1 9 2001 FINANCIAL ENGINEERING DERIVATIVES AND RISK MANAGEMENT J

1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE Dynamic Hedging and the Greeks ©K. Cuthbertson and D. Nitzsche 1

Topics Dynamic (Delta) Hedging The Greeks BOPM and the Greeks

Dynamic Hedging ©K. Cuthbertson and D. Nitzsche 3

Dynamic (Delta) Hedging Suppose we have written a call option for C 0 =10. 45 (with K=100, = 20%, r=5%, T=1) when the current stock price is S 0=100 and 0 = 0. 6368 At t=0, to hedge the call we buy 0 = 0. 6368 shares at So = 100 at a cost of $63. 68. hence we need to borrow (i. e. go into debt) Debt , D 0 = 0 S 0 - C 0 = 63. 68 – 10. 45 = $53. 23

Dynamic Delta Hedging At t = 1 the stock price has fallen to S 1= 99 with 1 = 0. 617. You therefore sell ( 1 - 0) shares at S 1 generating a cash inflow of $1. 958 which can be used to reduce your debt so that your debt position at t=1 is 53. 23 - 1. 958 = 51. 30 The value of your hedge portfolio at t = 1(including the market value of your written call): V 1 = = Value of shares held - Debt - Call premium = = 0. 0274 (approx zero) But as S falls (say) then you sell on a falling marker ending up with positive debt

Dynamic Delta Hedging OPTION ENDS UP OUT-OF-THE-MONEY ( T = 0 shares) $ Net cost at T: DT = 10. 19 % Net cost at T: (DT - C 0) / C 0 = 2. 46% OPTION ENDS UP IN THE-MONEY ( T = 1 share) $ Net cost at T: DT – K = 111. 29 – 100 = 11. 29 % Net cost at T: (DT – K - C 0) / C 0 = 8. 1% % Cost of the delta hedge = risk free rate %Hedge Performancer = sd( DT e-r. T - C 0) / C 0

THE GREEKS ©K. Cuthbertson and D. Nitzsche 7

Figure 9. 2 : Delta and gamma : long call Delta Gamma Stock Price (K = 50) ©K. Cuthbertson and D. Nitzsche 8

THE GREEKS: A RISK FREE HOLIDAY ON THE ISLANDS Gamma and Lamda df . d. S +(1/2) (d. S)2 + dt + r dr + d ©K. Cuthbertson and D. Nitzsche 9

HEDGING WITH THE GREEKS: Gamma Neutral Portfolio = gamma of existing portfolio T = gamma of “new” options port = NT T + = 0 therefore “buy” : NT = - / T “new” options Vega Neutral Portfolio Similarly : N = - / T “new” options ©K. Cuthbertson and D. Nitzsche 10

HEDGING WITH THE GREEKS ORDER OF CALCULATIONS 1) Make existing portfolio either vega or gamma neutral (or both simultaneously, if required in the hedge) by buying/selling “other” options. Call this portfolio-X 2) Portfolio-X is not delta neutral. Now make portfolio-X delta neutral by trading only the underlying stocks (can’t trade options because this would “break the gamma/vega neutrality). ©K. Cuthbertson and D. Nitzsche 11

Hedging With The Greeks: A Simple Example Portfolio–A: is delta neutral but = -300. A Call option “Z” with the same underlying (e. g. stock) has a delta = 0. 62 and gamma of 1. 5. How can you use Z to make the overall portfolio gamma and delta neutral? n z z + = O nz = - / z = -(-300)/1. 5 = 200 implies 200 long contracts in Z The delta of this ‘new’ portfolio is now = nz. z = 200(0. 62) = 124 Hence to maintain delta neutrality you must short 124 units of the underlying. We require: ©K. Cuthbertson and D. Nitzsche 12

BOPM and the Greeks ©K. Cuthbertson and D. Nitzsche 13

Figure 9. 5 : BOPM lattice Index, j 4, 4 3, 3 2, 2 1, 1 0, 0 0 1, 0 1 2, 0 2 4, 3 3, 2 4, 2 3, 1 4, 1 3, 0 3 ©K. Cuthbertson and D. Nitzsche 4, 0 4 Time, t 14

BOPM and the Greeks Gamma S* = (S 22 + S 21)/2 and in the lower part, S** = (S 21 + S 20)/2. Hence their difference is: [9. 32] = ] /2 ©K. Cuthbertson and D. Nitzsche = 15

End of Slides ©K. Cuthbertson and D. Nitzsche 16