Question

In: Finance

Stock XYZ is currently trading at $50. A JUN 60 call option on the stock is...

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).

Solutions

Expert Solution

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


Related Solutions

Assume a stock is currently trading at $150. One call option has a strike price of...
Assume a stock is currently trading at $150. One call option has a strike price of $145 and costs $10, and another call option has a strike price of $155 and costs $5. Show the gross and net pay-off table of a Bull spread on call option, and show the graph of the net payoff diagram. Hint: Consider the three possible stock positions: S ≤ K1,K1 < S < K2, S ≥ K2.
Consider a stock (XYZ Corporation) currently trading at $50, with annualized volatility of 65%. The continuously...
Consider a stock (XYZ Corporation) currently trading at $50, with annualized volatility of 65%. The continuously compounded annualized interest rate is 4%.   The stock does not pay dividends. a)     Fill in the following table of data for 1 year options on XYZ (based on the Black-Scholes Model, representing options on 1 share of stock). Indicate what source you used for the data, so that I can double check if something looks funny: Contract Strike   Price Call 45 Put   45 Call   50...
X Stock is currently trading for $32.80. If its January $35.00 strike call option has a...
X Stock is currently trading for $32.80. If its January $35.00 strike call option has a $1.50 premium and you want to sell 5 contracts, briefly explain what that entitles or potentially requires you to do. Is the contract in or out-of-the-money? Also, what would your dollar maximum gain, maximum loss, and break-even price (market price) be with this strategy (assume you don’t own HDS…and you hold the contract to expiration)? Last, briefly explain why this $1.50 premium is relative...
The put option of Joe Inc. is currently trading at $2.50 while the call option premium...
The put option of Joe Inc. is currently trading at $2.50 while the call option premium is $7.50. Both the put and the call have an exercise price of $25. Joe Inc. stock is currently trading at $32.25 and the risk free rate is 3%. The options will expire in one month. I. An investor applies a protective put strategy by buying the put option of Joe Inc. to protect his holding of the company’s stock. This strategy creates a...
You bought a call option with a strike price of $60. The underlying asset is trading...
You bought a call option with a strike price of $60. The underlying asset is trading for $53 and you paid a premium of $14. What is the most you could lose form this strategy?
7. An XYZ OCT 30 call option is trading at a premium of 2 and 1/2....
7. An XYZ OCT 30 call option is trading at a premium of 2 and 1/2. If XYZ is trading at 28, the option at which two of the following properties? An intrinsic value of 2 An Intrinsic value of 0 A time value of 1 A time value of 2 and 1/2. I and II I and III II and III II and IV 11. A customer who is losing an XYZ put may be intended to do which...
A stock that does not pay dividend is trading at $20. A European call option with...
A stock that does not pay dividend is trading at $20. A European call option with strike price of $15 and maturing in one year is trading at $6. An American call option with strike price of $15 and maturing in one year is trading at $8. You can borrow or lend money at any time at risk-free rate of 5% per annum with continuous compounding. Devise an arbitrage strategy.
Stock trades at $50. An at the money call option trades at $4. The maturity of...
Stock trades at $50. An at the money call option trades at $4. The maturity of the option is two years. The present value of the strike price over the two years is $45. The stock pays a dividend of 3 dollars in one year and no other dividends prior to expiration. The present value of the dividend is $2. The price of the European call option is $5. The price of the European put is $1.0. Construct a strategy...
Suppose that a March call option on a stock with a strike price of $50 costs...
Suppose that a March call option on a stock with a strike price of $50 costs $2.50 and is held until March. Under what circumstances will the option be exercised? Under what circumstances will the holder of the option make a gain? Under what circumstances will the seller of the option make a gain? What is the maximal gain that the seller of the option can make? Under what circumstances will the seller of the option make the maximal gain?
A European put option on the common stock of XYZ Ltd. is currently selling for a...
A European put option on the common stock of XYZ Ltd. is currently selling for a price of $2 per share. The expiration date of the put is 3 months from now. The relevant interest rate is 4% per annum. The exercise price of the put $30 per share and the size of one put is 100 shares. Suppose Ms. Parker sells 20 of these puts now, find her profit (loss), assuming the spot price of the stock on the...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT