Question

Stock XYZ is currently trading at $50. A JUN 60 call option on the stock is quoted at $12.65. The delta on the call is 0.5866, and its gamma is 0.0085. At the same time, a JUN 65 call option on this stock is quoted at $17.75 with a delta of 0.5432 and gamma of 0.0089. You currently have a short position on the JUN 60 call for 60 contracts. The risk-free rate is 5%. (1) Construct a hedging strategy according to both delta and gamma hedging (delta neutral and gamma neutral at the same time). Please specific how many stocks and calls you buy/sell? (2) How much initial investment do you need to construct this strategy? (3) The following date, the stock closes at $52. The JUN 60 call is trading at 14.55, while the JUN 65 call is trading at $19.65. How much is your investment now worth? How much you would end up with if you had invested all the money in a risk-free bond in the very beginning? (Assuming continuous compounding).

Answer 1

Part (1)

Let's add A number of stocks and B number of June 65 call option to the existing portfolio of short 60 contracts of Jun 60 call.

Hence the portfolio, P = AS + BC65 - 60C60

ΔP = AΔS + BΔC65 - 60ΔC60 = A x 1 + B x 0.5432 - 60 x 0.5866 = A + 0.5432B - 35.1960 = 0 as the portfolio is delta neutral

Hence, A + 0.5432B - 35.1960 = 0 ----------------- Eqn (1)

The portfolio is gamma neutral also,

Hence, Gamma of the portfolio = A x Gamma of S + B x gamma of C65 - 60 x Gamma of C60 = A x 0 + B x 0.0089 - 60 x 0.0085 = 0.0089B - 0.51 = 0

Hence, B = 0.51 / 0.0089 = 57.30

From eqn (1): A = 35.1960 - 0.5432B = 35.1960 - 0.5432 x 57.30 = 4.07

Hence, as part of the hedging strategy:

• Buy 4.07 nos. of stock
• Buy 57.30 number of Jun 65 call option

Part (2)

Initial investment needed = A x S0 + B x C65 price today = 4.07 x 50 + 57.30 x 17.75 = $ 1,220.58

Part (3)

Worth of investment now = A x S + B x C65 price today = 4.07 x 52 + 57.30 x 19.65 = $ 1,337.59

If you had invested all the money in a risk-free bond in the very beginning, you would end up with = Initial investment x ert = 1,220.58 x e5% x 3/12 = $ 1,235.93