Exercises to solve mathematically and financially without software: Part 1 3. Value a derivative that, up to the price of the underlying asset 55+62 pays at maturity as a call with exercise price E, and for larger price of the underlying, pays as a put with exercise price E2> E. 6. A former MIF student, very inclined to mathematics, designs an option with a bilateral barrier in the E/2 and 2E prices of the underlying (E being a fixed price) and with pay- off, in case the derivative survives until maturity T. given by V(s) = st* sin(2.4" In S-In E + In 2 In 4 With k-2r/c Value this derivative. Hint. Find a solution of separate variables. It must be said that the product will probably not be successful despite its "mathematical simplicity" 9. Let us consider the Black-Scholes equation for the value V (S1, S2, 1) of a derivative on two price assets S and S. volatilities of and 02 and correlation p that do not pay dividends: ve + as? V55, + {s} V55, +p0;629752V5,5, +rs,Vs, + rSzVs. - "V = 0. Value a "best-of-call" option that has a maturity pay-off consisting of the better of the two underlying assets: V(S1, S2,T) = max(S1,S2) assuming they are not correlated. Hint Find a solution to the equation and the final condition of the form V(S.S,t) = SW wer) prove that W (5, 1) satisfies the Black-Scholes equation with zero interest rate and volatility or a + oz and with final condition WE, 7) = max (, 1) which can be written as the sum of a call with exercise price 1 plus cash by value 1. 12. Value a European call "down and in, that is, a derivative that becomes a call option with exercise price E when (and only if the price of the underlying falls below a boundary X that we assume less than E, while it is worth 0 at maturity if the X barrier has not been crossed during the entire life of the product. Thus, it is the same as the vanilla call for S = X. For the same reason, only the region S > X needs to be considered. Exercises to solve mathematically and financially without software: Part 1 3. Value a derivative that, up to the price of the underlying asset 55+62 pays at maturity as a call with exercise price E, and for larger price of the underlying, pays as a put with exercise price E2> E. 6. A former MIF student, very inclined to mathematics, designs an option with a bilateral barrier in the E/2 and 2E prices of the underlying (E being a fixed price) and with pay- off, in case the derivative survives until maturity T. given by V(s) = st* sin(2.4" In S-In E + In 2 In 4 With k-2r/c Value this derivative. Hint. Find a solution of separate variables. It must be said that the product will probably not be successful despite its "mathematical simplicity" 9. Let us consider the Black-Scholes equation for the value V (S1, S2, 1) of a derivative on two price assets S and S. volatilities of and 02 and correlation p that do not pay dividends: ve + as? V55, + {s} V55, +p0;629752V5,5, +rs,Vs, + rSzVs. - "V = 0. Value a "best-of-call" option that has a maturity pay-off consisting of the better of the two underlying assets: V(S1, S2,T) = max(S1,S2) assuming they are not correlated. Hint Find a solution to the equation and the final condition of the form V(S.S,t) = SW wer) prove that W (5, 1) satisfies the Black-Scholes equation with zero interest rate and volatility or a + oz and with final condition WE, 7) = max (, 1) which can be written as the sum of a call with exercise price 1 plus cash by value 1. 12. Value a European call "down and in, that is, a derivative that becomes a call option with exercise price E when (and only if the price of the underlying falls below a boundary X that we assume less than E, while it is worth 0 at maturity if the X barrier has not been crossed during the entire life of the product. Thus, it is the same as the vanilla call for S = X. For the same reason, only the region S > X needs to be considered