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Greeks : Theory and Illustrations By A.V. Vedpuriswar June 14, 2014.

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Presentation on theme: "Greeks : Theory and Illustrations By A.V. Vedpuriswar June 14, 2014."— Presentation transcript:

1 Greeks : Theory and Illustrations By 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. 1

3 22 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(d 1 ) in the Black Scholes equation.  The delta of a European put option is N(d 1 ) – 1 in the Black Scholes equation.

4 33 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.

5 44 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.

6 55 Vega  Vega is the rate of change of the value of the portfolio with respect tothe 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.

7 66 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?

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/{ /2} = 4.86 years 7

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  $ 5,000,000 par value of 30 year bonds, duration of 14.81, price  Market value of Bond 1 = 2,000,000 x.955 = 1,910,000, weight =.19  Market value of Bond 2 = 3,000,000 x = 2, 658,825, weight =.26  Market value of Bond 3 = 5,000,000 x = 5,793,750, weight =.56  Portfolio duration = 6.95x x x.56 =

10 Problem  If all the spot interest rates are increased by one basis point, the value of a portfolio of swaps will increase by $ 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

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 = - 50,000  So she must buy 50,000 shares. 10

12 11 Problem  Suppose an existing short option position is delta neutral and has a gamma of - Here, gamma is negative because we have sold options. Assume there exists a traded option with a delta of 0.6 and gamma of 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.

13 12 Problem  A delta neutral position has a gamma of - 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

14 13  Suppose a portfolio is delta neutral, with gamma = and vega = 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 =  If there is a second traded option with gamma = 0.8, vega = 1.2, delta = 0.5.  if w 1 and w 2 are the weights in the portfolio,  w w 2 = w w 2 = 0  w 1 = 400 w 2 =  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.  Problem Ref : John C Hull, Options, Futures and Other Derivatives,

15 The Black Scholes Model and the Greeks  For a European call option on a non dividend paying stock,Delta = N(d 1 )  For Put, Delta = N(d 1 ) -1  For a dividend paying stock,  For Call, Delta = e -qt N(d 1 )  For Put, Delta = e -qt [N(d 1 ) – 1] 14

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 = 15

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/% 16

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. 17 Stock priceCall priceIntrinsic value DeltaGamma

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 S 0 = K, r = 0.1, σ = 0.25, and T = 0.5. Also,   The delta of the option is N(d 1 ) or  We can also calculate using Deriva Gem. 18 Ref : John C Hull, Options, Futures and Other Derivatives,

20 Problem  The Black-Scholes price of an out-of-the-money call option with an exercise price of $40 is $4. A trader who has written the option plans to use a stop-loss strategy. The trader’s plan is to buy at $40.10 and to sell at $ Estimate the expected number of times the stock will be bought or sold.  Solution  The strategy costs the trader $0.20 each time the stock is bought and sold.  The total expected cost of the strategy, in present value terms, must be $4.  So the expected number of times the stock will be bought and sold is approximately 20 each. 19 Ref : John C Hull, Options, Futures and Other Derivatives,

21 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(d 1 )  In this case F 0 = 8, K = 8, r = 0.12, σ = 0.18, T =  N(d 1 ) = and the delta is e -0.12x x =  The delta of a short position in 1000 futures options is therefore Ref : John C Hull, Options, Futures and Other Derivatives,


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