Assume that we are in the Black-Scholes world where am) 2 ,uS(t)dt + oS(t)dX(t) 0’3 = 5.3161: and choose the stock as the numeraire asset. (a) If the…

Black scholes, stock and bond, European call, numeraire

Assume that we are in the Black-Scholes world where am) 2 ,uS(t)dt + oS(t)dX(t)0’3 = 5.3161: and choose the stock as the numeraire asset. (a) If the stock is the numeraire asset, what is true about the process followed by the ratio of thestock price to the bond price, 51/6}? (b) By using your answer to part (a), Ito’s lemma, and Girsanov’s theorem, determine thestochastic process followed by 5′; when S; is the numeraire. (c) Now consider a European call option, Vt. Write down the value of the option in terms of anexpectation when S; is the numeraire. ((1) Using your answers to parts (b) and (0) write V; as the sum of two integrals and solve theintegral with S in it. Was this easier than evaluating this term when fit was numeraire‘?

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