Option strategies. Stock Price Payoff Profit. The butterfly spread leads to a loss when the final stock price is greater than $64 or less than $56.

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1 Option strategies Problem Three put options on a stock have the same expiration date and strike prices of $55, $60, and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy. For what range of stock prices would the butterfly spread lead to a loss? A butterfly spread is created by buying the $55 put, buying the $65 put and selling two of the $60 puts. This costs $ 1 initially. The following table shows the profit/loss from the strategy. Stock Price Payoff Profit S T S T ST 64 ST 55 S T 60 ST 55 ST 56 S T The butterfly spread leads to a loss when the final stock price is greater than $64 or less than $56. Problem What trading position is created from a long strangle and a short straddle when both have the same time to maturity? Assume that the strike price in the straddle is halfway between the two strike prices of the strangle. A butterfly spread (together with a cash position) is created. Problem (Excel file) Describe the trading position created in which a call option is bought with strike price K 1 and a put option is sold with strike price K 2 when both have the same time to maturity and K 2 > K 1. What does the position become when K 1 = K 2? The position is as shown in the diagram below (for K 1 = 25 and K 2 = 35). It is known as a range forward and is discussed further in Chapter 16. When K 1 =K 2, the position becomes a regular long forward.

2 Figure S11.6 Trading position in Problem 11.25

3 RNV and option pricing Problem A stock price is currently $50. It is known that at the end of six months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the value of a six-month European call option on the stock with an exercise price of $48. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers. At the end of six months the value of the option will be either $12 (if the stock price is $60) or $0 (if the stock price is $42). Consider a portfolio consisting of: shares 1 option The value of the portfolio is either 42 or in six months. If i.e., the value of the portfolio is certain to be 28. For this value of the portfolio is therefore riskless. The current value of the portfolio is: f where f is the value of the option. Since the portfolio must earn the risk-free rate of interest ( f) e 28 i.e., f 6 96 The value of the option is therefore $6.96. This can also be calculated using risk-neutral valuation. Suppose that p is the probability of an upward stock price movement in a risk-neutral world. We must have p 42(1 p) 50 e i.e., 18p or: p The expected value of the option in a risk-neutral world is: This has a present value of e 6 96 Hence the above answer is consistent with risk-neutral valuation. Problem Footnote 1 shows that the correct discount rate to use for the real world expected payoff in the case of the call option considered in Figure 12.1 is 42.6%. Show that if the option is a put rather than a call the discount rate is 52.5%. Explain why the two real-world discount rates are so different. The value of the put option is

4 ( ) e The expected payoff in the real world is ( ) The discount rate R that should be used in the real world is therefore given by solving 0 25R e The solution to this is or -52.5%. The underlying stock has positive systematic risk as its expected return is higher than the risk free rate. This means that the stock will tend to do well when the market does well. The call option has a high positive systematic risk because it is a leveraged position in the underlying asset and therefore it tends to do very well when the market does well. As a result a high discount rate is appropriate for its expected payoff. The put option is in the opposite position. It tends to provide a high return when the market does badly. As a result it is appropriate to use a highly negative discount rate for its expected payoff. Problem A futures price is currently 40. It is known that at the end of three months the price will be either 35 or 45. What is the value of a three-month European call option on the futures with a strike price of 42 if the risk-free interest rate is 7% per annum? In this case u and d The risk-neutral probability of an up move is (1 875) ( ) 0 5 The value of the option is e 25 [ ] 1 474

5 Option dynamic replication and hedging Problem A financial institution has the following portfolio of over-the-counter options on sterling: Type Position Delta of Option Gamma of Option Vega of Option Call 1, Call Put 2, Call A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8. (a) What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral? (b) What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral? The delta of the portfolio is ( 0 40) The gamma of the portfolio is The vega of the portfolio is (a) A long position in 4,000 traded options will give a gamma-neutral portfolio since the long position has a gamma of The delta of the whole portfolio (including traded options) is then: Hence, in addition to the 4,000 traded options, a short position of 1,950 in sterling is necessary so that the portfolio is both gamma and delta neutral. (b) A long position in 5,000 traded options will give a vega-neutral portfolio since the long position has a vega of The delta of the whole portfolio (including traded options) is then Hence, in addition to the 5,000 traded options, a short position of 2,550 in sterling is necessary so that the portfolio is both vega and delta neutral. Problem Consider again the situation in Problem Suppose that a second traded option with a delta of 0.1, a gamma of 0.5, and a vega of 0.6 is available. How could the portfolio be made delta, gamma, and vega neutral? Let w 1 be the position in the first traded option and w 2 be the position in the second traded option. We require: w 0 5w 1 2

6 w 0 6w 1 2 The solution to these equations can easily be seen to be w , w The whole portfolio then has a delta of Therefore the portfolio can be made delta, gamma and vega neutral by taking a long position in 3,200 of the first traded option, a long position in 2,400 of the second traded option and a short position of 1,710 in sterling. Problem (Excel file) Use the DerivaGem Application Builder functions to reproduce Table (Note that in Table 18.2 the stock position is rounded to the nearest 100 shares.) Calculate the gamma and theta of the position each week. Calculate the change in the value of the portfolio each week and check whether equation (18.3) is approximately satisfied. (Note: DerivaGem produces a value of theta per calendar day. The theta in equation (18.3) is per year. ) Consider the first week. The portfolio consists of a short position in 100,000 options and a long position in 52,200 shares. The value of the option changes from $240,053 at the beginning of the week to $188,760 at the end of the week for a gain of $51,293. The value of the shares changes from $ to $ for a loss of $45,936. The net gain is $ The gamma and theta (per year) of the portfolio are and 430,533 so that equation (18.3) predicts the gain as ( ) The results for all 20 weeks are shown in the following table which shows that, on average, the differential equation holds quite closely. Week Actual Gain Predicted Gain 1 5,357 5, ,689 6, ,742 21, ,941 1, ,706 3, ,320 9, ,249 5, ,491 9, ,380 18, ,643 2, ,645 1, ,365 10, ,876 3, ,936 12, ,566 8, ,880 2, ,764 6, ,295 5, ,806 4,805 Average 5,921 5,945

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