The variable S is an investment asset providing income at the rate q measured in
ID: 2766273 • Letter: T
Question
The variable S is an investment asset providing income at the rate q measured in currency A and it follows the process: dS = mu(subscript S)Sdt + sigma(subscript S)Sdz In the real world. Defining new variables as necessary, give the process followed by S, and the corresponding market price of risk, in:
1.) A world that is the traditional risk-neutral world for Currency A
2.) A world that is the traditional risk-neutral world for Currency B
3.) A world that is forward risk-neutral world with respect to a zero-coupon currency A bond maturing at time T
4.) A world that is forward risk-neutral world with respect to a zero-coupon currency B bond maturing at time T
Explanation / Answer
1) In the traditional risk- neutral world the process followed by S is:
dS= ( r- q )S dt +?s S dz
Where r is the instantaneous risk-free rate.The market price of dz-risk is Zero.
2)In the traditional risk- neutral world for currency B the process is
dS= ( r- q +PQS?S?Q )S dt +?s S dz
Where Q is the exchange rate (units of A per unit of B),?Q is the volatility of Q and PQS is the coefficient of correlation between Q and S. The market price of dz- risk is PQS?Q.
3) In a world that is forward risk - neutral with respect to zero-coupon currency A maturing at time T
dS= ( r- q +?S?P)S dt +?s S dz
Where ?P is the bond price volatility. The market price of dz- risk is ?P.
4) In a world that is forward risk - neutral with respect to zero-coupon currency B maturing at time T
dS= ( r- q +?S?P+PFS?S?F)S dt +?s S dz
Where F is forward exchange rate, ?F is the volatility of F (units of A per unit of B, and PFS is the correlation between F and S. The market price of dz- risk is ?P +PFS?F..
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