A major drawback of this model is that the interest

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(3) It is a famous model to model stochastic interest rates, called “Vasicek Model”. A major drawback of this model is that the interest rate can take negative values with a very small probability. Which property of the riskless asset is not satisfied if the interest rate can take negative values? Answer: The property not satisfied is that e t integraltext 0 r s ds must always be increasing. That is, the bank account should be increasing with respect to time (see first page of the lecture notes on multi- period). But the probability to get a negative interest rate is low, so that this model is in fact well accepted in the marketplace. Problem 4. Solution: (1) Applying Ito’s lemma to the function log( S t ) yields: d log S t = 1 S t dS t + 1 2 - 1 S 2 t ( dS t ) 2 = dS t S t - 1 2 parenleftbigg dS t S t parenrightbigg 2 = rdt + σdW t - 1 2 σ 2 dt = ( r - σ 2 2 ) dt + σdW t Taking the integral of both sides from 0 to t gives: t integraldisplay 0 d log S s = t integraldisplay 0 ( r - σ 2 2 ) ds + t integraldisplay 0 σdW s log S t - log S 0 = ( r - σ 2 2 )( t - 0) + σ ( W t - W 0 ) log S t S 0 = ( r - σ 2 2 ) t + σW t S t = S 0 e ( r - σ 2 2 ) t + σW t (2) E [ S T ] = E [ S 0 e ( r - σ 2 2 ) T + σW T ] = S 0 e rT e - σ 2 2 T E [ e σW T ] = S 0 e rT e - σ 2 2 T e σ 2 2 T = S 0 e rT
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because σW T is a gaussian N (0 2 T ). Then e σW T is lognormal. (3) E [ S 1 / 2 T ] = E [ S 1 / 2 0 e ( r - σ 2 2 ) T 2 + σ 2 Z T ] = S 1 / 2 0 e rT/ 2 e - σ 2 4 T E [ e σ Z T 2 ] = S 1 / 2 0 e rT/ 2 e - σ 2 T/ 4 e σ 2 T/ 8 = S 1 / 2 0 e rT/ 2 e - σ 2 T/ 8 because e σ W T 2 is a lognormal LN (0 , σ 2 T 4 ). (4) E [log S T ] = log S 0 + ( r - σ 2 2 ) T + σE [ W T ] = log S 0 + ( r - σ 2 2 ) T (5) Use the question (1) with T/ 2.
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