Question

Company NYY’s stock is trading at $27. The annual standard deviation of stock returns is

15%. The annual effective interest rate is 2%. Assuming no active option markets, an investor

can replicate the payoffs of a call option with $24 exercise price by:

Answer 1

The following steps show the calculations of a replicating portfolio based on the Black Scholes model (Formula view of the spreadsheet is provided in next screenshot):

А D E F G H I Given data: Spot price, s $ 27.00 Standard deviation (0) 15% Annual effective rate (r) 2% Strike price, K $ 24.00 t (time to expiry) = Time to expiry is not specified so let us replicate the call option having maturity in 1 year We can use the Black Scholes formula for the replication portfolio by calculating di & d2 as In + (r + o)t dı = " - Os Vt and d2 = dı - 03 Vt 0.9935536 d2 0.8435536 1 Now the replicating portfolio can be created by following 2 steps: Buy N(d1) shares of the stock. Borrow K*e^(-rt) N(d2) amount of money 2 Here N(x) is the cumulative distribution function of the standard normal distribution Shares to buy = N(d1)= 0.8398 Borrow money= $ 18.83 This is the required replicating portfolio

So the replication includes borrowing $18.83 and buying 0.8398 shares of NYY. This will have the same payoff as a call option.

Formula view:

Given data: Spot price, s Standard deviation (0) Annual effective rate (r) Strike price, K t (time to expiry) = Time to expiry is not specified so let us replicate the call option having maturity in 1 year 27 0.15 0.02 1 We can use the Black Scholes formula for the replication portfolio by calculating di & d2 as t di = In x + (r + Os vt and d2 = di -os vt =(LN(D4/D7)+(D6+(D5^2)/2)*D8)/(D5*SQRT(D8)) d2 =D22-D5*SQRT(D8) Now the replicating portfolio can be created by following 2 steps: Buy N(d1) shares of the stock. Borrow K*e^(-rt) N(02) amount of money Here N(x) is the cumulative distribution function of the standard normal distribution Shares to buy = N(d1)= Borrow money= This is the required replicating portfolio =NORM.S.DIST(D22, TRUE) =D7*EXP(-D6*D8) *NORM.S.DIST(D24,TRUE)

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