Latest Revision: October 30, 1995.
The normal distribution is a continuous bell--shaped probability distribution. The normal distribution is particularly convenient to use because the distribution can be completely described by two parameters -- the mean and the variance. Below is a diagram of a normal distribution and the actual U.S. stock returns. That is, this distribution is drawn with the historical mean and variance of the U.S. stock returns.
We will work with the "standard" normal. This distribution is called standard because the mean is assumed to be zero and the standard deviation is one. The area under the normal curve represents the probability. In all the options pricing examples we will be interested in cumulative probability. The interpretation is: what is the probability that the value is less than some number, x. This simply means that we are going to be summing up the area under the curve -- working from the left.
Let's consider an example. If I am given the number zero, what is the cumulative probability. Stated differently, what is the probability that the number that I draw from the distribution has the value of negative infinity to zero. What I want is the area under the curve to the left of zero. Since the curve is symmetric and since the area under the curve must sum to one, the probability is .50. If I am given the number -3, what is the probability? Viewing the diagram of the normal distribution, the -3 is way out on the left tail. The area from -3 to the left is very small (a guess would be .01). If we were given the point 3, this is way out on the right tail. The area to the left of 3 is almost the entire area under the curve (a guess would be .99).
In order to use the Black-Scholes formula, we need precise numbers for these probabilities. We usually use a computer or a statistical table to get these probabilities.
We will denote the cumulative normal by N(x)=prob. Some examples are
N(x=0) =.50 N(x=1) =.8413 N(x=-1)=1-N(1) =.1587 N(x=1.96) =.975 N(x=-1.96)=1-N(1.96) =.025The formula for the normal distribution is:
First, some definitions are needed.
The Black-Scholes Formula:
There are many ways of writing the Black-Scholes formula. All of them are equivalent to the above. For example, Sharpe's text presents the formula differently. However, with some algebra, the formula is identical to the Black and Scholes formula above.
Now to get used to using the formula, it is best to do some examples. I will show how to use Black Scholes tables to get the price of the call option. However, everyone should be able to code the formula into an EXCEL spreadsheet..
But before we work some examples, I will try to given some of the intuition behind the formula.
Usually, in Finance, we refer to people as risk averse. I defined this earlier as the preference for $50 for sure rather than taking a 50-50 bet with payoffs of $0 and $100. Now let's suppose that agents are risk neutral. This means that they are indifferent between the $50 for sure and the bet. If this is the case, then the value of the call can be thought of as the expected payoff of the call at expiration discounted back to present value. We could write:
Note that S^* is the stock price at expiration. This expression says that the call price is the present value of the expected expiration price minus the exercise price times the probability the call is in the money.
In terms of the Black-Scholes formula, N(x_2) can be interpreted as the probability that the call option will be in the money at expiration. The term N(x_1) is the present value of the expected terminal stock price conditional upon the call option being in the money at expiration times the probability that the call will be in the money at expiration. Finally, the term e^{-rT}kN(x_2) is the present value of the cost of exercising the option at expiration times the probability that the call will be in the money at expiration. So the Black-Scholes formula for call option has a fairly simple interpretation. The call price is simply the discounted expected value of the cash flows at expiration.
(b) Value a 5 month IBM call with the same terms.
(c) If the current prices of these two calls are $15.75 for the 5 month and $12.12 for the 2 month, then are these options overvalued or undervalued according to the Black-Scholes formula.
(d) What are the potential causes of these discrepancies?
(e) What standard deviation is implied by the current market price of the 5 month call option (approximately)?
(a) The information that we know is: S=265, k=260, T=2/12=.166, 'sigma'=.30, r=.07, dividend yield=.04. First, let's adjust the stock price for the dividend.
The 'delta' is .04 and we are assuming a continuous dividend payout. The 263.24 represents the stock price minus the present value of the dividend.
Next compute the present value of the exercise price.
Step One. Calculate the ex-dividend stock price divided by the present value of the exercise price.
Step Two. Calculate the standard deviation of the stock's return to maturity.
Step Three. Go to the option tables. Along the top of the table is find 1.02 and at the same time search for the row 0.123. Since the table does not have such a fine division, interpolate between 0.10 and 0.15. Black-Scholes Tables. The approximate value is 6.1%. This is represents the call option price as a percentage of the ex-dividend stock price. To get the value of the call option, multiply 6.1% by the adjusted stock price:
c=.061 x 263.23=$16.06
Since we are interpolating from a table this is only an approximation. It roughly matched the call price from our previous solution.
(b) Follow the same steps as (a). Now we are considering T=5/12.
Step One. Calculate the ex-dividend stock price divided by the present value of the exercise price.
Step Two. Calculate the standard deviation of the stock's return to maturity.
Step Three. Go to the tables. Along the top of the table is find 1.03 and at the same time search for the row 0.194. Since the table does not have such a fine division, interpolate between 1.02 and 1.04 along the top and use 0.20 along the top. Black-Scholes Tables. The approximate value is 9.3%. This is represents the call option price as a percentage of the ex-dividend stock price. To get the value of the call option, multiply 9.3% by the adjusted stock price:
c=.093 x 260.62=$24.23
Since we are interpolating from a table this is only an approximation. It almost exactly matches the call price from our previous solution ($24.31).
(c) According to these assumptions and the Black-Scholes formula, market is currently undervaluing these calls:
12.12<$15.83 2 month
and 15.75<$24.31 5 month
(d) The major sources for error are: (1) the Black-Scholes assumptions are incorrect or (2) the estimated 'sigma' is incorrect.
(e) This question asks us to calculate the 'sigma' that makes the Black-Scholes call price exactly equal to the market price. The sigma that we get will not equal the .30 that we were given. It will be called the implied standard deviation. It turns out that there is no direct way to solve for this implied measure. The implied standard deviation can be obtained using tables (which is the next approach). One can also use SOLVER in EXCEL to get the implied standard deviation.
Calculate the ratio of the actual call price to the ex-dividend stock price.
We know from before that ratio of the ex-dividend stock price to the present value of the exercise price was 1.024. From the table search for a 'sigma'\sqrt T that when matched with 1.024 delivers 4.6%. Black-Scholes Tables. That value turns out to be about .083. To get the annualized implied standard deviation divide by the square root of T.
We can compute the same quantity for the 5 month call. The ratio of the actual call price to the ex-dividend stock price is:
The ratio of the ex-dividend stock price to the present value of the exercise price is 1.032 from before. Black-Scholes Tables. The 'sigma'\sqrt T that approximately matches these two quantities is .11. So the annual standard deviation is
The option delta measures the change in the option price for a unit change in the stock price.
The delta is also known as the hedge ratio. This means that the delta tells how many shares I need to buy to hedge the sale of one call. We can calculate the delta of a call option the same way using the tables. Black-Scholes Tables.
Of course, we get more precise answers on the computer. Unfortunately, this year, we do not allow computers into the exam. However, this will change!
The precise formula for the delta is:
The elasticity or 'omega' is interpreted as the call option's elasticity to changes in the underlying asset's price. So if the stock price rises by 1% the call price should rise by 'omega'%. We can now see the link between the delta and the omega.
We can calculate the 'omega' of a call option the same way using the tables.
The delta of a call option is the first derivative of the Black-Scholes call formula with respect to the stock price:
The delta of a put option is the first derivative of the put formula with respect to the stock price. From put-call parity, we know.
The delta of the put is:
The omega of the call option is
Similarly, the omega of the put option is
