Probability
6.5 Pricing Stock Options.
In this section we shall consider stock options that give the holder the right to buy a certain stock at a certain price at a certain future time. The ones we consider are a simplified version of more common options that give the holder the right to buy a certain stock on or before a certain future time. The holder of the option does not have to buy the stock at the future time. The holder would only buy the stock at the future time if the actual price were above the price set by the option. One has to pay something to buy stock options.
Example 1. At the current time JetCo stock is selling for $2.50 a share. For 25 cents a share I can buy options to buy JetCo in six months for $2.60 a share.
If I bought some of these options then in six months if the price of JetCo stock is above $2.60, then I would exercise the option and buy the number of shares that I have options for at $2.60 a share and then sell them at the higher price making a profit. However, the price per share in six month would have to be above $2.60 + $0.25 = $2.85 for me to make an overall profit. For example, if the price in six months were $3.00 per share I would make an overall profit of 15 cents per share. If the price in six months were $2.70 per share I would exercise the option, but I would suffer an overall loss of 10 cents per share. If the price in six months was $2.40 I would not exercise the options and suffer an overall loss of 25 cents per share.
Purchasing a stock option offers some advantages to buying the actual stock. Usually the price of the option is a lot less than the price of the stock. So the buyer of the options would not lose as much as if he bought the stock if the price of the stock goes down a lot. However, the buyer of the option could make quite a bit if the price of the stock goes up a lot. Whether or not buying the option is a good idea depends on the price of the option and the likelihood that the stock price will go up in the period of the option
Here is some notation and terminology
X(t) = the price of the stock at time t. We assume this is a geometric Brownian motion with drift ( and volatility (.
x0 = X(0) = the current price of the stock. We assume this is known. It is $2.50 in Example 1.
K = strike price = the price the holder of the option has the right to buy the stock for at the future date. It is $2.60 in Example 1.
T = time until maturity = the future time at which the holder of the option can buy the stock at the strike price. It is 1/2 year in Example 1.
( = interest rate = the interest rate on guaranteed investments.
For the moment, let's ignore what one must pay to buy options of the stock in the first place and look at the expected amount of profit the holder of the option will make when it matures. The amount per share the purchaser will gain is
G =
This can be written as G = hK(X(T))X(T) – KhK(X(T)) where
hK(x) =
G is a random variable and there is a well-known formula for its expected value called the Black-Scholes option pricing formula. It is usually given for e-(tE{G}, the present value of the expected gain.
Proposition 1. Assume X(t) is a geometric Brownian motion with drift ( and volatility ( and ( = ( + (artitrage holds). Then
(1) e-(tE{G} = x0( - Ke-(t(
= x0((- b + () - Ke-(t((- b)
where
b =
Here ((z) = Pr{Z ( z} = is the distribution function of a standard normal random variable. The second term Ke-(t((- b) is the present value of the expected amount you will pay to purchase the stock. In the proof of the proposition below we shall show that ((- b) = Pr{Z ( b} = Pr{X(T) ( K} is just the probability the stock price will be above K. The first term x0((- (b - ()) is the present value of what you will get when you sell the stock. Before giving the proof of this proposition, let's give an example of its use.
Example 2. At the current time JetCo stock is selling for $2.50 a share. Suppose the price of JetCo stock is a geometric Brownian motion with volatility ( / yr1/2. Suppose the interest rate on guaranteed investments is 2% and the drift of JetCo's price has adjusted so that arbitrage holds. Consider an option that allows one to buy JetCo in six months for $2.60 a share. What is the present value of the expected gain per share of such an option.
We use formula (1) with x0 = $2.50, T = ½, K = $2.60, ( = 0.02 and ( = 0.4. One has e-(t = e-(0.02)(1/2) = e-0.01 ( 0.99005.
b = =
= = = = 0.245
((- b) = 1 - ((b) = 1 - ((0.245) = 0.4032
= probability price of JetCo stock will be above $2.60 in 6 months
Ke-(t((- b) = (2.60)(0.9905)(0.4032) = $1.038
= present value of expected amount you will pay to buy JetCo stock in 6 months
b - ( = 0.245 – (0.4)() = 0.245 – 0.283 = - 0.038
((- (b - ()) = ((0.038) = 0.515
x0((- (b - ()) = (2.50)(0.515) = $1.29
present value of expected gain = x0((-(b-()) - Ke-(t((-b) = $1.29 - $1.038 = $0.25
Proof of Proposition 1. It suffices to show that E{hK(X(T))X(T)} = x0e(t((- (b - ()) and E{hK(X(T))} = ((- b). Consider the second of these two formulas. Let pT(x) be the density function of X(T). Then
E{hK(X(T))} = = = Pr{X(T) > K}
Since X(T) is a geometric Brownian motion, ln(X(T)) is a regular Brownian motion with mean (T and standard deviation (. So ln(X(T) – ln(x0) is normal with mean (T and standard deviation (. So Z = (ln(X(T)) – ln(x0) - (T)/(() is a standard normal random variable. So
Pr{X(T) > K} = Pr{ln(X(T)) > ln(K)} = Pr{
= Pr{Z > b} = Pr{Z < - b} = ((- b)
Now we show that E{hK(X(T))X(T)} = x0e(t((- (b - ()). Since Z = (ln(X(T)) – ln(x0) - (T)/(() one has X(T) = x0e(Z + (T. Note that X(T) > K precisely if Z > b. Therefore hK(X(T))X(T) = x0hb(Z)e(Z + (T. So
E{hK(X(T))X(T)} = x0E{hb(Z)e(Z + (T} = x0
= x0e(T = x0e(T + (2T/2
= x0e(T = x0e(T = x0e(TPr{Z > b - (}
= x0e(TPr{Z < - (b - ()} = x0e(t((- (b - ())
This concludes the proof of Proposition 1. //
Problem 1. Suppose the current price of DTE stock is $49, the price has a volatility of ( = 0.15 / yr1/2 and the current interest rate is ( = 0.05 / yr. Assume that arbitrage is working so the drift ( is ( = ( - (2/2. What would be the theoretical price of a stock option that allowed the purchaser to buy the stock for $50 in 3 months?
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