To derive the Black-Scholes formula, we’ll first apply Ito’s Lemma to the Geometric Brownian Motion formula. Then, we’ll use the risk-neutral pricing approach to finally arrive at the Black-Scholes equation.
The Geometric Brownian Motion (GBM) formula for stock price S under the risk-neutral measure is given as:
dSt = rStdt + σStdWt
where:
- St is the stock price at time t
- r is the risk-free interest rate
- σ is the stock price’s volatility
- Wt is a standard Brownian motion under the risk-neutral measure
Let f(t, S) be a twice-differentiable function w.r.t t and S, Ito’s Lemma states that:
$$df(t, S) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial S} dS + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} dS^2$$
Now let’s consider a European call option C = C(t, S) with maturity T and strike price K. We will try to find an f(S, t) = C(S, t). Using Ito’s Lemma and plugging in our GBM formula, for dS we have:
$$dC = \frac{\partial C}{\partial t} dt + \frac{\partial C}{\partial S} dS + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} dS^2$$
$$dC = \frac{\partial C}{\partial t} dt + \frac{\partial C}{\partial S}(rSdt + \sigma SdW) + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} (\sigma^2 S^2 dW^2)$$
Using the fact that dWt2 = dt, we have:
$$dC = \left( \frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \right) dt + \sigma S \frac{\partial C}{\partial S_t} dW_t$$
Now, using the risk-neutral pricing for the European call option:
C(t, S) = e − r(T − t)𝔼t[max(ST−K,0)]
Under the risk-neutral measure, the payoff is discounted at the risk-free rate:
$$\frac{dC(t,S)}{dt} - rC(t, S) = \sigma S \frac{\partial C}{\partial S_t} dW_t$$
Comparing the two equations above (LHS and RHS), we have
$$\frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} -rC(t, S) = 0$$
This is the famous **Black-Scholes partial differential equation (PDE)**. Next, we need to solve this PDE with the boundary conditions for a European call option:
C(T, S) = max(S − K, 0)
The solution to this PDE with the given boundary condition is the Black-Scholes formula for a European call option:
C(t, S) = SN(d1) − Ke − r(T − t)N(d2)
where:
$$d_1 = \frac{\ln{\frac{S}{K}} + \left(r + \frac{\sigma^2}{2}\right)(T-t)}{\sigma\sqrt{T-t}}$$
$$d_2 = \frac{\ln{\frac{S}{K}} + \left(r - \frac{\sigma^2}{2}\right)(T-t)}{\sigma\sqrt{T-t}} = d1 - \sigma\sqrt{T-t}$$
and N(x) is the cumulative distribution function of the standard normal distribution.