Deriving a PDE For Call Option Prices with Black-Scholes

Brownian Motion

The building block of stochastic calculus is Brownian motion, or Weiner process. We take a stochastic process starting at zero that is then defined by independent, Gaussian increments:

$$ \begin{align*} B_0 &= 0 \\ B_t - B_s &\sim N(0, t - s), \end{align*} $$

or in shorthand, $dB_t \sim N(0, \delta t)$, where $d B_t$ is the increment and $\delta t$ is the time interval. We care about $\delta t \rightarrow 0$. We note that the differences $dB_t$ are all independent if they don’t overlap.

Quadratic Variation

Given a function $f$ we define the quadratic variation to be $(d f\; d f) = \lim_{\left|\Pi\right| \to 0} \sum_{j = 0}^{n - 1}(f(t_{j + 1}) - f(t_j))^2,$ where $\Pi$ is the sum of the square distance between the intervals on the time grid $t_j$.

If a function is differentiable, the mean value theorem implies that these differences go to zero as the intervals $[t_j, t_{j + 1}]$ go to zero. This is however not the case for Brownian motion. Indeed, Brownian motion is continuous but not differentiable (we will not prove this here).

Lemma (Quadratic Variation of Brownian Motion). $d B_t \; dB_t = dt$.

Proof. Omitted.

Covariance

We will now pause to consider the covariance of Brownian motion. We can compute (using our definition above)

$$ \begin{align*} \operatorname{Cov} (B_t, B_s) &= \mathbb{E}[B_t B_s] \\ &= \mathbb{E}[B_t (B_s - B_t) + B_t^2]\\ &= \mathbb{E}[B_t^2] = \min(t, s). \end{align*} $$

A Stochastic Differential Equation

We want to write down a stochastic differential equation which will define the increments of a stochastic process $S_t$. We will do this in the form of a drift-diffusion model

$$ dS_t = \mu(S_t, t) dt + \sigma(S_t, t) dW_t, $$

where $\mu$ and $\sigma$ are arbitrary (non-stochastic) functions defining the drift and volatility.

The Black-Scholes equation is

$$ d S_t = r s_t dt + \sigma S_t dW_t. $$

We wil find this yields asset prices $S_t$ that are log-normally distributed.

Ito’s Lemma

We want to ‘taylor expand’ a stochastic process by using a stochastic differential equation. Suppose we have an SDE of the form above,

$$ dS_t = \mu(S_t, t) dt + \sigma(S_t, t) dW_t, $$

We want to ignore ‘small’ terms. Normally we would ignore anything smaller than $O(dt)$ and $O(S_t)$, but quadratic variation implies the $d W_t^2$ term will be order $O(d_t)$ and can’t be ignored. So doing this gives us Ito’s Lemma:

$$ df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial S_t} d S_t + \frac{1}{2} \frac{\partial^2 f}{\partial S_t^2} dS_t^2. $$

For the drift-diffusion model above we have

$$ df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial S_t} \mu(S_t, t) d t + \frac{\partial f}{\partial S_t} \sigma(S_t, t) d W_t + \frac{1}{2} \frac{\partial^2 f}{\partial S_t^2} \sigma(S_t, t)^2 dt. $$

Pricing a Call Option

Now we can put together all of the building blocks above to come up with the price of a call option. We have

$$ dS_t = r S_t dt + \sigma S_t d W_t. $$

We want to compute the price of the call option with a strike $K$, that is, one whose terminal value is

$$ c(S_T, T) = \max(S_T - K, 0). $$

We can model this as a stochastic process $c(S_t, t)$ which has the above boundary condition and evolves according to the Black-Scholes equation above.

We want to write down a SDE for $c$. Applying Ito’s Lemma gives us

$$ dc = \left(\frac{\partial c}{\partial t} + \frac{\partial c}{\partial S_t}\mu S_t + \frac{1}{2} \frac{\partial^2 c}{\partial S_t^2} \sigma^2 S_t^2 \right) dt + \frac{\partial c}{\partial S_t} \sigma S_t d W_t $$

Note that this implies that if we hold the portfolio $\Pi(S_t, t) = c(S_t, t) - \frac{\partial c}{\partial S_t} S_t$, the portfolios is left with no exposure to the ‘randomness’ (Brownian motion) term, and so our portfolio is riskless\footnote{This, you may have guessed, is a delta hedged portfolio.}.

This portfolio evolves according to (after expanding)

$$ \begin{align*} d \Pi(S_t, t) = \left(\frac{\partial c}{\partial t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 c}{\partial s^2}\right) dt \end{align*} $$

Now we already know we have a risk free portfolio, and so it must evolve according to the risk free rate $r$ by

$$ \begin{align*} d \Pi(S_t, t) &= r \Pi dt \\ &= r (c(S_t, t) - \frac{\partial c}{\partial s} S_t) dt \end{align*} $$

These models must give us the same result ultimately, and equating them gives

$$ r c(S_t, t) - r\frac{\partial c}{\partial s} S_t = \frac{\partial c}{\partial t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 c}{\partial s^2} $$

This is a PDE describing the price of a call option.