The Greek Letters

1. The Greek Letters Finance (Derivative Securities) 312 Tuesday, 17 October 2006 Readings: Chapter 15

2. Hedging • Suppose that: • A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend paying stock • S0 = 49, K= 50, r = 5%, s = 20%, m = 13%, T = 20 weeks • Black-Scholes value of option is $240,000 • How does the bank hedge its risk?

3. Naked and Covered Positions • Naked position • If stock price < $50 at expiry, bank profits • If stock price > $50, bank must pay prevailing stock price x 100,000, no limit to loss • Covered position • If stock price > $50 at expiry, bank profits • If stock price < $50, bank loses on stock position

4. Stop-Loss Strategy • Ensures covered position if option closes in-the-money, naked position if it closes out-of-the-money • Buy 100,000 shares if price rises over $50 • Sell 100,000 shares if price falls below $50 • Cost of hedge would always be less than Black-Scholes price, leading to riskless profit • Ignores time value of money • Purchases and sales will not be made at K exactly

5. Delta Hedging • Delta (D) is the rate of change of the option price with respect to the underlying asset price Option price Slope = D B Stock price A

6. Delta Hedging • Suppose that: • Stock price is $100, option price is $10, D = 0.6 • Trader sells 20 calls on 2,000 shares • How can the trader employ delta hedging? • Buy 0.6 x 2,000 = 1,200 shares • Option D is 0.6 x –2,000 = –1,200 • Overall D is 0 (delta neutral)

7. European Stock Options • Call on non-dividend paying stock: • D = N(d1) • Put on non-dividend paying stock: • D = N(d1)– 1 • Call on stock paying dividends at rate q • D = N(d1)e–qT • Put on stock paying dividends at rate q • D = e–qT[N(d1) – 1]

8. Effect of Dividends • Suppose that: • A bank has sold six-month put options on £1m with strike price of 1.6000 • Current exchange rate is 1.6200, UK r = 13%, US r = 10%, volatility of sterling is 15% • How can the bank construct a delta neutral hedge?

9. Effect of Dividends • Put option D = -0.458 • Exchange rate rises by DS, price of put falls by 45.8% of DS • Bank must add £458,000 to its position to make it delta neutral • Note that deltas on a portfolio are a weighted average of individual derivative deltas

10. Delta of Futures • Futures price on non-dividend paying stock is S0erT • When stock price changes by DS, futures price changes by DSerT • Marking-to-market ensures investor realises profit/loss immediately, thus D = erT • With dividends, D = e(r-q)T • Not the case with forwards

11. Delta of Futures • To achieve delta neutrality • HF = e–rT HA • HF = e–(r–q)T HA(with dividends) • HF = e–(r–rf)T HA(with currency futures) • From earlier example, hedging using nine-month futures requires short position of: • e–(0.10–0.13)9/12x 458,000= £468,442 • Each contract = £62,500, no. of contracts = 7

12. Theta • Theta (Q) is the rate of change of value with respect to time • Usually negative for an option • For a call option, theta is • Close to zero when the stock price is very low • Large and negative when at-the-money, and approaches –rKe-rT as stock prices gets larger • Useful as a proxy for gamma

13. Gamma • Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset • Small gamma implies less frequent rebalancing required • Sensitivity of value of portfolio to DS • Delta, Theta, Gamma are connected

14. Gamma Call price C′′ C′ C Stock price S S′

15. Gamma Neutrality • Suppose that: • Delta neutral portfolio has a gamma of –3,000 • Delta and gamma of an option are 0.62 and 1.50 respectively • How can this portfolio be made gamma neutral?

16. Gamma Neutrality • Adding wT options with gamma GT to a portfolio with gamma G gives wT GT +G • wT must therefore be –G/ GT • Include long position of 3,000/1.5 = 2,000 call options • Delta will change from zero to 2,000 x 0.62 = 1,240 • Sell 1,240 units of the underlying asset

17. Theta as a Proxy for Gamma • For a delta neutral portfolio, DP»QDt + ½GDS 2 DP DP DS DS Positive Gamma Negative Gamma

18. Vega • Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility • High vega implies high portfolio sensitivity to small changes in volatility • To ensure both gamma and vega neutrality, at least two derivatives must be used

19. Vega Neutrality • Suppose that: • A delta neutral portfolio has gamma of –5,000 and vega of –8,000 • Option 1 has gamma of 0.5, vega of 2.0, delta of 0.6 • Option 2 has gamma of 0.8, vega of 1.2, delta 0.5 • How can this portfolio be made gamma, vega and delta neutral?

20. Vega Neutrality • Simultaneous equations • –5,000 + 0.5w1 + 0.8w2 = 0 • –8,000 + 2.0w1 + 1.2w2 = 0 • w1 = 400, w2 = 6,000 • Delta = 400 x 0.6 + 6,000 x 0.5 = 3,240 • Vega is always positive for a long position in either European or American options

21. Rho • Rho is the rate of change of the value of a derivative with respect to the interest rate • Long calls and short puts have positive Rhos (increase in interest rate would mean increase in call premium) • Rho becomes more significant the longer the time remaining to expiry of the options • For currency options there are two rhos corresponding to the two different interest rates

22. Hedging in Practice • Traders usually ensure that their portfolios are delta-neutral at least once a day • Whenever the opportunity arises, they improve gamma and vega • As portfolio becomes larger hedging becomes less expensive

23. Synthetic Positions • Strategy I: investing part of portfolio into the risk-free asset • Recall, for a put, D = e– qT [N(d1) – 1] • Ensure that at any time, a proportion of e– qT[N(d1) – 1] stocks in the portfolio has been sold and invested into riskless assets

24. Synthetic Positions • Suppose that: • A portfolio is worth $90m • Six-month European put is required with strike price of $87m • rf is 9%, dividend yield is 3%, volatility is 25% p.a. • How can the option be synthetically created?

25. Synthetic Positions • D = –0.3215 • 32.15% of portfolio should be sold initially and invested into riskless assets • If portfolio value falls to $88m after one day, delta becomes –0.3679 and further 4.64% should be sold

26. Synthetic Positions • Strategy II: use index futures • Using previous example: • D = eq(T–T*)e–rT* [N(d1) – 1] A1/A2 • T = 0.5, T* = 0.75, A1 = 100,000, A2 = 250, d1 = 0.4499 • D =122.95, ≈ 123 • 123 futures contracts should be shorted