Programming/Kdb/Labs/Option pricing

Background: the Black-Scholes formulae

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 $V(S,T)=C(S,T)=:\max\{S-K,0\}$, 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 for the European call:

$C(S_{t},t)=e^{-r\tau }[F_{t}N(d_{1})-KN(d_{2})]$

where $\tau =T-t$ is the time to maturity, $F=S_{t}e^{(r-q)\tau }$ is the forward price, and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_1 = \frac{1}{\sigma\sqrt{\tau}} \left[ \ln\left(\frac{S_t}{K}\right) + \left(r - q + \frac{1}{2}\sigma^2\right)\tau \right] }

and

$d_{2}=d_{1}-\sigma {\sqrt {\tau }}.$

Here we have used $N(x)$ to denote the standard normal cumulative distribution function,

$N(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-z^{2}/2}\,dz.$

Similarly, if the payoff is given by $V(S,T)=P(S,T)=:\max\{K-S,0\}$, in other words, the option is a European put option, then the value of the option at time $t$ is given by the Black-Scholes formula for the European put:

$P(S_{t},t)=e^{-r\tau }[KN(-d_{2})-F_{t}N(-d_{1})].$

We will implement the formulae for the European call and European put in q. However, our first task is to implement $N(x)$.

Task 1: Implementing the standard normal cumulative distribution function

$N(x)$ can be approximated by

$\left\{{\begin{array}{ll}1-\phi (x)\left[c_{1}k+c_{2}k^{2}+c_{3}k^{3}+c_{4}k^{4}+c_{5}k^{5}\right],&{\hbox{if,}}\\C,&{\hbox{D,}}\end{array}}\right.$ Failed to parse (unknown function "\begin{array}"): {\displaystyle \left\{ \begin{array}{ll} 1 - \phi(x) \left[ c_1 k + c_2 k^2 + c_3 k^3 + c_4 k^4 + c_5 k^5 \right], & \hbox{x \geq 0,} \\ 1 - N(-x), & \hbox{x < 0,} \end{array} \right. }

where

$\phi (x)=\exp(-x^{2}/2)/{\sqrt {2\pi }},$

$k=1/(1+0.2316419x),c_{1}=0.319381530,c_{2}=-0.356563782,c_{3}=1.781477937,c_{4}=-1.821255978,c_{5}=1.330274429$.