Greeks : Theory and Illustrations By A.V. Vedpuriswar

1. Greeks : Theory and IllustrationsBy A.V. Vedpuriswar June 14, 2014

2. Introduction • Greeks help us to measure the risk associated with derivative positions. • Greeks also come in handy when we do local valuation of instruments. • But Greeks are not useful to get an aggregated view of risk.

3. Delta • Delta is the rate of change in option price with respect to the price of the underlying asset. • It is the slope of the curve that relates the option price to the underlying asset price. • A position with Delta of zero is called Delta neutral. • Since Delta keeps changing, the investor’s position may remain delta neutral for only a relatively short period of time. • The hedge has to be adjusted periodically. • This is known as rebalancing. • The delta of European call option is N(d1) in the Black Scholes equation. • The delta of a European put option is N(d1) – 1 in the Black Scholes equation. 2

4. Gamma • The gamma is the rate of change of the portfolio’s delta with respect to the price of the underlying asset. • It is the second partial derivative of the portfolio price with respect to the asset price. • If gamma is small, it means delta is changing slowly. • So adjustments to keep a portfolio delta neutral can be made relatively infrequently. • However, if gamma is large, the delta is highly sensitive to the price of the underlying asst. • It is then quite risky to leave a delta neutral portfolio unchanged for any length of time. 3

5. Theta • Theta of a portfolio is the rate of change of value of the portfolio with respect to change of time. • Theta is also called the time decay of the portfolio. • Theta is usually negative for an option. • As time to maturity decreases with all else remaining the same, the option loses value. 4

6. Vega • Vega is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset. • High Vega means high sensitivity to small changes in volatility. • A position in the underlying asset has zero Vega. • The Vega can be changed by adding options. • To make the portfolio Gamma and Vega neutral, two traded derivatives dependent on the underlying asset are needed. 5

7. Rho • Rho of a portfolio of options is the rate of change of value of the portfolio with respect to the interest rate. • If interest rate increases, value of call increases. Why? 6

8. Problem • A bank has a $ 25 million par position in a 5 year zero coupon bond with a market value of $ 19,059,948. What is the modified duration of the bond? • 19,059,948=25,000,000/[(1+r)^5] • r = .0558 • Modified duration = 5/{1 + .0558/2} = 4.86 years

9. Problem • An investor holds the following bonds in her portfolio. Calculate the duration. • $ 2,000,000 par value of 10 year bonds, duration of 6.95 ,price 95.5 • $3,000,000 par value of 15 year bonds, duration of 9.77, price 88.6275 • $ 5,000,000 par value of 30 year bonds, duration of 14.81, price 115.875 • Market value of Bond 1 = 2,000,000 x .955 = 1,910,000, weight = .19 • Market value of Bond 2 = 3,000,000 x .886275 = 2, 658,825, weight = .26 • Market value of Bond 3 = 5,000,000 x 1.15875 = 5,793,750, weight = .56 • Portfolio duration = 6.95x.19 + 9.77x .26 + 14.81x.56 = 12.15

10. Problem • If all the spot interest rates are increased by one basis point, the value of a portfolio of swaps will increase by $ 1100. How many Euro dollar futures contracts are needed to hedge the portfolio? • A Eurodollar contract has a face value of $ 1 million and a maturity of 3 months. If rates change by 1 basis point, the value changes by (1,000,000) (.0001)/4= $ 25. • So the number of futures contracts needed = 1100/25=44

11. Problem • A bank has sold USD 300,000 of call options; with strike price of 50 on 100,000 shares currently trading at 49.5.How should the bank do delta hedging? • Current delta = -.5x 300,000 + 100,000 = - 50,000 • So she must buy 50,000 shares.

12. Problem • Suppose an existing short option position is delta neutral and has a gamma of －6000. Here, gamma is negative because we have sold options. Assume there exists a traded option with a delta of 0.6 and gamma of 1.25. Create a gamma and delta neutral position. • Solution • To gamma hedge, we must buy 6000/1.25 = 4800 options. • Then we must sell (4800) (.6) = 2880 shares to maintain a gamma neutral and original delta neutral position. 11

13. Problem • A delta neutral position has a gamma of －3200. There is an option trading with a delta of 0.5 and gamma of 1.5. How can we generate a gamma neutral position for the existing portfolio while maintaining a delta neutral hedge? • Solution • Buy 3200/1.5 = 2133 options • Sell (2133) (.5) = 1067 shares 12

14. Problem • Suppose a portfolio is delta neutral, with gamma= - 5000 and vega = - 8000. A traded option has gamma = .5, vega = 2.0 and delta = 0.6. How do we achieve vega neutrality? • To achieve Vega neutrality we can add 4000 options.  Delta increases by (.6) (4000) = 2400 • So we sell 2400 units of asset to maintain delta neutrality. • As the same time, Gamma changes from – 5000 to (.5) (4000) – 5000 = - 3000. • If there is a second traded option with gamma = 0.8, vega = 1.2, delta = 0.5. • if w1 and w2 are the weights in the portfolio, • - 5000 + 0.5w1 + 0.8w2 = 0 - 8000 + 2.0w1 + 1.2w2 = 0 • w1 = 400 w2 = 6000. • This makes the portfolio gamma and vega neutral. • But delta = (400) (.6) + (6000) (.5) = 3240 • 3240 units of the underlying asset will have to be sold to maintain delta neutrality. Ref : John C Hull, Options, Futures and Other Derivatives, 13

15. The Black Scholes Model and the Greeks • For a European call option on a non dividend paying stock, Delta = N(d1) • For Put, Delta = N(d1) -1 • For a dividend paying stock, • For Call, Delta = e-qt N(d1) • For Put, Delta = e-qt [N(d1) – 1]

16. The Black Scholes Model and the Greeks • For a European call or put option on a non dividend paying stock, • Gamma = • For a European call or put option on a dividend paying stock, • Gamma =

17. Problem • Stock price = 49 • Strike price = 50; Volatility = 20% • Risk free rate = 5%; Time to exercise = 20 weeks • Using Deriva Gem spreadsheet, we get : • Call option price = 2.40 • Delta = .522/$ • Gamma = .066/$/$ • Vega = .121/% • Theta = -.012/day • Rho = .089/%

18. Problem • Strike price = 25; Risk free rate of interest = 6% • Time to maturity = 0.5 years; Stock volatility = 30% • Establish the relationship between option price, delta, gamma and underlying price.

19. Problem • Calculate the delta of an at-the-money 6-month European call option on a non-dividend-paying stock when the risk-free interest rate is 10% per annum and the stock price volatility is 25% per annum. • In this case S0 = K, r = 0.1, σ = 0.25, and T = 0.5. Also, • The delta of the option is N(d1) or 0.6450. • We can also calculate using Deriva Gem. Ref : John C Hull, Options, Futures and Other Derivatives,

20. Problem • What is the delta of a short position in 1,000 European call options on silver futures? The options mature in 8 months, and the futures contract underlying the option matures in 9 months. The current 9-month futures price is $8 per ounce, the exercise price of the option is $8, the risk-free interest rate is 12% per annum, and the volatility of silver is 18% per annum. • The delta of a European futures option is usually defined as the rate of change of the option price with respect to the futures price (not the spot price). • It is e-rT N(d1) • In this case F0 = 8, K = 8, r = 0.12, σ = 0.18, T = 0.6667 • N(d1) = 0.5293 and the delta is e-0.12x0.6667 x 0.5293 = 0.4886 • The delta of a short position in 1000 futures options is therefore -488.6. Ref : John C Hull, Options, Futures and Other Derivatives,