VIII ARBITRAGE AND HEDGING WITH OPTIONS 8 1

VIII. ARBITRAGE AND HEDGING WITH OPTIONS

8. 1. Derivative Securities Markets and Hedging • As we discussed earlier, a derivative security is simply a financial instrument whose value is derived from that of another security, financial index or rate.

8. 2. Put-Call Parity • Call and put Payoff functions: p. T = MAX[0, X – ST] • Subtract and re-write: c. T - p. T = MAX[0, ST - X] - MAX[0, X – ST] = ST - X p. T = c. T + X – ST

The Collar • Collar underlying to protect downside, give up upside to finance: ST +p. T -c. T = ST +MAX[0, X–ST] - MAX[0, ST -X] = X e. g. , if X = S 0: ST +p. T -c. T = ST +MAX[0, S 0–ST] - MAX[0, ST - S 0] = S 0 • Zero Cost Collar: S 0 +p 0 –c 0 = p 0(Xp, S 0, T, σ, rf ) - c 0(Xc, S 0, T, σ, rf ) = 0 • Select Xp and Xc so that p 0 = c 0

8. 3. Options and Hedging in a Binomial Environment • The Binomial Option Pricing Model is based on the assumption that the underlying stock follows a binomial return generating process. • This means that for any period during the life of the option, the stock's value will be only one of two potential constant values.

One-Time-Period Binomial Model

Valuing the One-Period Option

Extending the Binomial Model to Two Periods • First, we substitute for the hedge ratio: • Some algebra then substitute hedging probabilities:

Two-Time-Period Binomial Model

Two Time Periods The hedge ratio for time zero is -. 75 and the hedge ratio in time one is either -. 1875 or -1, depending on whether the share price increases or decreases in the first period

Extending the Binomial Model to n Time Periods

Illustration: Three Time Periods

Three-Time-Period Binomial Model

Convergence of the Binomial Model to the Black-Scholes Model

Obtaining Multiplicative Upward and Downward Movement Values • One difficulty in applying the binomial model is obtaining estimates for u and d that are required for p; all other inputs are normally quite easily obtained.

8. 4. The Greeks and Hedging in a Black-Scholes Environment

Black Scholes Illustration T =. 5 X = 80 =. 4 rf =. 10 2 =. 16 S 0 = 75

Greeks Calculation

Put Sensitivities

Portfolio Delta Neutrality • Suppose that we wish to create a delta-neutral portfolio. • Based on our calculations, we can do so with a single call with X = 80 and shorting. 536 shares of stock. • Or, we could do so by shorting a single share and buying 1/. 536 calls with X = 80. • Of course, these portfolios would have to be continually rebalanced because delta is continually changing.

Delta and Gamma Neutrality • In the example above, we set a delta-neutral portfolio with. 536 shares of stock and a single call with X = 80. • Now, suppose that we wish to create a delta-gammaneutral portfolio. • Same example as above, but add a second call with X = 75 in order to create a delta-gamma neutral portfolio. • Obviously, the delta of the stock is 1, the delta of the call with X = 80 is. 536 and the delta of the call with X = 75 is. 626. • The gamma of the stock is 0, the gamma of the call with X = 80 is. 0187 and the gamma of the call with X = 75 is. 0178.

Delta-Gamma Neutral Portfolio • Thus, if we were to long one share, and with portfolio delta and gamma equal to zero, we solve for option holdings as follows:

8. 5. Exchange Options Suppose that the prices S 1 and S 2 of two assets follow geometric Brownian motion processes with 1 and 2 as standard deviations of logarithms of price relatives (or returns) for each of the two securities: And the variance of logarithms of price relatives of the two assets relative to one another is 2: where E[d. Z 1 d. Z 2] = 1, 2 dt, where 1, 2 is the correlation coefficient between logarithms of price relatives ln(S 1/S 2) between the two securities.

Currency Options • Interest rates may differ between the foreign and domestic countries. • We quote two interest rates r(f) and r(d), one each for the foreign and domestic currencies.

Currency Option Valuation • Garman and Köhlagen [1983] and Grabbe [1983] currency option pricing model:

Currency Option Illustration • Consider the following example where call options are traded on Swiss francs (CHF). • One U. S. dollar is currently worth 1. 8 Swiss francs (s 0 =. 5556). • We wish to evaluate a 6 -month European call and put on CHF with exercise prices equal to X =. 5556. • U. S. and Swiss interest rates are both. 10. • The annual standard deviation of exchange rates is. 40.

Currency Option Illustration, Continued

8. 6. Hedging Exchange Exposure with Currency Options • Suppose that the Dayton Company of America (See Chapter 7) is considering hedging a payoff of £ 1, 000 in three months, which it intends to convert to dollars. Further assume: – – – Spot exchange rate: $1. 7640/£ Three month forward exchange rate: $1. 7540/£ U. K. Borrowing interest rate: 10. 0% U. S. Borrowing interest rate: 8. 0% U. K. Lending interest rate: 8. 0% U. S. Lending interest rate: 6. 0 %

Hedging with FX Derivatives • There are call and put options and forward contracts with the following terms: – – – Term to options expiration: 3 months Exercise price: $1. 75 per £ Put Premium: $0. 025 per £ Call Premium: $. 065 per £ Brokerage cost per options contract on £ 31, 250: $50

The Put Hedge Strategy • Here, the company will purchase 3 -month put options on £ 1, 000 with an exercise price of $1. 75/£ with a total premium of $25, 000. • Time zero brokerage costs total $1, 600 (32 contracts at $50 per contract). • Thus the total time zero cash outlay is $26, 600. • Forgone interest on the sum of the premium and brokerage costs totals $399. • Expressed in terms of future value, the total cash outlay is $26, 999. The result of this strategy is that the firm receives one of the following in three months: 1. An unlimited maximum less the $26, 999 premium and brokerage fees. The dollar value of this strategy increases as the value of the dollar drops against the pound. Since cash flows are not certain, this hedge is considered partial. 2. A minimum of $1, 750, 000 less $26, 999 for a net of $1, 723, 001. This minimum value to be received may be unacceptably low; however, there is upside cash flow potential.

The Conversion Strategy (The Call and Put Hedge) • Writing calls with an exercise price of $1. 75 expiring in 3 months. • Buy puts with the same terms (The collar). The time zero cash flows are summarized as follows: Put Premium. . . . - $25, 000 Call Premium. . . . + $65, 000 Put brokerage fee. - $ 1, 600 Call brokerage fee. - $ 1, 600 Net Time zero cash flows + $36, 800 • Interest earned on the net time zero outlay is $552. • If the three month exchange rate is less than $1. 75/£, the exchange rate of $1. 75/£ is locked in by the put. • If the exchange rate exceeds $1. 75/£, the obligation incurred by the short position in the call is exercised. • Thus, the firm's exchange rate of $1. 75/£ is locked in no matter what the market exchange rate is.

The Conversion Strategy, Continued • Cash flows in 3 months are summarized as follows: Put cash flows (£ 1, 000 × MAX[1. 75 - s 1, 0]) Call cash flows (£ 1, 000 × MIN[1. 75 - s 1, 0]) Total of option transactions: £ 1, 000 × (1. 75 - s 1) = $1, 750, 000 - (£ 1, 000 × s 1) Exchange of Currency = (£ 1, 000 × s 1) Time zero cash flows = $ 36, 800 Interest on Time zero flows = $ 552 TOTAL TIME ONE CASH FLOWS= $1, 787, 352 • This cash flow of $1, 787, 352 is assured in the absence of default risk.