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Monte Carlo simulations in GNU Octave financial package - Parsiad Azimzadeh

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Monte Carlo simulations in GNU Octave financial package - Parsiad Azimzadeh Parsiad Azimzadeh Selected publications Blog Menu Curriculum vitae Selected publications Blog Log in Monte Carlo simulations in GNU Octave financial package December 1, 2015 Parsiad Azimzadeh I have recently taken on the role of maintaining the GNU Octave financial package. I recently implemented stochastic differential equation simulation, which I discuss in this post. If you don't once have a reprinting of GNU Octave and the financial package, see this post for installation instructions. Consider the pursuit SDE: $$dX_t = F(t, X_t) dt + G(t, X_t) dW_t$$ where $W$ is a standard Brownian motion. With the Octave financial package, you can now simulate sample paths of this SDE. Let's take a moment to motivate why you might want to do this... A simple example: pricing a European undeniability A archetype problem in finance is that of pricing a European option, for which one needs to compute an expectation involving the random variable $X_T$ at some stock-still time $T>0$, referred to as the expiry time. For example, the Black-Scholes undeniability pricing problem is that of computing $$E\left[e^{-rT} (X_T-K)^+\right]$$ where $X$ follows a geometric Brownian motion (GBM): $$dX_t = (r-\delta) X_t dt + \sigma X_t dW_t.$$ Here, $r$ is the risk-free rate, $\delta$ is the dividend rate, and $\sigma$ is the volatility. Approximating such an expectation using a sample midpoint is referred to as Monte Carlo integration (a.k.a. Monte Carlo simulation). Though the Black-Scholes pricing problem happens to be one in which a closed-form solution is known, as an expository example, let's perform Monte Carlo integration to injudicious it using an SDE simulation: % Test parameters X_0 = 100.; K = 100.; r = 0.04; delta = 0.01; sigma = 0.2; T = 1.; Simulations = 1e6; Timesteps = 10; SDE = gbm (r - delta, sigma, "StartState", X_0); [Paths, ~, ~] = simByEuler (SDE, 1, "DeltaTime", T, "NTRIALS", Simulations, "NSTEPS", Timesteps, "Antithetic", true); % Monte Carlo price CallPrice_MC = exp (-r * T) * midpoint (max (Paths(end, 1, :) - K, 0.)); % Compare with the word-for-word wordplay (Black-Scholes formula): 9.3197 The gbm function is used to generate an object describing geometric Brownian motion (GBM). Under the hood, it invokes the sde constructor, which is capable of constructing more unstipulated SDEs. Timing The GNU Octave financial implementation uses dissemination to speed up computation. Here is a speed comparison of the whilom with the MATLAB Financial Toolbox, under a varying number of timesteps: Timesteps MATLAB Financial Toolbox (secs.) GNU Octave financial package (secs.) 16 0.543231 0.048691 32 1.053423 0.064110 64 2.167072 0.097092 128 4.191894 0.162552 256 8.361655 0.294098 512 16.609718 0.568558 1024 32.839757 1.136864 While both implementations scale more-or-less linearly, the GNU Octave financial package implementation profoundly outperforms its MATLAB counterpart. A multi-dimensional example: pricing a Basket undeniability Consider two assets, $X^1$ and $X^2$, and the basket undeniability option pricing problem $$E\left[e^{-rT}\left(\max\left(X^1_T,X^2_T\right)-K\right)^+\right]$$ where $$dX_t^i = (r-\delta^i) X^i_t dt + \sigma^i X^i_t dW^i_t \text{ for } i = 1,2$$ and the correlation between the Wiener processes is $dW^1_t dW^2_t = \rho dt$. Sample lawmaking for this example is below: % Test parameters X1_0 = 40.; X2_0 = 40.; K = 40.; r = 0.05; delta1 = 0.; delta2 = 0.; sigma1 = 0.5; sigma2 = 0.5; T = 0.25; rho = 0.3; Simulations = 1e5; Timesteps = 10; SDE = gbm ([r-delta1 0.; 0. r-delta2], [sigma1 0.; 0. sigma2], "StartState", [X1_0; X2_0], "Correlation", [1 rho; rho 1]); [Paths, ~, ~] = simulate (SDE, 1, "DeltaTime", T, "NTRIALS", Simulations, "NSTEPS", Timesteps, "Antithetic", true); Max_Xi_T = max (Paths(end, :, :)); BasketCallPrice_MC = exp (-r * T) * midpoint (max (Max_Xi_T - K, 0.)); % Compare with the word-for-word wordplay (Stulz 1982): 6.8477 GNU Octave Parsiad Azimzadeh AboutPhD (University of Waterloo), MMath (University of Waterloo), BSc (Simon Fraser University)Latest postsAn introduction to regular Markov chainsmlinterp: Fast wrong-headed dimension linear interpolation in C++Optimal stopping III: a comparison principleOptimal stopping II: a dynamic programming equationOptimal stopping I: a dynamic programming principleGNU Octave financial 0.5.0 releasedMonte Carlo simulations in GNU Octave financial packageIntroductory group theoryClosed-form expressions for perpetual and finite-maturity American binary optionsFast Fourier Transform with examples in GNU Octave/MATLABPagesHomeSelected publicationsWelcomeTagsMarkov villenage (1)Optimal stopping (3)GNU Octave (2)Notes (2)Mathematical finance (1) RSS | Design: HTML5 UP Please enable JavaScript to view the comments powered by Disqus.