A Simple Simulation to Value Options

Douglas Ulmer

University of Arizona

AWS 2002

Key points

• Standard topics can be made more exciting and relevant through the use of an interesting application.

• Technology makes it possible to compute things that are analytically out of reach of students at this level.

Overview

• The students and the topics they are supposed to learn

• A business problem that ties together all the topics

• A simulation to solve the problem

• Discussion and revision of the simulation

The students and the topics

The students: Typically second semester freshman majoring in business. Their preparation is pre-calculus at best. They do have some computer experience.

The topics discussed here form about half of the first semester Business Math course. The course emphasizes large scale projects and team work.

The students and the topics

• Math:– Compound interest (discrete & continuous

exponential growth)– Simple descriptive statistics (range, mean,

frequency, relative frequency, histograms)– Random variables (Bernoulli, binomial, uniform,

exponential) and their distribution function (PMF/PDF/CDF)

– Expected Value– Random samples, sample mean– Estimating E(X), fX from a sample

The students and the topics

• Technology– Excel functions to compute algebraic

expressions, exponentials, logs, averages, etc.

– Histogram and chart generation– Random number generation– Simulations

The project: business background

• A share of stock is fractional ownership of a company.

• An option is the right, but not the obligation, to buy or sell a certain number of shares of the underlying stock at a fixed price during a fixed period of time.

• Characteristics:– Underlying stock– Strike price– Expiration date– Call = right to buy, Put = right to sell

Business background: Example

• A call on 100 shares of Microsoft expiring July 19, 2002 with a strike price of $75– If Microsoft is selling for more than $75 on

7/19, say at S, then the option is worth S-75 per share, or (S-75)*100.

– If Microsoft is selling for less than $75 on 7/19, the option expires worthless.

Business background: Example

• A put on 100 shares of Apple expiring 7/19/2002 with a strike price of $25– If Apple is selling for more than $25 on

7/19, the option expires worthless.– If Apple is selling for less than $25 on 7/19,

say S, then the option is worth 25-S per share, or (25-S)*100.

Business background

C

P

Business background

• The above is from the point of view of the buyer of the option. A seller (one who “writes” an option) has the opposite perspective.

• American vs. European options• The “risk-free” rate of interest is what can be

earned on an investment free of risk (I.e., with known future value). In practice, this means the rate of return on US government bonds.

Business background: Exercise

• The value of a call is:– an increasing function of the expiration

date– a decreasing function of the strike price– a decreasing function of the “risk-free”

interest rate– an increasing function of the volatility of the

underlying stock

• Similarly with puts

Business background: More information

• The Chicago Board of Options Exchange has a web site with lots of information: www.cboe.com

• Here is a useful intro: Understanding Stock Options

• And this is the book everybody gets when they open an options account: The Characteristics and Risks of Standardized Options

The project: Overview

• Each team has a call option on a certain stock, with a certain strike price and expiration date. They are also given a risk-free interest rate and a period of historical data (weekly closing prices for the underlying stock) to use.

• The problem is to find the fair value of the option.

The project: Our case

• We are to value a call option with the following characteristics:– Underlying stock: Stillwater Mining Co.

(NYSE symbol “SWC”)– Expiration date: July 19, 2002– Strike Price: $15

• We are to use 8 years of historical data.• The relevant “risk-free” rate of interest is

1.8%.

The project: A strategy

• We can think of the present value of the option as a random variable. Simulation can be used to determine its expected value:– The weekly relative price change is a random

variable R. (R for ratio.) Our historical data gives us a large sample of observations of R.

– The stock closing price S in n weeks can be determined from the current price and n observations of R.

The project: A strategy

– The value of the option at expiration is FV = max(S-K,0) (K=strike price).

– This can be discounted (using the risk free rate) to give a present value PV.

– Doing this many times and averaging gives the expected value of PV, which is our option price.

The project: Carrying out the strategy

• Download historical data.• Generate observations of the weekly gain

R.• Use random number generation and the

sample of R to simulate observations of the future value of the stock and the option. Then discount these future values and average to get an expected present value of the option.

The project: A variant

• We used historical data to generate an empirical probability distribution for R.

• Another possibility is to model R.

• A standard model of stock prices is “geometric Brownian motion” which means that R should have a normal distribution. We can estimate its mean and variance from our sample.

The project: A variant

• How reasonable is this model? Let’s check.

• Now we can run our simulation using this model for R.

Comparison with reality

• On 3/6/02, SWC was trading at 17.95 and our simulation gives option values of C = 3.58 And P = 1.46

• Market prices on that day (average of bid and ask at 1:03 pm) were …

• C=2.35 And P=1.83. So our simulation overvalues the call and undervalues the put. Why?

What we tell our students

Our model attempts to predict future prices of SWC based on past prices. But in the past 8 years, SWC grew at about 26% per year, well above the risk-free rate. But if everyone thought SWC would continue to grow at this rate, they would buy it, driving up the price and driving down the future rate of return.

What we tell our students

This suggests that the only reasonable assumption about the future returns of SWC is that they will be at the risk-free rate. So, we tell the students to normalize R (by subtracting a constant) so that E(R) is the risk-free rate.

The real story: Black-Scholes

• Their key observation is that “expectation pricing” must be replaced with “arbitrage pricing.”

• Basically what this means is that one constructs a portfolio (“long” one share and “short” a certain number of options) which, in the short run, is risk-free; i.e., the future value is independent of changes in the stock price.

• The value of such a portfolio must grow at the risk-free rate of interest.

Black-Scholes

• Model stock prices as geometric Brownian motion:

• ( a standard normal RV)

• Then the option value C satisfies a stochastic PDE which miraculously isn’t stochastic and is independent of €

ΔSS= μΔt +σ (Δt)1/ 2ε

Black-Scholes

• The PDE is essentially the heat equation and the formula C=max(S-K,0) for the option value at expiration gives boundary values.

• The explicit solution is the famous Black-Scholes formula, which gives the option value as a function of the expiration date, the strike price, the current stock price, the stock volatility, and the risk-free rate of interest.

Black-Scholes

• A byproduct of the B-S analysis is that our simulation model gives the correct answer if we normalize R so that E(R) is the risk-free rate.

• So, although we are not giving students the full story, what we tell them is not too misleading.

• Here is a 1-page note which explains the key idea of B-S and why the simulation gives the right answer, in a simple discrete situation.

Comparison with reality, part 2

• The revised simulation agrees well with the Black-Scholes formula.

• Both agree reasonably well with reality, i.e., market data.

Some references

• The UA Business Math web site: http://business.math.arizona.edu

• F. Black & M. Scholes, Journal of Political Economy, 1973 (Available on JSTOR)

• J. C. Hull, “Options, Futures, and Other Derivatives,” Prentice Hall 1999

• M. Baxter & A. Rennie, “Financial Calculus,” Cambridge UP, 1996