Question

(Delta-Hedge / No Rebalancing) Suppose a stock price is $50, a call option has a strike price of $50 and the call’s market price is $4. A dealer sells 10 call option contracts (for 1000 option-shares).   The original Delta is .55

(a) What does our basic hedging logic say is the Dealers’ real risk and what should be generally done.

(b) To start a Delta Hedge, what should the dealer do NOW, and what should it cost ? (Hint-550 shares).

(c) After (b), if there is an immediate $0.50 rise in the stock price, what is the exact “delta expected” change in the dealer’s value of her “long shares + written calls”, and the combined overall change.

(d) Assume the dealer hedges as indicated in (b and c), BUT, the ACTUAL change in the option price is $0.30, to $4.30, not to the $4.275 you should have computed in (c). What is the exact ACTUAL change in the dealer’s value of her “long shares + written calls”, and the combined overall change.

(e) Again assume (b, not c), but the stock price rises by $5.00 to $55. The Non-ReBalanced delta hedge has a problem. Explain with logic and perhaps graph illustration what might happen due to the Gamma problem. An exact “combined overall change” is not needed.

Answer 1

a) Delta is the percentage change isn value of option divided by the percentage change in value of the underlying.In the above question, the dealer sells its call options, therefore he is doing a C-. C- means selling call option. to hedge the option we should so S+ or buy Stocks to hedge the current position.Our existing position is C-, to hedge it we must do S+. Because he is expecting that in future price of stock will not Rise,for this assumption he is doin C- but for hedging his current position he should buy stocks so that in future if his assumption goes wrong, he could compensate it by the stock which he has bought to hedge himself.

b) Given delta is 0.55. Therefore no. of stocks required to hedge its current position is equal to 0.55*1000 = 550 shares.Therefore he should long on 550 shares, given the stock price = $50. His total cost will be = 550*50 = $27500.

c) Given stock price rises by $0.50, therefore the price of call will also rise. Previously, delta euals to 0.55.After increase in price of the stock the delta will remain the same because the increase in stock price, increases the call option price therefore the current delta should be change in option price divided by change in price of stock. Given stock price increased by $0.50, the price of call option should increase by $.0275 (by backcalculating the call option value i.e. 0.55*0.5). NOW, if the dealer wants to hedge its position, the cost will be 550 shares* $50.5 = $27775.

d) Given that the stock value rises by $0.30, and the stock price by $0.50, therefore new delta equal to 0.30/.0.50 = 0.60. Now, the delta has increased therefore we nee more share to hedge our position. To hedge we need 1000*0.6 = 600 shares i.e. 50 more shares (50 S+) to rebalance it.

e) Gamma is negative for short options. therefore here Gamma is negative. Gamma illustrate thechange in delta divided by change in price if stock. Gamma for short call option ranges from -1 to 0.