Programming/Kdb/Labs/Option pricing

Background: the Black-Scholes formulae

Recall the celebrated Black-Scholes equation

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 \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} - r V = 0. }

Here

• 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 t} is a time in years; we generally use 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 t = 0} as now;
• 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 V(S, t)} is the value of the option;
• 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 S(t)} is the price of the underlying asset at time 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 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

$d_{1}={\frac {1}{\sigma {\sqrt {\tau }}}}\left[\ln \left({\frac {S_{t}}{K}}\right)+\left(r-q+{\frac {1}{2}}\sigma ^{2}\right)\tau \right]$

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left\{{\begin{array}{ll}A,&{\hbox{B,}}\\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{if $x \geq 0$,} \\ ?1 - N(-x), & \hbox{if $x < 0$,} \end{array} \right. }

where

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

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 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} .