What is an implied volatility? I know that it has something to do with the value of a call option and something called the Black Scholes model but other than that I don't have a clue. 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. Most importantly, note that in Cell B2 the volatility of the stock underlying the option is set to. The output for the Black-Scholes formula reportted in Cell E9 shows that the Black Scholes model price based on a volatility of 0. Thus, it appears that the Black Scholes model price for the option, based on an implied volatility estimate of 32 percent 0.
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. Notice that when you change the volatility input in Cell B2 from.
Let's close in on the "implied volatility" for the option. Let's increase our volatility estimate in Cell B2 to 0. 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. Since a volatility estimate of 0. 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.
Since that is pretty technical, think of 0 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.
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.More...