Sam Somani’s project: Understanding Black-Scholes
How are stock derivatives priced? In order to understand this question, we must first know what a derivative is. It is an instrument whose value is derived from that of another. The Black-Scholes formula we derive in this post is an equation to find the price of derivatives whose values are derived from the price of an underlying stock. The particular derivative we examine in this post is an option—a contract that allows the owner to buy or sell a stock at a predetermined price at a predetermined time. Thus, its price depends on the price of the underlying stock and the time to maturity. Of course, our next question is: “what is a stock?” A stock is a contract that essentially gives the holder ownership in a company.
Logically, we now ask how the price of a stock is determined. We expect that our stock will increase in value by a given percentage. We will call this our expected rate of return. We also expect the value of our stock to increase as more time passes. Therefore, stock price is proportional to the chance in time. However, our rate of return is not guaranteed: Sometimes the change in value will be lower than what we expect; sometimes it will be higher. We call this volatility. While we cannot know the exact difference, we have an idea of how much of a difference from the expected change in value we can anticipate. Although we can use this value as a constant measure of volatility, the stock rarely moves up or down this exact value. Of course, prices swings due to volatility are greater in higher-priced stocks than they are in lower-priced stocks. Thus, our random price changes term must include stock price in it. We model the unpredictability of the change in the price of the stock by adding in another variable whose change is determined by a drawing from a normal distribution with a mean of zero and a variance of one. This means the, although the value of this variable can vary from negative infinity to positive infinity, it is likely to be relatively close to zero. Multiplying this random variable by our constant measure of volatility gives us a way to consider changes in the price of stock due to random events. Combining all these statements gives us the following model of change in stock price:
where is the price of the stock in question, represents the rate of change in the price of the stock, is our expected rate of return, is our constant measure of volatility, and is the change in the variable we use to model price changes due to random events.
As an example, let us take a share of stock in the fictional company AZBY. The current price of a share in this company is $100. The expected rate of return is 10% per annum. The volatility is also 10%. Properties of the normal distribution and the fact that stock prices are distributed log normally (i.e. the natural logarithm of stock prices follows a normal distribution) allows us to make a prediction for the value of the random variable. What will be the value of our stock a year from now?
So, after one year, we can say with 95% of certainty, that the natural log of the value of our stock will be between 4.505 and 4.896 (which means that the price will be between $90.37 and $133.75.)
Now we return to our original question—How are derivatives priced? First, what are derivatives valuable? Derivatives allow one to hedge their bets. Options allow the bearer, if he or she so chooses, to buy or sell a stock at a predetermined price at a predetermined time. In other words, owning an option for a stock one holds makes it such that the bearer earns a profit whether the price of the stock rises or falls. Clearly, the value of a derivative will depend on the price of the underlying stock. Higher values stocks will have higher valued derivatives simply because there is greater risk involved with higher valued stocks: Prices will swing more as shown by our formula for the change in stock price. However, the value of a derivative also depends on time—premiums must be paid for longer times to maturity. As time to maturity goes to zero, it becomes clearer whether the stock price will be higher or lower than the strike price, meaning lower uncertainty, or in other words, lower risk. As time to maturity and risk decrease, the premium is no longer necessary. However, the price of the stock still varies with time, and this price can increase or decrease with time, so the effect of time on the price of the derivative can be positive or negative. So while we know the price of the derivative depends on time and the price of the stock, we do not know the exact relationship. Even though we do not know the exact formula, Itō’s lemma tells us the function for the change in price of the derivative: