# hw3-3 - 48 2 Basic Options is a linear function of c for c...

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48 2 Basic Options is a linear function of c for c (3 . 2081 , 4 . 8784). From the data we know σ (3 . 2081) = 0 . 3 and σ (4 . 8784) = 0 . 4. Thus σ ( c ) = σ (3 . 2081) + σ (4 . 8784) - σ (3 . 2081) 4 . 8784 - 3 . 2081 · ( c - 3 . 2081) = 0 . 3 + 0 . 1 4 . 8784 - 3 . 2081 · ( c - 3 . 2081) . Consequently σ (4 . 5) = 0 . 3 + 0 . 1 4 . 8784 - 3 . 2081 · (4 . 5 - 3 . 2081) = 0 . 3773 . That is, the implied volatility is about 0.3773. 35. Consider a European option on a non-dividend-paying stock. The stock price is \$37, the exercise price is \$34, the risk-free interest rate is 5% per annum, the volatility is 30% per annum, and the time to maturity is six months. Find the call and put option prices by using the Black-Scholes formulae and verify that the put-call parity holds. Solution : For this case c ( S, t ) = 5 . 27 , p ( S, t ) = 1 . 43 and Ee - r ( T - t ) = 34 e - 0 . 05 · 0 . 5 = 33 . 16 . Thus c ( S, t ) - p ( S, t ) - S + Ee - r ( T - t ) = 5 . 27 - 1 . 43 - 37 + 33 . 16 = 0 . That is, the put-call parity relation holds. 36. Suppose that c ( S, t ) and p ( S, t ) are the prices of European call and put options with the same parameters respectively. Show the put-call parity c ( S, t ) - p ( S, t ) = Se - D 0 ( T - t ) - Ee - r ( T - t ) without using the Black-Scholes formulae. Solution : Let us look at the portfolio Π = Se - D 0 ( T - t ) - Ee - r ( T - t ) - c ( S, t ) + p ( S, t ) . The payo± for this portfolio at expiry is Π ( S, T ) = S - E - max( S - E, 0) + max( E - S, 0) = 0 . Therefore we have Π ( S, t ) = 0 for any t . As we know

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2 Basic Options 49 Π ( S, t ) = e - r ( T - t ) Z 0 [ S 0 - E - max( S 0 - E, 0) + max( E - S 0 , 0)] G ( S 0 , T ; S, t ) dS 0 = Se - D 0 ( T - t ) - Ee - r ( T - t ) - c ( S, t ) + p ( S, t ) . Consequently, from Π ( S, t ) = 0 we can have the put-call parity relation: c ( S, t ) - p ( S, t ) = Se - D 0 ( T - t ) - Ee - r ( T - t ) . 37. By using the put-call parity relation of European options
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## This note was uploaded on 02/11/2010 for the course MATH 6203 taught by Professor Zhu during the Spring '10 term at University of North Carolina Wilmington.

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hw3-3 - 48 2 Basic Options is a linear function of c for c...

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