dstefan
03-22-2008, 01:20 PM
Chapter 1, Problem 17:
You expect that an asset with spot price $35 will trade in the $40--$45 range in one year. One year at-the-money calls on the asset can be bought for $4. To act on the expected stock price appreciation, you decide to either buy the asset, or to buy ATM calls. Which strategy is better, depending on where the asset price will be in a year?
Solution: (posted on 8/9/2008)
For every $1000 invested, the payoff in one year of the first strategy, i.e., of buying the asset, is
V_1(T) ~=~ \frac{1000}{35} S(T),
where S(T) is the spot price of the asset in one year.
The payoff of the second strategy in one year for every $1000 invested, i.e., of investing everything in buying call options, is
V_2(T) ~=~ \frac{1000}{4} \max(S(T)-35,0) ~=~ \left\{ \begin{array}{cl} \frac{1000}{4} (S(T)-35), & \mbox{if}~~ S(T) \geq 35; \\ 0, & \mbox{if}~~ S(T) < 35. \end{array} \right
It is easy to see that, if S(T) is less than 35, than the calls expire worthless and the first strategy of buying the asset will not lose all its value, while the calls expire worthless and the speculative strategy of investing everything in call options will loose all the money invested in it.
However, investing everything in the call options is very profitable if the asset appreciates in value, i.e., if S(T) is significantly larger than $35.
The breakeven point of the two strategies, i.e., the spot price at maturity of the underlying asset where both strategies have the same payoff is $39.5161, when
\frac{1000}{35} S(T) ~=~ \frac{1000}{4} (S(T)-35)
If the price of the asset will, indeed, be in the $40-$45 range in one year, then buying the call options will be the more profitable strategy.
You expect that an asset with spot price $35 will trade in the $40--$45 range in one year. One year at-the-money calls on the asset can be bought for $4. To act on the expected stock price appreciation, you decide to either buy the asset, or to buy ATM calls. Which strategy is better, depending on where the asset price will be in a year?
Solution: (posted on 8/9/2008)
For every $1000 invested, the payoff in one year of the first strategy, i.e., of buying the asset, is
V_1(T) ~=~ \frac{1000}{35} S(T),
where S(T) is the spot price of the asset in one year.
The payoff of the second strategy in one year for every $1000 invested, i.e., of investing everything in buying call options, is
V_2(T) ~=~ \frac{1000}{4} \max(S(T)-35,0) ~=~ \left\{ \begin{array}{cl} \frac{1000}{4} (S(T)-35), & \mbox{if}~~ S(T) \geq 35; \\ 0, & \mbox{if}~~ S(T) < 35. \end{array} \right
It is easy to see that, if S(T) is less than 35, than the calls expire worthless and the first strategy of buying the asset will not lose all its value, while the calls expire worthless and the speculative strategy of investing everything in call options will loose all the money invested in it.
However, investing everything in the call options is very profitable if the asset appreciates in value, i.e., if S(T) is significantly larger than $35.
The breakeven point of the two strategies, i.e., the spot price at maturity of the underlying asset where both strategies have the same payoff is $39.5161, when
\frac{1000}{35} S(T) ~=~ \frac{1000}{4} (S(T)-35)
If the price of the asset will, indeed, be in the $40-$45 range in one year, then buying the call options will be the more profitable strategy.