Question 5. (a) List the assumptions of the Black-Scholes model. E Consider a derivative whose payoff at expiry time T is given by CT(ST) = max (S K, 0), where K > 0 is the strike price and St is the price of the underlying share at expiry. This type of derivative is known as a power option, since the payoff depends on St to the power of some fixed number (2 in this case). Derivatives of this type have to be cash-settled. (b) Carefully sketch the graph of Ct as a function of St. E (c) We showed in the lectures that the Black-Scholes price V at time 0 of a vanilla European option with payoff function VT(ST) is given by Vo= L (S. exp [(x -012)T +oVT])e='/?dz, assuming that the underlying share pays no dividends during the lifetime of the option. The various symbols here have their usual meanings, and you may assume that So, o and T all are strictly positive. By using this result, or otherwise, prove that the Black-Scholes price Co at time 0 of the power option described above is given by Co = S exp [(r +0)T] (h+) Ke="To(h_), where log(So/VK) + (r + 302/2)T h+ OVT h_ = h+ 20VT, and (2) is the cumulative distribution function of the standard normal distribution. LE Hint: For full marks in parts (d), (e) and (f), your final answers should each be written in the most compact form possible. (d) By (partially) differentiating the expression for Co with respect to So, find the formula for the delta A of this power option. [. (e) By differentiating once again, find the formula for the gamma I. E aco (f) Finally, find the formula for the theta O, defined here as [ at . Question 5. (a) List the assumptions of the Black-Scholes model. E Consider a derivative whose payoff at expiry time T is given by CT(ST) = max (S K, 0), where K > 0 is the strike price and St is the price of the underlying share at expiry. This type of derivative is known as a power option, since the payoff depends on St to the power of some fixed number (2 in this case). Derivatives of this type have to be cash-settled. (b) Carefully sketch the graph of Ct as a function of St. E (c) We showed in the lectures that the Black-Scholes price V at time 0 of a vanilla European option with payoff function VT(ST) is given by Vo= L (S. exp [(x -012)T +oVT])e='/?dz, assuming that the underlying share pays no dividends during the lifetime of the option. The various symbols here have their usual meanings, and you may assume that So, o and T all are strictly positive. By using this result, or otherwise, prove that the Black-Scholes price Co at time 0 of the power option described above is given by Co = S exp [(r +0)T] (h+) Ke="To(h_), where log(So/VK) + (r + 302/2)T h+ OVT h_ = h+ 20VT, and (2) is the cumulative distribution function of the standard normal distribution. LE Hint: For full marks in parts (d), (e) and (f), your final answers should each be written in the most compact form possible. (d) By (partially) differentiating the expression for Co with respect to So, find the formula for the delta A of this power option. [. (e) By differentiating once again, find the formula for the gamma I. E aco (f) Finally, find the formula for the theta O, defined here as [ at
