FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche

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

Topics Dynamic (Delta) Hedging The Greeks BOPM and the Greeks ©K.Cuthbertson and D.Nitzsche

Dynamic Hedging ©K.Cuthbertson and D.Nitzsche

Dynamic (Delta) Hedging Suppose we have written a call option for C0 =10.45 (with K=100,  = 20%, r=5%, T=1) when the current stock price is S0=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 , D0 = 0S0 - C0 = 63.68 – 10.45 = $53.23 ©K.Cuthbertson and D.Nitzsche

Dynamic Delta Hedging At t = 1 the stock price has fallen to S1= 99 with 1 = 0.617. You therefore sell (1 - 0) shares at S1generating 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): V1 = = 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 ©K.Cuthbertson and D.Nitzsche

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 - C0) / C0 = 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 - C0) / C0 = 8.1% % Cost of the delta hedge = risk free rate %Hedge Performancer = sd( DT e-rT - C0) / C0 ©K.Cuthbertson and D.Nitzsche

THE GREEKS ©K.Cuthbertson and D.Nitzsche

Figure 9.2 : Delta and gamma : long call Delta Gamma Stock Price (K = 50) ©K.Cuthbertson and D.Nitzsche

THE GREEKS: A RISK FREE HOLIDAY ON THE ISLANDS Gamma and Lamda • df D.dS +(1/2)  (dS)2 +  dt + r dr +  ds ©K.Cuthbertson and D.Nitzsche

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

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

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? • We require: nz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. ©K.Cuthbertson and D.Nitzsche

Figure 9.5 : BOPM lattice Index, j 4,4 3,3 4,3 2,2 3,2 4,2 1,1 2,1 3,1 4,1 1,0 2,0 3,0 4,0 0,0 Time, t 0 2 3 4 1 ©K.Cuthbertson and D.Nitzsche

BOPM and the Greeks Gamma S* = (S22 + S21)/2 and in the lower part, S** = (S21 + S20)/2. Hence their difference is: [9.32]  = ] /2 = ©K.Cuthbertson and D.Nitzsche

FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche