Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.15$ and $l = 0.95$. The interest rate is $R = .05$.
a.) If the asset pays 10% of its value as dividend in the first period and 20% in the second period, find the price of the ATM call option.
b.) Consider a more complicated dividend strategy which pays 10% dividend only if the price moves up and no dividend if the price moves down at each period. Find the price of the ATM call option.
Forgive me my professor can have many errors in his problems. I am not sure how to solve this problem and what ATM call option mean. We have never covered dividends in regards to binomial model. I need some help with this any suggestions is greatly appreciated.
Answer
(a) First of all, ATM means strike price $K=S_0$. By the end of second period, the risky asset has values (from top to down) $S_0 u^2 (1-d_1) (1-d_2)=0.95$, $S_0 ul (1-d_1) (1-d_2)=0.79$ and $S_0 l^2 (1-d_1) (1-d_2)=0.65$. The risk neutral probabilities are calculated as $\hat{\pi}_u=(1+R-l)/(u-l)=1/2$ and $\hat{\pi}_l=(u-R-1)/(u-l)=1/2$. Hence three stages have probabilities 1/4, 1/2 and 1/4 respectively.
The option is determined by the terminal price $$ V(S_T)=(S_T-K)_+ $$ However, if $K=S_0=1$, then $V(S_T)=0$ for sure. Then price of the ATM call will be $$ C=\frac{1}{(1+R)^2} \hat{E}(V(S_T))=0. $$ If you change $K=0.8$, then $$ C=\frac{1}{(1+R)^2} \hat{E}(V(S_T))= \frac{1}{4}(0.95-0.8) + \frac{1}{2}0 + \frac{1}{4}0. $$
(b)
Terminal stages are $S_0 u^2 (1-d)^2=0.93$, $S_0 u l (1-d)=0.98$ and $S_0 l^2=0.9$. Neutral probabilities won't change. Just repeat steps in part (a) you will get zero for $K=S_0$ and a nonzero price for $K=0.8$.
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