Deriving the Black-Scholes Equation

Now that we have derived Ito's Lemma, we are in a position to derive the Black-Scholes equation.

Suppose we wish to price a vanilla European contingent claim C, on a time-varying asset S, which is set to mature at T. We shall assume that S follows a geometric Brownian motion with mean growth rate of $\mu$ and volatility $\sigma$. $r$ will represent the continuously compounding risk-free interest rate. $r$, $\mu$, and $\sigma$ are not functions of time, t, or the asset price S, and so are fixed for the duration of the option's lifetime.

Since our option price, C, is a function of time t and the price of the asset S, we will use the notation C = C(S,t) to represent the price of the option. Note that we are assuming at this stage that C exists and is well-defined. We will later show this to be a justified claim.

The first step is to utilise Ito's Lemma on the function C(S,t) to give us an SDE:

$$\begin{array}{r}dC=\frac{\partial C}{\partial t}dt+\frac{\partial C}{\partial S}(S,t)dS+\frac{1}{2}\frac{{\partial }^{2}C}{\partial S^2}(S,t)d{S}^2\end{array}$$

Our asset price is modelled by a geometric Brownian motion, the expression for which is recalled here. Note that $\mu$ and $\sigma$ are constant - i.e. not functions of S or t:

$$\begin{array}{r}dS(t)=\mu S(t)dt+\sigma S(t)dX(t)\end{array}$$

We can substitute this expression into Ito's Lemma to obtain:

$$\begin{array}{r}dC=(\frac{\partial C}{\partial t}(S,t)+\mu S\frac{\partial C}{\partial S}(S,t)+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}C}{\partial S^2}(S,t))dt+\sigma S\frac{\partial C}{\partial S}(S,t)dX\end{array}$$

The thrust of our derivation argument will essentially be to say that a fully hedged portfolio, with all risk eliminated, will grow at the risk-free rate. Thus, we need to determine how our portfolio changes over time. Specifically, we are interested in the infinitesimal change of a mixture of a call option and a quantity of assets. The quantity will be denoted by $\mathrm{\Delta }$. Hence:

$$\begin{array}{r}d(C+\mathrm{\Delta }S)=(\frac{\partial C}{\partial t}(S,t)+\mu S\frac{\partial C}{\partial S}(S,t)+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}C}{\partial S^2}(S,t)+\mathrm{\Delta }\mu S)dt+\mathrm{\Delta }S(\frac{\partial C}{\partial S}+\mathrm{\Delta })dX\end{array}$$

This leads us to a choice for $\mathrm{\Delta }$, which will eliminate the term associated with the randomness. If we set $\mathrm{\Delta }=-\frac{\partial C}{\partial S}(S,t)$ we receive:

$$\begin{array}{r}d(C+\mathrm{\Delta }S)=(\frac{\partial C}{\partial t}(S,t)+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}C}{\partial S^2}(S,t))dt\end{array}$$

Note that we have glossed over the issue of what the derivative of $\mathrm{\Delta }$ is. We will return to this later.

This technique is known as Delta-Hedging and provides us with a portfolio that is free of randomness. This is how we can apply the argument that it should grow at the risk-free rate; otherwise, as with our previous arguments, we would have an arbitrage opportunity. Hence, the growth rate of our delta-hedged portfolio must be equal to the continuously compounding risk-free rate, $r$. Thus, we are able to state that:

$$\begin{array}{r}\frac{\partial C}{\partial t}(S,t)+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}C}{\partial S^2}(S,t)=r(C-S\frac{\partial C}{\partial S})\end{array}$$

If we rearrange this equation, and using shorthand notation to drop the dependence on (S,t) we arrive at the famous Black-Scholes equation for the value of our contingent claim:

$$\begin{array}{r}\frac{\partial C}{\partial t}+rS\frac{\partial C}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}C}{\partial S^2}-rC=0\end{array}$$

Although we have derived the equation, we do not yet possess enough conditions in order to provide a unique solution. The equation is a second-order linear partial differential equation (PDE) and without boundary conditions (such as a payoff function for our contingent claim), we will not be able to solve it.

One payoff function we can use is that of a European call option struck at K. This has a payoff function at expiry, T, of:

$$\begin{array}{r}C(S,T)=max(S-K,0)\end{array}$$

We are now in a position to solve the Black-Scholes equation.

Reference: Deriving the Black-Scholes Equation

From https://www.quantstart.com/articles/Deriving-the-Black-Scholes-Equation/