1 / 16

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

dawson
Télécharger la présentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

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

  3. Dynamic Hedging ©K.Cuthbertson and D.Nitzsche

  4. 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

  5. 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

  6. 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

  7. THE GREEKS ©K.Cuthbertson and D.Nitzsche

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

  9. 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

  10. 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

  11. 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

  12. 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

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

  14. 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

  15. 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

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

More Related