## GNU Octave financial 0.5.0 released

I am happy to announce the release of the GNU Octave financial package version 0.5.0. This is the first version to be released since I took on the role of maintainer.

If you do not already have GNU Octave, you can grab a free copy here.

To install the package, launch Octave and run the following commands:

pkg install -forge io
pkg install -forge financial

Perhaps the most exciting addition in this version is the Monte Carlo simulation framework, which is significantly faster than its MATLAB counterpart. A brief tutorial (along with benchmarking information) are available in a previous post. Other additions include Black-Scholes options and greeks valuation routines, implied volatility calculations, and general bug fixes. Some useful links are below:

## Monte Carlo simulations in GNU Octave financial package

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 already have a copy of GNU Octave and the financial package, see this post for installation instructions.

Consider the following 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 call

A classic 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 fixed time $T>0$, referred to as the expiry time. For example, the Black-Scholes call 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 mean 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 approximate 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) * mean (max (Paths(end, 1, :) - K, 0.));

% Compare with the exact answer (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 general SDEs.

### Timing

The GNU Octave financial implementation uses broadcasting to speed up computation. Here is a speed comparison of the above 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 greatly outperforms its MATLAB counterpart.

### A multi-dimensional example: pricing a Basket call

Consider two assets, $X^1$ and $X^2$, and the basket call 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 code 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) * mean (max (Max_Xi_T - K, 0.));

% Compare with the exact answer (Stulz 1982): 6.8477