Programming/Kdb/Labs/Option pricing

Recall the celebrated Black-Scholes equation

${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+(r-q)S{\frac {\partial V}{\partial S}}-rV=0.$

Here

• $t$ is a time in years; we generally use $t=0$ as now;
• $V(S,t)$ is the value of the option;
• $S(t)$ is the price of the underlying asset at time $t$;
• $\sigma$ is the volatility — the standard deviation of the asset's returns;
• $r$ is the annualized risk-free interest rate, continuously compounded;
• $q$ is the annualized (continuous) dividend yield.

The solution of this equation depends on the payoff of the option — the terminal condition. In particular, if at the time of expiration, $T$, the payoff is given by $V(S,T)=\max\{S-K,0\}$, in other words, the option is a European call option, then the value of the option at time $t$ is given by the Black-Scholes formula