Definition

Black-Scholes PDE for call-option on non-dividend paying stock KK with strike KK and maturity TT

rSt∂C∂S+∂C∂t+12σ2S2∂2C∂S2−rC=0rS_t \frac{\partial C}{\partial S} + \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} -rC = 0

Derivation

Assume the stock follows geometric Brownian motion, such that

dSt=μStdt+σStdWtdS_t = \mu S_t \, dt + \sigma S_t \, dW_t

where WtW_t is standard Brownian motion, with constants μ\mu (percentage drift) and σ\sigma (percentage volatility).

Assume constant interest rate such that 11 unit of currency will be worth Bt:=exp⁡(rt)B_t := \exp(rt) at time tt. For value of call option C(S,t)C(S,t) at time tt, by Itô's lemma, #incomplete

References

  1. https://www.columbia.edu/~mh2078/FoundationsFE/BlackScholes.pdf
  2. https://tobydriscoll.net/fnc-julia/diffusion/blackscholes.html