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Question 1. A European Logarithmic Option is a type of derivative contract where the payoff depends on the natural logarithm of the underlying asset at expiry. Consider a European Logarithmic Option which has a payoff defined by V (ST) - In(S/E) (1) where E is the given strike price. Determine the fair pricing formula for a European Logarithmic Option by looking for a solution of Black-Scholes Partial Differential Equation of the form V (S. 1) = f () In(S) + S2 () Question 2. A digital Call, Ca is an option that is equivalent to a straight bet on an asset price, S. It pays Il at expiry time, T, if the value of the given asset is greater than or equal to the the strike price E, but expires worthless if S E. (a) Draw payoff diagrams for the digital Call and Put. Assuming that a digital Put and digital Call both share the same expiry date T, use a simple financial argument to derive the Put-Call parity relation. [B Marks) (b) By drawing payoff diagrams, show that the Delta, A, for a European Vanilla Call option is the same as the payoff for a digital Call. (3 Marks (c) By differentiating Black Scholes partial differential equation with respect to S and subsituting for A = S, show that the value of the digital Call written on an underlying asset which pays no dividend is equal to the Delta of a European Vanilla Call on an underlying that pays a continuous dividend yield q' and has an interest rater. Determine q* and r. 17 Marks 12 Marks (d) Briefly explain why an investor would buy a digital option. Question 3. 1 Consider the Black-Scholes equation for an option, in the usual notation av av at +rS as - PV = 0. where the volatility o and the interest rater are both constants. (a) By using the substitution V (S.t) = svi (S.t), where a = 1 - 2r/o?, find the equation satisfied by Vi (S.). 15 Marks + 2v as2 (2) (b) By using the further substitution Vi (S. t) = V2 (6.6), where E. (a) Draw payoff diagrams for the digital Call and Put. Assuming that a digital Put and digital Call both share the same expiry date T, use a simple financial argument to derive the Put-Call parity relation. [B Marks) (b) By drawing payoff diagrams, show that the Delta, A, for a European Vanilla Call option is the same as the payoff for a digital Call. (3 Marks (c) By differentiating Black Scholes partial differential equation with respect to S and subsituting for A = S, show that the value of the digital Call written on an underlying asset which pays no dividend is equal to the Delta of a European Vanilla Call on an underlying that pays a continuous dividend yield q' and has an interest rater. Determine q* and r. 17 Marks 12 Marks (d) Briefly explain why an investor would buy a digital option. Question 3. 1 Consider the Black-Scholes equation for an option, in the usual notation av av at +rS as - PV = 0. where the volatility o and the interest rater are both constants. (a) By using the substitution V (S.t) = svi (S.t), where a = 1 - 2r/o?, find the equation satisfied by Vi (S.). 15 Marks + 2v as2 (2) (b) By using the further substitution Vi (S. t) = V2 (6.6), where

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