A call option gives an investor the right to purchase a stock (or other asset) at a fixed/guaranteed price on or before a given future date. For example, you might purchase a call option giving you the right to purchase one share of IBM at a price of $100 per share at any time during the eight monnths prior to January 17, 200Y). Since an investor is never forced to exercise a call option, any loss is limited to the original cost of purchasing the call option (if IBM is trading at $88, the cost of a call option with eight months to expiration might cost $3 per share). For example, if IBM is at $80 in eight months the investor lets the call option expire and the loss is limited to the purchase price of the call option. However, if IBM is at $120, the investor makes $20 per share by exercising the call option to purchase the stock at $100 and then selling for $120.

The fact that an investor's loss is limited to the purchase price for the call option implies that investors will always be willing to pay more for options on more volatilie stocks (as measured by the standard deviation of the return on the underlying stock). The relatively greater value attaching to call options on more volatile stocks is due to the fact that the loss from purchasing a call option is limited to the original cost of the option, while the upside potential is unlimited. Thus, an increase in the volatility of the stock underlying a call option has no impact on the potential loss to the investor (given the original cost of the call option) but can have a dramatic on the potential payoff for the call option.

The Black Scholes model values a call option as a function of the current stock price, the strike price and time to expiration for the call option, the risk-free rate of interest, and of course the yearly standard deviation for the return on the underlying stock. Of the five variables that are needed to determine the Black Scholes "model price" for the value of a call option, the only variable that is not readily observable is the volatility of the return on the stock underlying the call option (where we measure volatility by the standard deviation of the yearly return).

Given the important role of volatility in determining the value of a call option, it is quite natural to infer the volatility of a stock from the prices of call options, provided of course that we are reasonably confident that the call option is fairly priced (as is usually the case for actively traded at-the-money options). That is, since we can observe the market price for a call option, the Black Scholes model can be used to determine the standard deviation (volatility) for the underlying stock that would make the Black Scholes model price for the call option equal to the current market price of the option (given the current stock price, the risk-free rate of interest, as well as the strike price and time to expiration for the option).

An estimate of volatility that is inferred from the price of a call option is usually referred to as an "implied volatility". Computation of such an estimate for volatility is similar in many respects to computing the yield to maturity for a bond, where we search for the discount rate that makes the present value of the cash flows from the bond equal to the current market price. While the Black Scholes model is somewhat more complicated than the annuity factor and discount factor that are used to value a bond, the underlying principal of searching for the vale of a financial variable (be it yield or volatility) that equates a market price with a model price is the same.

Setting up the Black Scholes formula for the value of a call option in an Excel Spreadsheet is fairly easy to do. However, in order to follow along with the discussion on computing implied volatilities below, you may wish to download the linked Excel Implied Volatility Spreadsheet. Download the spreadsheet and open it up. You will see that in column B, the stock price in Cell B1 is set to 59, the risk free rate (in Cell B4) is set to .02 (2 percent), for an option with a strike price of 60 (Cell B5) and 91 days to expiration (Cell B6). Most importantly, note that in Cell B2 the volatility of the stock underlying the option is set to .32 (32 percent). The output for the Black-Scholes formula reportted in Cell E9 shows that the Black Scholes model price (based on a volatility of 0.32) is equal to $3.42.

Unfortunately, the MARKET PRICE for this call option is currently $4.54. Thus, it appears that the Black Scholes model price for the option, based on an implied volatility estimate of 32 percent (0.32) underestimates the value of the call option.

How could you get the Black Scholes model price to equal the market price??? We know that the the Black Scholes model captures the fact that the value of a call option increases with the volatility of the underlying stock. Further, we know that the Black Scholes model price is well below the market of the call option. Suppose that we change the volatility input to the model from .32 to .40 (indicating that the standard deviation of the return for the stock is 40 percent). Try this. Notice that when you change the volatility input in Cell B2 from .32 to .40, the model price in Cell B9 changes from $3.42 to $4.36. Although a model price of $4.36, is still less than the market price of $4.52, we are gettin closer to the estimate of volatility that the market is (apparently) using to determine the value of the option (assuming of course that the market is indeed using the Black Scholes model.

Let's close in on the "implied volatility" for the option. Since the "model price" of $4.36 is still less than the market price of $4.54, we know that our volatility estimate of .40 is still less than the volatility being used by the market to price the call option. In other words, an increase in our volatility estimate is required to increase the value of our Black Scholes model price up to the market price of $4.54. Let's increase our volatility estimate in Cell B2 to 0.42 (42 percent). Notice that the Black Scholes model price increases to $4.60, which is now somewhat greater than the market price of $4.54. Therefore, we need to reduce our volatility estimate somewhat to bring our "model price" in line with market price of the call option. Change the volatility estimate from 0.42 to 0.415, which will reduce the "model price" for the option to $4.54, which is exactly equal to the hypothetical market price for the call option. Since a volatility estimate of 0.415 makes the Black Scholes model price equal to the market price for the call option, 0.415 is the implied volatility for the 91-day options having a 60 strike price.

As long as we are discussing the Black Scholes model and the prices of call options, lets talk about the "hedge ratio" or "delta", the difference between N(d1) and N(d2), and the use of options to hedge positions in a stock.

First lets talk about the difference in d1 and N(d1). Think of it this way, -d2 is the value of a mean 0 unit standard deviation random variable that would permit the option to finish just at the money. We showed in class that N(d2) can be thought of as the probability that the option finishes in the money. Basically, the N(*)'s are an integral over a probability density whose upper limit is d2. Since that is pretty technical, think of 0 < N(d2) < 1 as the probability of finishing in the money. It can also be shown that N(d1) is the number of shares of stock that are required to match the short run impact of stock price moves on the value of the call option. This makes sense since the first part of the Black Scholes formula for the value of a call option is the current stock price of S multiplied by N(d1), which gives us the dollar of the stock required to match the short run prices move in the value of the call option. Check the spreadsheet template to be sure that you can compute d2 (or its cousin d1). Once you are able to compute these quantities, the Excel function NORMSDIST(d1) will compute the value of N(d1) and N(d2).

The spreadsheet package will provide estimates of N(d1) and N(d2) for you, provided that you input the contractual features of the call option, the current stock price, volatility, and an estimate for the risk free rate of interest. Note that if you are selling options to hedge the price of the stock (you are Mark Cuban and you just sold your company for shares of Yahoo that you are prohibited by contract from trading). You know that an option moves less than a dollar for a one dollar move in the stock ( N(d1) dollars to be precise). So you know that you need to sell more than one option for each share of stock that you wish to hedge. The answer to how many more is 1/N(d1) for each share of stock. To solve any hedging problem, us the spreadsheet template to find N(d1) for the option that you are going to hedge with. For example, suppose that you find that N(d1) is about 0.40 so that 1/.40 = 2.50 . If the stock price moves a dollar, then each option moves by $0.40 and if so if you sell 2.50 call options they should collectively move by $1 = 2.50 x $0.40 in response to a $1 move in the price of the stock.

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