# Assume the Black-Scholes framework. Consider two stocks, each of which pays dividends continuously at a rate proportional

## Question:

Assume the Black-Scholes framework. Consider two stocks, each of which pays dividends continuously at a rate proportional to its price. For j = 1, 2 and t ≥ 0, let S_{j} (t) be the time-t price of one share of stock j.

You are given:

(i) S_{1}(0) = S_{2}(0) = 200.

(ii) Stock 1 and Stock 2 share the same dividend yield of 5%.

(iii) Stock 1’s volatility is 30%.

(iv) Stock 2’s volatility is 20%.

Consider a 6-month European exchange option to exchange Stock 2 for Stock 1.

(a) Calculate the price of the exchange option when ρ = 0.

(b) Calculate the price of the exchange option when ρ = 0.7.

(c) In the light of the results in parts (a) and (b), describe how the price of an exchange option behaves as a function of the correlation ρ.

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