a) Price a down-and-in call option with final payoff as follows: Payoff (T) = [max(ST -...

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a) Price a "down-and-in" call option with final payoff as follows: Payoff (T) = [max(ST - K,0); TH <T], where the event Ty is described as: (1) TH= inf{t>0: S, <H}, and H being the barrier level, K the strike price, S, the underlying price and T the maturity. In other words, the payoff equals that of a standard call option as long as the spot price S, is below (at least once) the barrier level H, otherwise 0. You know that the price of a standard call option on same underlying with same strike level and maturity worths $12.15 and the equivalent "down-and-out" call option worths $12.09. Note that a "down-and-out" option is a standard call option if event TH=inf{t>0: S₁ 2 H) occurs, otherwise 0. b) Draw a relationship between C(t) the price of a standard call option, Co(t) the price of a "down-and-in" call option and Code(t) the price of a "down-and-out" call option. Show graphically the following three scenarios and calculate the value of each option's final payoff. Based on the final payoff derive your relationship for C(t), Codi(t), Codo(t). • A scenario for S, starting at 100 and not cross the barrier level (H) of 70 until maturity T. • A scenario for S, starting at 100 and crossing the barrier level of 70 at T/3. Final value for ST is above strike price K = 100. . A scenario for S, starting at 100 and crossing the barrier level of 70 at T/2. Final value for Sr is below H. c) Can you derive a similar relationship for "up-and-out" and "up-and-in" call options with standard vanilla call options? Elaborate your answer. d) Price the following "knock-in, knock-out" option having a payoff at T equal to: ⚫0, if H₁ = 140 was not hit at any time between today and T, • 2 x max(ST - 120, 0), if H₁ was hit but H₂ = 160 is not hit in (to, T), max(ST 120, 0), if H₂ is hit. The following table provides "up-and-out" call option prices for different barrier levels, K = 120 and same maturity T as the option we want to price above. Barrier level Price 140 0.3986 150 1.0607 160 1.8897 170 2.7140 180 3.4286 00 5.3686 a) Price a "down-and-in" call option with final payoff as follows: Payoff (T) = [max(ST - K,0); TH <T], where the event Ty is described as: (1) TH= inf{t>0: S, <H}, and H being the barrier level, K the strike price, S, the underlying price and T the maturity. In other words, the payoff equals that of a standard call option as long as the spot price S, is below (at least once) the barrier level H, otherwise 0. You know that the price of a standard call option on same underlying with same strike level and maturity worths $12.15 and the equivalent "down-and-out" call option worths $12.09. Note that a "down-and-out" option is a standard call option if event TH=inf{t>0: S₁ 2 H) occurs, otherwise 0. b) Draw a relationship between C(t) the price of a standard call option, Co(t) the price of a "down-and-in" call option and Code(t) the price of a "down-and-out" call option. Show graphically the following three scenarios and calculate the value of each option's final payoff. Based on the final payoff derive your relationship for C(t), Codi(t), Codo(t). • A scenario for S, starting at 100 and not cross the barrier level (H) of 70 until maturity T. • A scenario for S, starting at 100 and crossing the barrier level of 70 at T/3. Final value for ST is above strike price K = 100. . A scenario for S, starting at 100 and crossing the barrier level of 70 at T/2. Final value for Sr is below H. c) Can you derive a similar relationship for "up-and-out" and "up-and-in" call options with standard vanilla call options? Elaborate your answer. d) Price the following "knock-in, knock-out" option having a payoff at T equal to: ⚫0, if H₁ = 140 was not hit at any time between today and T, • 2 x max(ST - 120, 0), if H₁ was hit but H₂ = 160 is not hit in (to, T), max(ST 120, 0), if H₂ is hit. The following table provides "up-and-out" call option prices for different barrier levels, K = 120 and same maturity T as the option we want to price above. Barrier level Price 140 0.3986 150 1.0607 160 1.8897 170 2.7140 180 3.4286 00 5.3686

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DownandIn Call Option Pricing and Relationships a b c a DownandIn Call Option Price Unfortunately due to the path dependency of the downandin option a... View the full answer