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(1) The prices of three put options with strikes 40, 50, and 70, but otherwise identical, are $10, $20, and $30, respectively. (i) Show that the upward convexity of the price of the put with respect to the strike price is violated in this case; (ii) Find an arbitrage opportunity. (2) Show that xty 2 , 2 (3) Compute the integral of the function f(x,y) = x2 - 2y on the region bounded by the parabola y - (x+1)² and the line y - 5.x - 1. (4) Which is bigger, et or ITe? Provide the proof for your answer without use of calculators! (5) Compute 1 dac dy, D where D = + + 4 < 1 } (6) Recall that the vega of a plain vanilla European call is - 1 e 2 , where In ( + - T d1 = OVI (i) Show that S lim vega(S) = 0 and Jim vega(S) = 0. (ii) Show that, as a function of the spot price S of the underlying asset, vega(C) is first increasing and then decreasing. (iii) Find the spot price corresponding to the maximum value of vega(C). (7) (i) Find the value of a six months down-and-out call on a non-dividend-paying asset with price following a lognormal distribution with 30% volatility and spot price 40. The barrier is B - 35 and the strike for the call is K - 40. The risk-free interest rate is constant at 5%. 1 (ii) Show that the value of a down-and-out call with B < K converges to the value of a plain vanilla call with strike K, as B goes down to zero. For simplicity, assume that the underlying asset does not pay dividends and that interest rates are zero. (8) Consider a knock-out call option with barrier B and strike K with B - K < S(0) expiring at time T. Derive a formula for the value of this call option. Derive the gamma and vega of this option. (9) (i) A European asset-or-nothing call option has a payoff equal to the asset price, as long as the asset price exceeds the strike price K at maturity T. Use the assumptions behind the Black-Scholes model and the risk-neutral valuation to derive the formula for the value of such option.

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