Black-Scholes Equation

This section requires some understanding of stochastic calculus. The course on stochastic calculus will give you the background you need. For the lognormal model, the underlying asset is a geometric Brownian motion $$ \text{d}S_t = \mu S_t \text{d}t + \sigma S_t \text{d}W_t. $$ We want to price a call option for this process. The call option is a function only of the stock price, $S_t$, and the time to expiry, $T$. It also has a parameter for the strike price, $K$. Let's apply Ito's Lemma to the call price $$ \text{d}C = \frac{\partial C}{\partial t} \text{d}t + \frac{\partial C}{\partial S} \text{d}S + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} \text{d}S^2. $$ The original derivation of Black-Scholes uses a hedging argument, which we will go through here. If we substitute d$S$ into the above equation, we get $$ \text{d}C = \left( \frac{\partial C}{\partial t} + \mu S \frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) \text{d}t + \sigma S \text{d}W_t. $$ Now, let's consider a portfolio $$ \Pi = -C + \int \frac{\partial C}{\partial S} \text{d}S. $$ Here, we are short one call option and long some quantity of shares. In the interval $[t, t + \delta t]$, consider the change in the value of the portfolio $$ \Delta \Pi = - \Delta C + \frac{\partial C}{\partial S} \Delta S. $$ We can obtain a value for $\Delta C$ by discretising the stochastic differential equation $$ \Delta C = \left( \mu S \frac{\partial C}{\partial S} + \frac{\partial C}{ \partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) \Delta t + \sigma S \frac{\partial C}{\partial S} \Delta W_t. $$ Substituting this back into the portfolio gives $$ \Delta \Pi = - \left[ \left( \mu S \frac{\partial C}{\partial S} + \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) \Delta t + \sigma S \frac{\partial C}{\partial S} \Delta W_t \right] + \frac{\partial C}{\partial S} \left[ \mu S \Delta t + \sigma S \Delta W_t \right] $$ $$ \Delta \Pi = - \left( \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) \Delta t $$ Since the portfolio is fully hedged, it can only grow at the risk-free rate. $$ \Delta \Pi = r \Pi \Delta t $$ $$ -\left( \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) \Delta t = r \Pi \Delta t $$ $$ -\left( \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) \Delta t = r \left( -C + \int \frac{\partial C}{\partial S} \text{d}S \right) \Delta t $$ $$ -\left( \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) \Delta t = r \left( -C + \frac{\partial C}{\partial S} S \right) \Delta t $$ If we now collect all terms on one side, we get $$ \left( \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS \frac{\partial C}{\partial S} - rC \right) \Delta t = 0 $$ and dividing by $\Delta t$ gives us the Black-Scholes equation $$ \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS \frac{\partial C}{\partial S} - rC = 0. $$ Notice how the drift of the asset, $\mu$ does not appear in the Black-Scholes equation. Only the risk-free rate remains. As the drift of the asset disappeared in our working, we see that it is not relevant to determine the price of an option. This is a very important fact to remember. The drift of an asset does not affect it's option price, only the risk-free rate does. In the next section, we will solve this equation and get an analytic formula for pricing options.

Assumptions of Black-Scholes

There are six key assumptions to the Black-Scholes model.
European options: the Black-Scholes model is specifically designed to price European options, which can only be exercised at the option expiry.
Underlying asset model: the model assumes that the underlying asset obeys the lognormal model.
No dividends: the derivation assumes that no dividends of the underlying asset will be paid during the lifetime of the option. Extensions of Black-Scholes do exist which incorporate dividends.
Constant parameters: the risk-free rate and volatility is assumed to be constant. Due to the fact that you can sum the variance for uncorrelated random variables, time-dependent volatility can be converted to a constant parameter by taking it's historical root mean square.
No transaction costs: the model assumes frictionless markets, meaning there are no transaction costs or taxes, allowing for continuous risk-free trading. This is necessary for the hedging condition in our derivation - that we can continuously buy or sell the underlying to maintain a delta-hedge.
Arbitrage-free markets: markets are assumed to be efficient, with prices reflecting all available information. There is no room for arbitrage opportunities (risk-free profit). This condition enables us to impose that the hedged portfolio must grow at the risk-free rate in our derivation.