

St+dt

St (1+d)

Chapter 10 The Black-Scholes model

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:

dSt = µ St dt + σ St dWt



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.

Chapter 10 The Black-Scholes model

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: ∂f 1 ∂f   ∂f  2 ∂² f  dDt = df =  + µ St + σ ² St  dt +  σ St  dWt ∂S 2 ∂S ²  ∂S   ∂t 

Chapter 10 The Black-Scholes model

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: ∂f 1  ∂f 2 ∂² f  + σ ² St  + µ St ÷dt ∂S 2 ∂S ²   ∂t



∂f   σ S and to a volatility risk  t ÷dWt ∂S   This risk is proportional to the volatility risk of the

underlying itself σ St dWt , and the coefficient of ∂f proportionality is ∂S Sébastien 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

Figure p.169: Drift & distribution of the underlying and the derivative Final Spot Final Option Value 250

200

Distribution of the underlying

Distribution of the derivative one-to-one correspondence 125

Initial Spot 100

Initial Option Value

25 Drift of the

derivative

Drift of the underlying 100

SĂŠbastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright ÂŠ 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 (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 δ units of underlying. Define Pt as the value at time t of a portfolio long one single unit of derivative and short δ units of underlying: Pt = f(t, St) – δSt At time t + dt, the portfolio value is: Pt + dt = f(t + dt, St + dt) – δSt + dt

Chapter 10 The Black-Scholes model

1. The Black-Scholes Partial Differential Equation (6) 

After the Ito-Doeblin theorem and the price model equation for S: dPt = Pt + dt − Pt = df − δ dSt  ∂f  ∂f 1 ∂² f  ∂f   =  + µ St + σ ² St2 − δµ S dt + σ S − δσ S t t t  dWt   ∂S 2 ∂S ²  ∂S    ∂t 



The portfolio is riskless between t and t + dt if and only if the change in value dPt has no random ∂f component: σS − δσ S = 0 t

∂S

Chapter 10 The Black-Scholes model

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

Dividing both sides by σSt: δ=



∂f ∂S

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 ∂f (or hedge ratio) ∂S

Chapter 10 The Black-Scholes model

1. The Black-Scholes Partial Differential Equation (8) 

Substituting δ with its value in our equation for dPt yields:  ∂f 1 2 ∂² f  dPt =  + σ ² St ÷dt ∂S ²   ∂t 2



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

Chapter 10 The Black-Scholes model

1. The Black-Scholes Partial Differential Equation (8) 

Since Pt = f (t, St) – δSt, we obtain another equation for dPt: dPt = r ( f − δ St ) dt ∂f   =  rf − r St  dt ∂S  



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.

Chapter 10 The Black-Scholes model

1. The Black-Scholes Partial Differential Equation (9) 

This yields the Black-Scholes partial differential equation: ∂f ∂f 1 2 ∂² f rf = + rSt + σ ² St ∂t ∂S 2 ∂S ²



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).

Chapter 10 The Black-Scholes model

1. The Black-Scholes Partial Differential Equation (10) 



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.

Chapter 10 The Black-Scholes model

2. Black-Scholes Formulas 

For vanilla European calls and puts, there are closedform solutions to the Black-Scholes partial differential equation:

c0 = S0 N (d1 ) − Ke − rT N (d 2 ) p0 = Ke − rT N (−d 2 ) − S0 N (−d1 ) where: S0  S0  σ² σ² ln +  r + ÷T ln +  r − ÷T K  2  K  2  d1 = , d2 = = d1 − σ T σ T σ T Sébastien 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

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 lognormal 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.

Chapter 10 The Black-Scholes model

3. Volatility 

Recall the model of asset prices for (St): dSt = µ dt + σ dWt St



The drift coefficient µ 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 µ increases, St will usually increase as well, and this change will be reflected in the option price.

Chapter 10 The Black-Scholes model

3. Volatility (2) 

The volatility coefficient σ 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 σ: 1. 2.

historical volatility implied volatility

Chapter 10 The Black-Scholes model

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.

Chapter 10 The Black-Scholes model

3.2. Implied Volatility 



The implied approach consists in finding the value of the parameter σ which matches the Black-Scholes value of an option with its present market price, Such value for σ is called implied volatility.

Chapter 10 The Black-Scholes model

Figure p.174: Black-Scholes pricing Parameters S K T r σ

Black Scholes

Theoretical Value

Chapter 10 The Black-Scholes model

Figure p.174: Implied volatility Parameters S K T r

Black Scholes

Theoretical Value

ﾏナmp Sﾃｩbastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright ﾂｩ Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

Chapter 10 The Black-Scholes model

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.