1 Black-Scholes Formula
References: Quantpie Video Series.
The Black-Scholes formula provides the theoretical price of a European call or put option under the Black-Scholes-Merton Model, assuming no dividends and a constant volatility environment.
In the Black-Scholes model setting the price of a stock \((S_{t})\) is assumed to follow a Geometric Brownian Motion with constant drift and diffusion terms \(\mu,\sigma\) and the interest rate \(r\) that is constant \[ \begin{cases} dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t} \\ dB_{t}=rB_{t}dt. \end{cases} \] The payoff function for a European call option given some maturity \(T\) and some strike price \(K\) is given by \(V(T-t, S_{t})=V_{t}\) which itself is some stochastic process and a function of \(t\) and \(S_{t}\). From Itô Formula we have that \[ \begin{align} dV_{t} & = \frac{\partial V_{t}}{\partial t}dt + \frac{\partial V_{t}}{\partial S_{t}}dS_{t} + \frac{1}{2} \frac{\partial ^2V_{t}}{\partial S_{t}^2} d\left< S,S \right> _{t} \\ & =\frac{\partial V_{t}}{\partial t}dt + \frac{{\partial V_{t}}}{\partial S_{t}}(\mu S_{t}+\sigma S_{t}dW_{t})+ \frac{1}{2} \frac{{\partial ^2 V_{t}}}{\partial S^2_{t}}\sigma^2S_{t}^2dt \\ & = \left( \frac{{\partial V_{t}}}{\partial t}+\mu S_{t} \frac{{\partial V_{t}}}{\partial S_{t}}+ \frac{1}{2} \sigma^2 S_{t}^2 \frac{{\partial^2 V_{t}}}{\partial S_{t}^2} \right)dt + \sigma S_{t} \frac{{\partial V_{t}}}{\partial S_{t}}dW_{t}, \end{align} \] where we have grouped the drift and diffusion terms respectively. The idea of Black-Scholes is to use a [[delta-hedging]] strategy to eliminate the stochastic element (i.e. trading in the underlying stock to eliminate the risk of the option).
Assume we are hedging a short position in a call option, so our strategy is to buy stocks assuming we have unlimited money in the form of bank loans / surplus subject to the interest rate. Lets suppose we bought \(\Delta\) units of stock and borrowed \(\alpha\) units of cash. Our portfolio \(\Pi\) is summarized by the stochastic process \[ \Pi_{t} = \Delta_{t} S_{t}+\alpha_{t} B_{t} \] Again applying Itô Formula we obtain \[ \begin{align} d\Pi & = \Delta dS+\alpha dB \\ & =\Delta(\mu Sdt + \sigma SdW)+\alpha rBdt \\ & =(\Delta \mu S + \alpha rB) dt + \Delta \sigma SdW. \end{align} \] Note: For this step we have assumed that the \(\Delta, \alpha\) are deterministic i.e. that the portfolio is self-financing where the change of value of the portfolio is down to only the change of value of the assets within without which the result after applying Itô’s Lemma would be \[ d\Pi = (\Delta dS + Sd\Delta + dS d\Delta) + (\alpha dB + Bd\alpha). \]
Now our total portfolio is given by our option plus of hedging stock and cash holdings. Our aim is to choose \(\Delta\) such that the coefficient of the stochastic term becomes zero, i.e. \[ \Delta \sigma S + \sigma S \frac{{\partial V}}{\partial S}=0 \implies \Delta = - \frac{{\partial V}}{\partial S}. \] Substituting in our result we obtain \[ \begin{align} dV + d\Pi & = \left( \frac{{\partial V}}{\partial t}+ \mu S \frac{{\partial V}}{\partial S}+ \frac{1}{2}\sigma^2 S^2 \frac{{\partial^2}V}{\partial S^2}- \frac{{\partial V}}{\partial S}\mu S + \alpha r B \right) dt \\ d(V + \Pi) & = \left( \frac{{\partial V}}{\partial t}+ \frac{1}{2}\sigma^2S^2 \frac{{\partial ^2 V}}{\partial S^2}+\alpha rB \right) dt. \end{align} \] Now substituting our \(\Delta\) into our expression for \(\Pi\) we have \[ \Pi = -\frac{{\partial V}}{\partial S}S+\alpha B. \] The combined portfolio is deterministic and therefore must grow at the risk free rate to avoid arbitrage, hence \[ \begin{align} d(V + \Pi) & = (V + \Pi)r dt \\ & =\left( V- \frac{{\partial V}}{\partial S}S + \alpha B \right)r dt \\ \implies \frac{{\partial V}}{\partial t}+ \frac{1}{2}\sigma^2 S^2 \frac{{\partial^2V}}{\partial S^2}+\alpha rB & =rV-r \frac{{\partial V}}{\partial S}S + \alpha rB \\ \frac{{\partial V}}{\partial t}+ \frac{1}{2}\sigma^2S^2 \frac{{\partial ^2V}}{\partial S^2}+ rS \frac{{\partial V}}{\partial S}-rV & =0, \end{align} \] the Black Scholes PDE, which we specifically define below.
Consider a derivative on a stock paying no dividends that has a fixed maturity \(T\) and payoff \(K(S_T)\), then the price of the derivative satisfies \[ \begin{cases} \frac{\partial V}{\partial t}+ \frac{1}{2}\sigma^2S^2 \frac{{\partial^2V}}{\partial S^2}+rS\frac{{\partial V}}{\partial S}-rV=0 \\ V(T,s)=K(s)\quad\forall s\end{cases} \] where \(V(t,S)\) is the price of the option as a function of stock price \(S\) and time \(t\), \(r\) is the risk-free interest rate and \(\sigma\) is the stock volatility.
Although the BS PDE looks complicated we can provide the following intuitive breakdown. Firstly, rewriting the expression as \[ \frac{{\partial V}}{\partial t}+ \frac{1}{2}\sigma^2S^2 \frac{{\partial ^2V}}{\partial S^2} = rV - rS \frac{{\partial V}}{\partial S}=r\left( V - S \frac{{\partial V}}{\partial S} \right), \] we see that the RHS represents the return on the bank account which is equal to the option premium minus the amount borrowed to finance the stock units, which in an infinitesimal period is equal to the balance times the interest rate. The LHS essentially represents how the value of the delta-hedged option changes in an infinitesimal period where the first term captures the impact of the shortening maturity, and the second term captures the gamma impact, the remaining risk after the delta is hedged. From the world of probability this is essentially the [[backwards-diffusion-equation]].