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Greeks : Theory and Illustrations By A.V. Vedpuriswar June 14, 2014 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. 1 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 Gamma The gamma is the rate of change of the portfolio’s respect to the price of the underlying asset. delta with 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 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 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 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 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 7 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 8 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 9 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. 10 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 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 Problem Suppose a portfolio is delta = - 5000 and vega = - 8000. A traded option has delta = 0.6. How do we achieve vega neutrality? neutral, with gamma gamma = .5, vega = 2.0 and 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, w1 = 400 - 5000 + 0.5w1 + 0.8w2 = 0 - 8000 + 2.0w1 + 1.2w2 = 0 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 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] 14 The Black Scholes Model and the Greeks For a European call or put option on a non dividend paying stock, e d1 2 2 S 0 T 2 1 Gamma = For a European call or put option on a dividend paying 2 stock, qT d1 2 e e Gamma = S 0 T 2 15 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/% 16 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. Stock price Call price Intrinsic value Delta Gamma 10 .00001 0 0 0 15 .01626 0 0.0153 .0121 20 .45875 0 0.2106 .0680 25 2.47066 0 .5977 .0730 27.5 4.17428 2.5 .757 .0536 30 6.21317 5 .8658 .0340 32.5 8.46707 7.5 .9311 .0192 35 10.844409 10.0 .9666 .0100 40 15.75 15 .9931 .0023 45 20.7425 20 .9987 .0004 17 Problem Calculate the delta of an at-the-money 6-month European call option on a nondividend-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, ln(S0 / K ) (0.1 0.252 / 2)0.5 0.3712 d1 0.25 0.5 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, 18 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 ln(8 / 8) (0.182 / 2) 0.6667 d1 0.0735 0.18 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, 20