Exercise 15.4.2 A forward-start option is like a standard option except that it becomes effective only at time τ

∗ from now and with the strike price set at the stock price then (the option thus starts at the money). Let C(S) denote the value of an at-the-money European forward-start call, given the stock price S. (1) Show that C(S) is a linear function in S under the Black–Scholes model. (2) Argue that the value of a forward-start option is e−rτ

∗C(Eπ [ S(τ

∗) ]) = e−rτ

∗C(Se(r−q) τ

∗ ), where q is the dividend yield.