Geometric Brownian Motion

From ThetaWiki

Jump to: navigation, search

Contents

[edit] Overview

Geometric Brownian motion is the underlying assumption for Black-Scholes Option Pricing. It dictates that future stock prices are not predictable and evolve as

dSt = μStdt + σStdWt

where μ is the drift rate, σ the volatility of S and dWt the increment of a Wiener process.

[edit] Process simulation

[edit] Implementation as Theta.m

[edit] 1-Dimensional 

function state = Theta(dt, state)
 
if nargin == 0        
    state.S.comment     = 'Stock price';
    state.EUR.comment   = 'Numeraire';
    state.sigma.comment = 'Volatility';     
    state.r.comment     = 'Risk-free rate';
else
    % risk-free rate
    r = state.r(1);
 
    % excess return
    mu = r; 
 
    % volatility
    sigma = state.sigma(1);
 
    state.S   = state.S .* exp( (mu-0.5*sigma^2)*dt + sqrt(dt)*sigma*randn(size(state.S)));
    state.EUR = state.EUR* exp(-r*dt);
end

[edit] N-Dimensional 

function state = Theta(dt, state)
 
if nargin == 0
 
     state.S.comment    = 'Stock price';
     state.EUR.comment  = 'Numeraire';
     state.rho.comment  = 'Correlation';
     state.sigma.comment= 'Volatility';
 
else
    r = 0.05;
    mu = r;
    rho=mean(state.rho);
    sigma = ones(1,size(state.S,2))*mean(state.sigma);
 
    Z=zeros(length(sigma),length(sigma));
    for i=1:length(sigma)
        for k=1:length(sigma)
            if i==k
                Z(i,i)=sigma(i)^2;
            else
                Z(i,k)=rho*sigma(i)*sigma(k);
            end
        end
    end
 
    A=(chol(Z))';
    B=randn(size(state.S'));
    for i = 1:length(sigma)
       state.S(:,i)  = state.S(:,i) .* exp( (mu-0.5*sigma(i)^2)*dt + sqrt(dt)*(A(i,:)*B)');
    end
    state.EUR = state.EUR*exp(-r*dt);
 
 end
 end

[edit] Implementation in ThetaScript 

Model BS
	%Black Scholes model
	import So 	"Initial Stock Price"
	import EURo "Initial value of Numeraire"
	import r 	"Risk free rate"
	import vola "volatility"
 
	export S  	"Stock Price"
	export EUR 	"Numeraire"
 
	S=So
	EUR=EURo
	fork
		loop inf
		theta $dt
		S=S*exp(-0.5*vola^2*$dt+vola*sqrt($dt)*randn())
		EUR=EUR*exp(-r*$dt)
		end
	end
end

[edit] Analytic pricing

The price of European puts and calls can be obtained by calling theta_bls() 

function result = theta_bls(S, K, sigma, r, T, call, greek)
%function result = theta_bls(S, K, sigma, r, T, call, greek)
%
% returns option characteristics European Put and Call options with
%
% S : asset price
% K : strike
% sigma: volatility
% r : risk-free rate
% T : maturity time
% call: true for call, otherwise put
% greek: "value", "delta" or "gamma"
 
if T<=0
    if call
        V = max(S-K,0);
    else
        V = max(K-S,0);
    end
    return
end
 
d1 = ( log(S./K) + (r + (sigma.^2)/2)*T ) ./ (sigma * sqrt(T));
d2 = d1 - sigma*sqrt(T);
 
switch call
    case true
        switch greek
            case 'value'
                result = S.*normcdf(d1,0,1) - K .* exp(-r.*T).*normcdf(d2);
            case 'delta'
                result = normcdf(d1,0,1)';
            case 'gamma'
                result = (normpdf(d1,0,1)./(S.*sigma.*sqrt(T)))';
        end
    otherwise
        switch greek
            case 'value'
                result = -S.*normcdf(-d1,0,1) + K .* exp(-r.*T).*normcdf(-d2);
            case 'delta'
                result = normcdf(d1,0,1)'-1;
            case 'gamma'
                result = (normpdf(d1,0,1)./(S.*sigma.*sqrt(T)))';
        end
end
Retrieved from "http://wiki.thetaris.com/index.php/Geometric_Brownian_Motion"
Personal tools