23 Stochastic Ordinary Differential Equations with Examples from Finance

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1 23 Stochastic Ordinary Differential Equations with Examples from Finance Scraping Financial Data from the Web The MATLAB/Octave yahoo function below returns daily open, high, low, close, and adjusted close price data, along with trading volume data for a specified financial instrument and date range. function A = yahoo(symbol, startdate, enddate) % YAHOO: Retrieve daily HLOC financial data from Yahoo! Finance. % A = yahoo(symbol, startdate, enddate) % symbol A string-valued financial instrument name. % startdate The starting date string in the form YYYY-MM-DD % enddate The ending date string in the form YYYY-MM-DD % The return value A is an nx9 numeric matrix, where n is the number of % days data returned, and the columns are: % year, month, day, open, high, low, close, volume, adjusted close a=sscanf(startdate, %f-%f-%f ); b=sscanf(enddate, %f-%f-%f ); request=strcat( &a=,num2str(a(2)), &b=,num2str(a(3)), &c=,num2str(a(1)),... &d=,num2str(b(2)), &e=,num2str(b(3)), &f=,num2str(b(1)),... &g=d&ignore=.csv ); x=urlread(request); y=x(43:end); n=length(regexp(y, \n )); A=sscanf(y, %f-%f-%f,%f,%f,%f,%f,%f,%f,[9,n]); A=flipud(A ); We can, for example, plot the daily adjusted closing price for Google s stock over all of 2009 with: goog=yahoo( GOOG, , )(:,9); plot(goog) Consider the simple initial value problem from Example 22.1: 0 Version April 24, 2014 d dt u(t) = λu(t), λ 0, T t 0, u(0) = u 0, 1

2 or, equivalently in differential form, du(t) = λu(t)dt. (1) This simple ODE models the value of an asset value u(t) as it appreciates over time with a continually compounded interest rate λ. There is no uncertainty in the initial value problem (1); the value of the asset can be computed precisely at any time t. The model is appropriate for financial instruments such as interest-bearing savings accounts and fixed-interest loans. But equation (1) is not good at modeling instruments with risk such as stock prices, as illustrated in the following example. Example 1 Model Google s stock price over 2009 with equation (1). goog=yahoo( GOOG, , )(:,9); n=length(goog); mu = log(goog(n)/goog(1)); t=linspace(0,1,n) ; u=goog(1)*exp(mu*t); plot(1:n,goog, -b,1:n,u, -r ) Although the elementary model derived from equation (1) captures the overall growth in the price of Google s stock over the period, it obviously lacks the many small up and down movements along the way. We will explore modifications to equation (1) that can be used to model a much wider class of problems that admit some level of uncertainty (risk). Brownian motion The botanist Robert Brown observed the erratic motion of pollen particles floating in water in He conducted experiments to rule out self-locomotion and described his results. Although similar observations had been made earlier in history, the random motion of particles is generally called Brownian motion. Some of the greatest thinkers of the 19th and 20th century worked out mathematical models to describe Brownian motion, including Thiele, Einstein, Wiener, and many others. One mathematical formulation of Brownian motion is given by the Wiener process. A Wiener process is a random variable W(t) that depends continuously on t 0 and satisfies the conditions: 1. W(0) = 0 2. W(t) is a continuous function of t. 3. For any 0 s < t, the Brownian increment W(t) W(s) is a normally-distributed random variable with zero mean and variance t s. 2

3 Figure 1: Google s stock price over 2009 (blue) and a simple ODE model (red). 4. Over non-overlapping intervals, the Brownian increments are mutually independent random variables. Computationally, we approximate realizations of W(t) over discrete time intervals by taking a step in a direction sampled from a normal distribution with mean 0 and variance one, scaled by the square root of the step length. The following MATLAB code illustrates this procedure: n=1000; % Number of time steps t=linspace(0,1,n) ; % Time t from 0 to 1. z=randn(n,1); % z contains n pseudorandom variables sampled % from N(0,1) W=(1/sqrt(n)) * cumsum(z); % W is an approximate realized sample path plot (h,b); % Plot the result 3

4 Figure 2: Ten sample paths of the standard Brownian motion process. Figure 2 displays ten sample paths of the standard Brownian motion process computed using the above method. The sample paths are extremely jagged, visually and mathematically. In fact, at no point is W(t) differentiable with respect to t. Louis Bachelier noticed around 1900 that sample paths of Brownian motion looked somewhat like stock prices, and he was the first to use formal ideas from mathematics to model financial markets. Stochastic Differential Equations and Financial Models Recall from equation (22.4) that the solution to the initial value problem (1) is equivalent to evaluating the integral T u(t) = u(0) + λu(t)dt. (2) 0 4

5 Define δt = T/N, for T > 0 and positive integer N, and let t j = jδt. The integral term in (2) can be defined by taking the limit as δt 0 of the Riemann sum N 1 j=0 λu(t j )(t j+1 t j ). Now consider a sum that includes a stochastic component involving the Wiener process W: N 1 j=0 σu(t j )(W(t j+1 ) W(t j )). Similar to the definition of the Riemann integral, the stochastic Itô integral may be formally defined by: T N 1 σu(t)dw(t) = lim σu(t j )(W(t j+1 ) W(t j )). (3) δt 0 0 j=0 The Itô integral (3) represents integration with respect to Brownian motion. We can use it to formulate a stochastic analog of the initial value problem (1): u(t) = u(0) + t 0 λu(s)ds + Expressing this relation in differential form yields t 0 σu(s)dw(s), 0 t T. du(t) = λu(t)dt + σu(t)dw(t), u(0) = u 0, W(0) = 0. (4) Equation (4) is a historically important example from finance. This stochastic differential equation (SDE) represents a modification of the initial value problem (1) that includes a diffusion component driven by Brownian motion. The parameter λ is often referred to as the drift coefficient and σ the diffusion coefficient. The celebrated Black Scholes partial differential equation for pricing options can be derived from (4). The analytic solution of (4) can be shown to be the geometric Brownian motion equation ( u(t) = u 0 exp (λ 1 ) 2 σ2 )t + σw(t). (5) Equation (5) can provide a more realistic model of stock price movements than equation (1). We revisit example (1) to illustrate this, using the sample variance of Google s stock price to derive an an estimate of σ: Example 2 Model Google s stock price over 2009 with equation (4). 5

6 goog = yahoo( GOOG, , )(:,9); n = length(goog); sigma = sqrt(var(log(goog))); lambda = log(goog(n)/goog(1)) + 0.5*sigma^2; t = linspace(0,1,n) ; z = randn(n,1); W = (1/sqrt(n)) * cumsum(z); u = goog(1)*exp((lambda-0.5*sigma^2)*t + sigma*w); plot(1:n,goog, -b,1:n,u, -r ) Figure 3: Google s stock price over 2009 (blue) and a model based on equation (4) (red). We see from figure (3) that the new model presents a more realistic-looking one than that of example (1). The new model is stochastic, thus repeated runs of example (2) will yield different 6

7 approximate solutions. Numerical Methods for SDEs Numerical solutions to equation (4) can be computed using a variation of Euler s method called the Euler Maruyama (EM) method. Define δt = T/N, for T > 0 and positive integer N, and let t j = jδt. The EM method proceeds as: u 0 = u(0) u j+1 = u j + λu j δt + σu j W j, where W j = W(t j ) W(t j 1 ), for j = 1, 2,...,N. Each random Brownian increment W j is computed using W j = t j z j, where z j is sampled from a Gaussian distribution of zero mean and unit variance. Strong and Weak Convergence Computed EM solutions of (4) match the true solution more closely as δt decreases. Note that a solution of (4) is a random variable, as is the difference between it and the exact solution. In order to investigate the convergence properties of numerical methods for SDEs, we need notions of convergence that can handle random variables. One such approach is strong convergence. The numeric method converges strongly of order γ if there exists a constant C so that its solution {u j } N 1 j=0 satisfies E u j u(t) Cδt γ, (6) for any fixed t = jδt [0, T]. The notation EX indicates expected value of the random variable X. It was shown by Gikhman and Skorokhod that the strong order convergence of the EM method is γ = 1/2. We can experimentally investigate strong convergence properties of the EM method by considering the average error at a single fixed point (for example, at the endpoint) for many repeated runs of the EM algorithm. A variation of the standard Runge-Kutta method for ODEs can be applied to problem (4) that has strong order convergence γ = 1. The method proceeds as: u 0 = u(0) u j+1 = u j + λu j δt + σu j W j + 1 ) (σ(u j + σu j δt) σuj ( Wj 2 δt). 2 7

8 Many applications do not require accuracy of specific solutions, but instead focus on solution statistics. The notion of weak convergence order is a useful one for such applications. A method converges weakly with order γ if there exists a constant C such that Ef(u j ) Ef(u(t)) Cδt γ, for all polynomial functions f and for any fixed t = jδt [0, T]. We consider the case in which f is the identity function. The EM method has weak order convergence γ = 1. Monte-Carlo Simulation and Option Pricing Assume that a stock price u(t) evolves according to (4). Consider the European call option with value at expiration time T defined by max{u(t) K, 0}, where K the strike price. Assuming no arbitrage 1 and no short-selling, it can be shown that the expected present value of the option is given by exp( λt)e(max{u(t) K, 0}). We can estimate this quantity by simply averaging max{u(t) K, 0} for many solutions u(t) computed by repeated runs of the EM method. The exact solution of the value European call option, assuming that the present time t = 0, can be shown to be given by the solution of the Black Scholes equation: where, C(u(0)) = u(0)n(d 1 ) K exp( λt)n(d 2 ), (7) d 1 = log(u(t)/k) + (λ + 0.5σ2 )T) σ, T d 2 = d 1 σ T, N(x) = 1 x exp( z 2 /2)dz. 2π inf In light of the exact Black Scholes solution, the so-called Monte-Carlo approach of computing many EM solutions to estimate the value of an option seems overly computationally expensive. However, although an exact solution is known for European-style options contracts, closed-form solutions like (7) are generally not available in options pricing. Such cases are very common and arise, for example, in pricing American options. Methods based on the Monte-Carlo approach similar to the above simple example are widely used to price options in these cases. 1 An arbitrage opportunity exists when there are two or more distinct prices for the same financial instrument available simultaneously. 8

9 Exercises 1. Code the Euler Maruyama method and graphically compare its solution to the analytic solution given by geometric Brownian motion for a stock price series. 2. Using the same stock series chosen for the previous exercise, approximate the strong convergence order of the EM method by computing many solutions and averaging the error against the true solution at the endpoint for several values of δt. Your experiments should compute a value of γ of approximately 1/2. 3. Code the example Runge-Kutta method for SDEs and similarly to the last exercise, investigate its strong order convergence properties. 4. Use the definition of weak order convergence and experimentally estimate the weak order of the EM method as in Exercise 2. Compute many runs of the EM method and compute the solution mean, comparing with the true solution mean for each δt. 5. Let a hypothetical option contract be defined by parameters u(0) = 10, K = 12, λ = 0.05, σ = 0.5, and T = 0.5. Compare the computed option value using the Monte Carlo approach against the exact option value computed using formula (7) for 10, 100, and 1000 runs of the EM method. References [1] Chuck Gartland and Kazim Khan, Lectures on Computational Finance, unpublished notes, Kent State University, [2] Desmond Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Rev. Volume 43, Issue 3, pp (2001). [3] Timothy Sauer, Numerical solution of stochastic differential equations in finance. To appear, Handbook of Computational Finance, Springer. 9