Question

A “straddle” option is a portfolio of a long call option and a long put option in which both options have the same exercise price and are written on the same underlying asset. For an exercise price of 1 and an asset price of S, the maturity value of a straddle is (very) roughly like (S − 1)2 or like e^(S−1) + e^(1−S) − 2.

Using e^x = 1 + x + x^2/2! + x^3/3! + · · · + x^n/n! + · · · , expand e^(S−1) + e^(1−S) − 2 about S = 1 up to O ( (S − 1)3 ) .

Assume that dS = µ S dt + σ S dW where dW is a Wiener process, and use Itˆo’s lemma to compute df1 and df2 where f1 = (S − 1)^2 and f2 = e^(S−1) + e^(1−S) − 2.

A) Compute Taylor series expansions of df1 and of df2 about S = 1 to O ( (S − 1)2 , and compare the results of the two expansions.

Answer 1