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 a SDE:

\begin{eqnarray} 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)dS^2 \end{eqnarray}

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{eqnarray*} dS(t) = \mu S(t)dt + \sigma S(t) dX(t) \end{eqnarray*}

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

\begin{eqnarray*} dC = \left(\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)\right)dt + \sigma S \frac{\partial C}{\partial S} (S,t) dX \end{eqnarray*}

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 in 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 $\Delta$. Hence:

\begin{eqnarray*} d(C+\Delta S) = \left(\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) + \Delta \mu S\right) dt + \Delta S \left(\frac{\partial C}{\partial S}+\Delta\right) dX \end{eqnarray*}

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

\begin{eqnarray*} d(C+\Delta S) = \left(\frac{\partial C}{\partial t}(S,t) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}(S,t)\right)dt \end{eqnarray*}

Note that we have glossed over the issue of what the derivative of $\Delta$ 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 the continuously compounding risk free rate, $r$. Thus we are able to state that:

\begin{eqnarray*} \frac{\partial C}{\partial t}(S,t) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}(S,t) = r\left(C-S\frac{\partial C}{\partial S}\right) \end{eqnarray*}

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{eqnarray*} \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{eqnarray*}

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{eqnarray*} C(S,T)=\max(S-K,0) \end{eqnarray*}

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

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