dstefan
07-01-2008, 05:30 PM
Chapter 4, Problem 8
Consider a put option with strike 55 and maturity 4 months on a non-dividend paying asset with spot price 60 which follows a lognormal model with drift \mu = 0.1 and volatility \sigma = 0.3. Assume that the risk-free rate is constant equal to 0.05.
(i) Find the probability that the put will expire in the money.
(ii) Find the risk--neutral probability that the put will expire in the money.
(iii) Compute N(-d_2).
Solution:
(i) The probability that the put option will expire in the money is equal to the probability that the spot price at maturity is lower than the strike price, i.e., to P(S(T) < K). Recall that
\ln \left( \frac{S(T)}{S(0)} \right) ~=~ \left( \mu - q - \frac{\sigma^2}{2} \right) T ~+~ \sigma \sqrt{T} Z
Then,
P(S(T) < K) = P\left( \frac{S(T)}{S(0)} < \frac{K}{S(0)} \right) ~=~ P\left( \ln \left( \frac{S(T)}{S(0)} \right) < \ln \left( \frac{ K}{S(0)} \right) \right) = P \left( \left( \mu - q - \frac{\sigma^2}{2} \right) T + \sigma \sqrt{T} Z ~<~ \ln \left( \frac{ K}{S(0)} \right) \right)
= P \left( Z ~<~ \frac{\ln \left( \frac{K}{S(0)} \right) - \left( \mu - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}} \right) = N\left( \frac{\ln \left( \frac{K}{S(0)} \right) - \left( \mu - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}} \right)
For S=60, K=55, T=1/3, \mu = 0.1, q=0, \sigma = 0.3, and r = 0.05, we obtain that the probability that the put will expire in the money is 27.1525%.
(ii) The risk-neutral probability that the put option will expire in the money is obtained just like the probability that the put expires in the money, by substituting the risk-free rate r for \mu, i.e.,
P_{RN}(S(T) < K) ~=~ P \left( Z ~<~ \frac{\ln \left( \frac{K}{S(0)} \right) - \left( r - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}} \right) ~=~ 0.304331
(iii) Recall that
d_2 ~=~ \frac{\ln \left( \frac{S(0)}{K} \right) + \left( r - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}}
Then, d_2 = 0.511983, and N(-d_2) ~=~ 0.304331, which is the same as the risk-neutral probability that the put option will expire in the money.
To understand this result, note that
P_{RN}(S(T) < K) ~=~ P \left( Z ~<~ - \frac{\ln \left( \frac{S(0)}{K} \right) + \left( r - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}} \right) ~=~ P(-d_2)
Consider a put option with strike 55 and maturity 4 months on a non-dividend paying asset with spot price 60 which follows a lognormal model with drift \mu = 0.1 and volatility \sigma = 0.3. Assume that the risk-free rate is constant equal to 0.05.
(i) Find the probability that the put will expire in the money.
(ii) Find the risk--neutral probability that the put will expire in the money.
(iii) Compute N(-d_2).
Solution:
(i) The probability that the put option will expire in the money is equal to the probability that the spot price at maturity is lower than the strike price, i.e., to P(S(T) < K). Recall that
\ln \left( \frac{S(T)}{S(0)} \right) ~=~ \left( \mu - q - \frac{\sigma^2}{2} \right) T ~+~ \sigma \sqrt{T} Z
Then,
P(S(T) < K) = P\left( \frac{S(T)}{S(0)} < \frac{K}{S(0)} \right) ~=~ P\left( \ln \left( \frac{S(T)}{S(0)} \right) < \ln \left( \frac{ K}{S(0)} \right) \right) = P \left( \left( \mu - q - \frac{\sigma^2}{2} \right) T + \sigma \sqrt{T} Z ~<~ \ln \left( \frac{ K}{S(0)} \right) \right)
= P \left( Z ~<~ \frac{\ln \left( \frac{K}{S(0)} \right) - \left( \mu - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}} \right) = N\left( \frac{\ln \left( \frac{K}{S(0)} \right) - \left( \mu - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}} \right)
For S=60, K=55, T=1/3, \mu = 0.1, q=0, \sigma = 0.3, and r = 0.05, we obtain that the probability that the put will expire in the money is 27.1525%.
(ii) The risk-neutral probability that the put option will expire in the money is obtained just like the probability that the put expires in the money, by substituting the risk-free rate r for \mu, i.e.,
P_{RN}(S(T) < K) ~=~ P \left( Z ~<~ \frac{\ln \left( \frac{K}{S(0)} \right) - \left( r - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}} \right) ~=~ 0.304331
(iii) Recall that
d_2 ~=~ \frac{\ln \left( \frac{S(0)}{K} \right) + \left( r - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}}
Then, d_2 = 0.511983, and N(-d_2) ~=~ 0.304331, which is the same as the risk-neutral probability that the put option will expire in the money.
To understand this result, note that
P_{RN}(S(T) < K) ~=~ P \left( Z ~<~ - \frac{\ln \left( \frac{S(0)}{K} \right) + \left( r - q - \frac{\sigma^2}{2} \right) T}{\sigma \sqrt{T}} \right) ~=~ P(-d_2)