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

• $t$ is a time in years; we generally use $t=0$ 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} ;
• 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 \sigma} is the volatility — the standard deviation of the asset's returns;
• 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 r} is the annualized risk-free interest rate, continuously compounded;
• 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 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, 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} , the payoff is given by 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) = 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 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 given by the Black-Scholes formula for the European call:

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 C(S_t, t) = e^{-r\tau} [F_t N(d_1) - K N(d_2)] }

where 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 \tau = T - t} is the time to maturity, 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 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

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_2 = d_1 - \sigma\sqrt{\tau}. }

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

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

Similarly, if the payoff is given by 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) = 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 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 given by the Black-Scholes formula for the European put:

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 P(S_t, t) = e^{-r\tau} [K N(-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 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 N(x)} .

Task 1: Implementing the standard normal cumulative distribution function

As mentioned in the Handbook of Mathematical Functions, 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 N(x)} can be approximated by

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 \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], & x \geq 0, \\ 1 - N(-x), & x < 0, \end{array} \right. }

where

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 \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.}

Can you implement this function in q?

First, we need

pi:acos -1;

One (terse) implementation would be

normal_cdf:{abs(x>0)-(exp[-.5*x*x]%sqrt 2*pi)*t*.31938153+t*-.356563782+t*1.781477937+t*-1.821255978+1.330274429*t:1%1+.2316419*abs x};

Task 2: Implement the Black-Scholes formula for the European call and put

Equipped with our implementation of normal_cdf, can you implement the Black-Scholes formula for the European call?

First, we need

compute_d1:{[S;K;r;q;sigma;tau](log[S%K]+((r-q)+.5*sigma*sigma)*tau)%sigma*sqrt tau};

Then,

call_price:{[S;K;r;q;sigma;tau]
  d1:compute_d1[S;K;r;q;sigma;tau];
  d2:d1-sigma*sqrt tau;
  F:S*exp tau*r-q;
  (exp neg r*tau)*(F*normal_cdf d1)-K*normal_cdf d2};

and

put_price:{[S;K;r;q;sigma;tau]
  d1:compute_d1[S;K;r;q;sigma;tau];
  d2:d1-sigma*sqrt tau;
  F:S*exp tau*r-q;
  (exp neg r*tau)*(K*normal_cdf neg d2)-F*normal_cdf neg d1};

We can test these implementations on a few sets of parameters, for example

call_price[100f;105f;.05;.07;.1;.5]

gives 0.7991363, whereas

put_price[100f;105f;.05;.07;.1;.5]

gives 6.646135.

Background: a simple Monte Carlo model

The Black-Scholes equation may not have analytic solutions for all derivatives that we are interested in. However, it may still be possible to solve it numerically using Monte Carlo methods.

The model for stock price evolution is

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 dS_t = \mu S_t \, dt + \sigma S_t \, dW_t, }

where 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 \mu} is the drift. The Black-Scholes pricing theory then tells us that the price of a vanilla option with pay-off 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 f} is equal to

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 e^{-r\tau}\mathbb{E}[f(S_T)], }

where the expectation is taken under the associated risk-neutral process,

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 dS_t = (r - q) S_t \, dt + \sigma S_t \, dW_t. }

We solve this equation by passing to the log and using Ito's lemma; we compute

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 \ln S_t = \left(r - q - \frac{1}{2}\sigma^2\right) \, dt + \sigma \, dW_t. }

As this process is constant-coefficient, it has the solution

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 \ln S_t = \ln S_0 + \left(r - q - \frac{q}{2}\sigma^2\right) t + \sigma W_t. }

Since 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 W_t} is a Brownian motion, 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 W_T} is distributed as a Gaussian with mean zero and variance 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} , so we can write

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 W_T = \sqrt{T} N(0, 1), }

and hence

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 \ln S_T = \ln S_0 + \left(r - q - \frac{q}{2}\sigma^2\right) T + \sigma \sqrt{T} N(0, 1), }