dstefan
06-14-2008, 04:40 PM
Chapter 3, Problem 18
You buy 1000 six months ATM Call options on a non-dividend-paying asset with spot price 100, following a lognormal process with volatility 30%. Assume the interest rates are constant at 5%.
(i) How much money do you pay for the options?
(ii) What Delta--hedging position do you have to take?
(iii) On the next trading day, the asset opens at 98. What is the value of your position (the option and shares position)?
(iv) Had you not Delta-hedged, how much would you have lost due to the increase in the price of the asset?
Solution:
(i) Using the Black--Scholes formula with input S_1=K=100, T=1/2, \sigma=0.3, r=0.05, q=0, we find that the value of one call option is C_1 = 9.634870, and therefore $9634.87 must be paid for 1000 calls.
(ii) The Delta-hedging position for long 1000 calls is short 1000 \Delta(C) = 1000 e^{-q T} N(d_1) = 588.59 units of the underlying. Therefore, 589 units of the underlying must be shorted.
(iii) The new spot price and maturity of the option are S_2=98 and T_2=125/252 (there are 252 trading days in one year). The price of the call option is $8.453134 and the value of the portfolio is 1000 C_2 - 589 S_2 ~=~ -49268.876.
(iv) If the long call position is not Delta-hedged, the loss incurred due to the decrease in the spot price of the underlying asset is
1000(C_2-C_1) ~=~ -\$1181.74
For the Delta--hedged portfolio, the loss incurred is
(1000 C_2 - 589 S_2) - (1000 C_1 - 589 S_1) ~=~ - \$3.74
You buy 1000 six months ATM Call options on a non-dividend-paying asset with spot price 100, following a lognormal process with volatility 30%. Assume the interest rates are constant at 5%.
(i) How much money do you pay for the options?
(ii) What Delta--hedging position do you have to take?
(iii) On the next trading day, the asset opens at 98. What is the value of your position (the option and shares position)?
(iv) Had you not Delta-hedged, how much would you have lost due to the increase in the price of the asset?
Solution:
(i) Using the Black--Scholes formula with input S_1=K=100, T=1/2, \sigma=0.3, r=0.05, q=0, we find that the value of one call option is C_1 = 9.634870, and therefore $9634.87 must be paid for 1000 calls.
(ii) The Delta-hedging position for long 1000 calls is short 1000 \Delta(C) = 1000 e^{-q T} N(d_1) = 588.59 units of the underlying. Therefore, 589 units of the underlying must be shorted.
(iii) The new spot price and maturity of the option are S_2=98 and T_2=125/252 (there are 252 trading days in one year). The price of the call option is $8.453134 and the value of the portfolio is 1000 C_2 - 589 S_2 ~=~ -49268.876.
(iv) If the long call position is not Delta-hedged, the loss incurred due to the decrease in the spot price of the underlying asset is
1000(C_2-C_1) ~=~ -\$1181.74
For the Delta--hedged portfolio, the loss incurred is
(1000 C_2 - 589 S_2) - (1000 C_1 - 589 S_1) ~=~ - \$3.74