Dr Scholes

1.3 The Black - Scholes partial differential equation

Another way to find the Black - Scholes option value is to solve the partial differential equation (PDE). We assume that the underlying stock follows a geometric Brownian motion. This is

T dS = MS t + dt sS t dW t

Where t W is a Weiner process.

Now let V be a kind of option on S, V is a function of S and T. V (S, T) is the value of the option at time t if the price of the underlying stock at time t is S. The value of the option when the option expires is known. To determine its value in an earlier era, we need to know how the value changes as we go back in time. To solve the Black - Scholes PDE, we must use Ito's lemma.

Ito's lemma states that if an asset price follows the Ito process

dS = m (S, T) dt + s (S, T) dW

Thus, according to Ito's lemma, we

dV = (aˆ‚ V aˆ‚ V mS + + ½ s ² ² ² S aˆ‚ V) dt + dW sS aˆ‚ V

aˆ‚ aˆ‚ aˆ‚ aˆ‚ S t S ² S

Now consider a trading strategy under which it holds an option and continuous trades in the stock to hold - aˆ‚ V part. At time t, the value of these holdings will

aˆ‚ S

P = V - V aˆ‚ S

aˆ‚ S

The composition of this portfolio, called delta - hedging portfolio may vary from time to time stage stage. Let R be the cumulative gain or loss to follow that strategy. Then, during the [period t, t + dt], the profit or loss is instantaneous

dR dV = - aˆ‚ V DS

aˆ‚ S

Substituting in the equations above we get

dR = (aˆ‚ V + ½ s ² ² ² S aˆ‚ V) dt

aˆ‚ aˆ‚ t S ²

This equation does not contain term dW. In other words, it presents no risk - less. Thus, since there is no arbitrage, the rate of return of this portfolio must be equal to the yield of any other risk - less instrument. Now, assuming the risk - free rate of return is r, we have during the [period t, t + dt]

rPdt = dR = (aˆ‚ V + ½ s ² ² ² S aˆ‚ V) dt

aˆ‚ aˆ‚ t S ²

If we now substitute for P and dividing by dt, we obtain by using the Black - Scholes PDE:

aˆ‚ V + ½ s ² ² ² S aˆ‚ V aˆ‚ V + R - RV = 0

aˆ‚ aˆ‚ aˆ‚ t SS ²

That is the law of evolution of the value of the option. With the assumptions of the Black - Scholes model, this partial differential equation holds whenever V is twice differentiable with respect to S and once with respect to t.

Consider the Black - Scholes price of a call option on a stock currently trading at price S. The option has an exercise price of K, ie the right to purchase one share at price K T years in the future. The interest rate is r and the constant stock volatility is s.

V (0, t) = 0 for all t

V (S, T) SS ~ ® ¥

V (S, T) = max (SK, 0)

The last condition gives the value of the option when the option expires. The solution of the PDE gives the value of the option at any time earlier. To solve the PDE, we transform the equation into a diffusion equation that can be solved using standard methods. We introduce the change of variable change

x = ln (S / K) + (r - s ² / 2) (T - t)

t = T - t

u = r V (T - t)

Then the Black - Scholes PDE becomes a diffusion equation

u = ¶ ¶ s ² ² u

¶ ¶ x ² t 2

The terminal condition V (S, T) = max (S - K, 0) now becomes an initial condition

u (x, 0) = u0 (x) K º max (e <.