# Assignment 2: Option Pricing and the Black-Scholes formula The University of British Columbia Science One CS Instructor: Michael Gelbart

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2 14 13 stock price (\$) time (arbitrary units) Figure 1: Simulating a stock price using geometric Brownian motion with S 0 = \$10, T = 1000, and σ = In this particular run of the simulation, S T is just under \$12. the initial (current) stock price, which we will call S 0, the strike price X, and expiration time T. One other parameter is relevant here, namely the volatility of the stock, which we denote as σ. The volatility is analogous to the step size in a random walk: a larger volatility means more movement in a given amount of time. The Monte Carlo approach. Although the problem of option pricing has been addressed theoretically, we will tackle it by simulating the stock price as it changes over time, and then computing the option value P T with eq. (1). Sometimes the simulation will yield a worthless option, and other times it will be worth something. By repeating the simulation many times and taking the average, we can compute an expected or average value of the option, which we will take to be its fair price (P 0 ). This general approach of simulating a random process in order to understand its characteristics is called Monte Carlo (after the famous Monte Carlo casino, where there is randomness aplenty). Geometric Brownian motion. Our particular approach for simulating the stock price over time will be to assume the stock price follows a geometric Brownian motion. In Assignment 1 you simulated a random walk which is essentially Brownian motion: at each time step you add a small random amount to your current position. In geometric Brownian motion, at each time step you multiply your current state by a (positive) random amount. Note that in regular Brownian motion the position can be both positive or negative, but in geometric Brownian motion the position is always positive; this is good because stock prices are never negative. (If this helps, you can also think of geometric Brownian motion as an exponentiated Brownian motion.) An example is shown in fig. 1. The update rule for our stock price model is as follows: S t = S t 1 exp ( 0.5σ 2 + σz ), (2) where Z is a random number drawn from a Gaussian distribution (also called a Normal distribution or a bell curve ). You can generate these random draws with numpy.random.randn() and you can 2

3 14 13 stock price (\$) time (arbitrary units) Figure 2: The same simulation as in Figure 1 (blue), but with an overlay (red) showing a smoothed version generated using a moving average window with W = 25. perform most of the math operations with numpy as well; for example, numpy.exp(x) computes exp(x), etc. Note that if σ = 0 in eq. (2), then the stock price never changes, whereas if σ is large the stock price can change wildly; this matches our intuition that σ represents the volatility of the stock. Note: the mathematical notation exp(x) means e x. Note: what we are doing is similar to numerically solving a differential equation (in fact, we are numerically solving something called a stochastic differential equation). When we numerically solve differential equations there is usually a time step t, but to keep things simpler I have scaled the units of time so that t = 1. Also for the sake of simplicity, I have neglected the interest rate, which usually plays an important role in financial models. Smoothing a curve. Figure 1 shows the stock price at every time step, and is therefore very bumpy. Investors are often more interested in a smoothed depiction of the stock price over time, in which small ups and downs are removed, leaving only the salient trends. Figure 2 shows a smoothed version of fig. 1. The smoothing is achieved by replacing the price at time t with an average of all the prices from times t W to t + W (inclusive), for some positive integer W (thus, the total width of the window is 2W + 1). Near the edges of the plot, where we do not have enough data for the entire window, the average computed only over the portion of the window that is within the data range. For example, if T = 1000 and W = 5 then the smoothed curve at t = 0 is the average of the prices from t = 0 to t = 5, the smoothed curve at t = 1 is the average of the prices from t = 0 to t = 5, the smoothed curve at t = 10 is the average of the prices from t = 5 to t = 15, and the smoothed curve at t = 997 is the average of the prices from t = 992 to t = The Black-Scholes formula. The famous Black-Scholes formula was derived by a theoretical analysis of the geometric Brownian motion model discussed above. It turns out that (in our oversimplified 3

4 case of zero interest rate) the expected option price, P 0, is given by: where P 0 = Φ(d 1 )S 0 Φ(d 2 )X, (3) d 1 = 1 [ σ log T d 2 = 1 σ T [ log ( S0 ) ] + σ2 X 2 T, ) ] σ2 2 T, ( S0 X and Φ( ) is the cumulative distribution function of the Gaussian distribution, which can be computed with scipy.stats.norm.cdf(x) (after importing the module with import scipy.stats). Note: unless otherwise specified, log refers to the base-e logarithm, also known as the natural logarithm and also sometimes written as ln(x). Your task Write a program OptionPrice.py that takes in six command-line arguments: the initial stock price S 0, the volatility σ, the number of time steps T, the option s strike price X, the smoothing window size W, and the number of repeated experiments N. Your program should contain the following functions: 1. A function simulate that takes as arguments S 0, σ, and T. This function should simulate the geometric Brownian motion for T steps using eq. (2) and then return an array of size T + 1 containing the stock price at each time step. 2. A function smooth that takes in an array of values and the window width W. This function should smooth the input array using the sliding window averaging scheme described above with window width W, and then return the smoothed result. The returned array should be the same size as the input array; you can access the length of an array with len(array). 3. A function blackscholes that takes as arguments S 0, X, σ, and T and returns the Black-Scholes option price given by eq. (3). Making use of the functions described above, your program should perform the following tasks: I. Simulate the stock price and plot the results (S t vs. t) using matplotlib. Make sure to include the axis labels as in figs. 1 and 2; you can set these with plt.xlabel and plt.ylabel (assuming you defined plt by typing import matplotlib.pyplot as plt at the top of your file. IMPORTANT NOTE: now that we have learned about arrays, you can use plt.plot(x,y) where x and y are arrays; this means you do not need to create your plot one point at a time as you did earlier(!!!), but rather you can create the entire curve with one call to plt.plot. You also no longer want the.b because you are no longer plotting individual dots. You can still include b if you want, but it will not do anything since blue is the default colour for plotting. II. Produce a smoothed version of the simulated results using the sliding window averaging scheme described above and window width W. Plot this on top of your original plot in red, using, for example, plt.plot(x,smooth, r ). Save the plot to the file stockprice.pdf using the command plt.savefig( stockprice.pdf ). IMPORTANT NOTE: You do not need to generate a figure like fig. 1, just one like fig. 2. III. Call the simulate function N times and compute the option value P T in each case; see eq. (1). Print out the option value averaged over the N trials as an estimate of P 0. IV. Print the Black-Scholes option price, which is the theoretical or correct value of P 0. 4

6 4. All import statements appear at the top of your program, just below your initial comment. 5. Your program contains the functions simulate, smooth, and blackscholes, and: (a) These functions conform exactly to the interfaces described above. (b) These functions are located at the top of your file, just below your import statements but above the rest of your code. (c) Each of these three functions is preceded with a comment describing the inputs and outputs of the function (variable names, variable types, descriptions), as well as a brief explanation of what the function does. As an example, here is an acceptable comment for the simulate function (which you are free to copy if you wish): # This function simulates geometric Brownian motion. # Inputs: S_0 (float), the initial stock price # sigma (float), the stock volatility # T (int), the number of time steps to simulate # Output: the simulated stock price over time (array of size T+1) Note that I specified the names of the inputs but not the output. This is because only the names of the inputs are specified when you define a function with def. However, I still specified the type of the output. Sample Syntax To help you remember the Python syntax for functions, arrays, and loops, here is an example function: # This function counts the number of positive elements in an array # Inputs : a (array), the input array # Outputs : num (int), the number of positive elements def count_positive(a): num = 0 for a_i in a: if a_i > 0: num = num + 1 return num Here is the same function written with a while loop instead of a for loop: # This function counts the number of positive elements in an array # Inputs : a (array), the input array # Outputs : num (int), the number of positive elements def count_positive(a): num = 0 i = 0 while i < len(a): if a[i] > 0: num = num + 1 i = i + 1 return num In general, anything you can do with a for loop can also be done with a while loop. 6

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