Black-Scholes Partial Differential Equation

Contents (hide)

  1. 1. Overview
  2. 2. Derivation
  3. 3. Applications

1.  Overview

Solving the Black-Scholes [BMS] partial differential equation (PDE) was the original method used to derive the Black-Scholes formula for the prices of European options. Constructing the PDE takes the following steps:

  1. Say that we want to price a derivative with a generic payoff \fs4 g(S). In our disposal we have the underlying asset, with price process \fs4 S(t), and a risk-free bank account, with an instantaneous interest rate \fs4 r.
  2. We construct portfolios using the asset and the bank account, that are self financing.
  3. We examine the conditions under which such portfolios are replicating.
  4. We argue that the value of the replicating portfolio should equal the value of the derivative contract, otherwise arbitrage opportunities would arise.
  5. This equalities result in the PDE that the price of the derivative has to satisfy.

Note that in the process we also prove that the market is complete, since derivatives with generic payoffs can be replicated.

2.  Derivation

The starting point is a filtered probability space \fs4 (\Omega, \mathbb{F}, \mathcal{F}, P). Consider a portfolio \fs4 \mathbf{H}=\{ H(t) \}_{t\geq 0} which is an (\fs4 \mathcal{F}(t)-) adapted process. \fs4 H(t) denotes the number of shares that we are holding at time \fs4 t; the rest of our money will be invested in the money market account, giving us a constant risk free rate of return \fs4 r.

Following the assumptions, the stock price, \fs4 S(t), follows a geometric Brownian motion

\fs4 \frac{dS(t)}{S(t)}=\mu dt+\sigma dB(t)

Then, the wealth of the investor, \fs4 X(t), will follow a diffusion given by [with time suppressed]

\fs4 dX = HdS+r( X-HS) dt = [ rX+HS( \mu -r) ] dt+HS\sigma dB

where one can observe the presence of the risk premium \fs4 \mu -r.

We can value a derivative which has payoffs \fs4 g[S(T)] at time \fs4 T. Suppose that the value of this claim at time \fs4 t is given by \fs4 G(t) =f( S,t), where the price \fs4 S=S(t). Applying the Ito formula to this function yields

\fs4 dG=[ f_{t}+\mu Sf_{S}+\frac{1}{2}\sigma ^{2}S^{2}f_{SS}]dt+\sigma Sf_{S}dB

A replicating portfolio \fs4 \mathbf{H} would track the value of \fs4 G(t) for all \fs4 t\in [0,T]. We saw above that the value of such a portfolio (i.e. the wealth of the investor) will be equal to

\fs4 dX=[ rX+HS( \mu -r)] dt+HS\sigma dB

Observe that both diffusions are (as expected) driven by the same Brownian motion. In order for the portfolio to track \fs4 G(t) at all times (mathematically \fs4 X(t) =G(t) =f(S,t) for all \fs4 t\in [0,T]) one can just equate the coefficients. The fact that this is feasible implies that the market under these assumptions is complete, since a claim with a generic payoff function \fs4 g[S(t)] can be replicated. The resulting relations are

\fs4 f_{t}+\mu Sf_{S}+\frac{1}{2}\sigma ^{2}S^{2}f_{SS} =rX+HS( \mu-r)
\fs4 \sigma Sf_{S} = HS\sigma

The second equation will give the delta-hedging rule

\fs4 H( t) =f_{S}( S,t)

while the first equation, using the fact that \fs4 X(t) =f(S,t) yields that

\fs4 f_{t}( S,t) +rSf_{S}( S,t) +\frac{1}{2}\sigma^{2}S^{2}f_{SS}(S,t) =rf(S,t)

This is the BMS partial differential equation. The terminal condition that has to be satisfied is that \fs4 f(S,T) =g(S)

3.  Applications

Apparently the Black-Scholes PDE can be used for all European contracts, depending on the payoff function \fs4 g(S). For example

  • European Calls : \fs4 g(S) =( S-K) ^{+}
  • European Puts : \fs4 g(S) =( K-S) ^{+}
  • Digital Calls : \fs4 g(S) = I[ S\geq K]
  • and many others

Black and Scholes (1973, Journal of Political Economy) give the solution of the PDE above for the European call, which is known as the Black-Scholes formula. In general, the PDE cannot be solved in closed form, giving explicit formulas. In such cases numerical methods such as finite differences have to be employed.

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