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8/13/2019 Chp09 Hedging and Volatility
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K.Cuthbertson and D.Nitzsche
FINANCIAL ENGINEERING:
DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)
K. Cuthbertson and D. Nitzsche
LECTURE
Dynamic Hedging and the Greeks
1/9/2001
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K.Cuthbertson and D.Nitzsche
Topics
Dynamic (Delta) Hedging
The Greeks
BOPM and the Greeks
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K.Cuthbertson and D.Nitzsche
Dynamic Hedging
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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 isS0=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 intodebt)
Debt , D0= 0S0- C0= 63.6810.45 = $53.23
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Dynamic Delta Hedging
At t = 1the 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 yourdebt position at t=1is
53.23 - 1.958 = 51.30
The value of your hedge portfolio at t = 1(including the marketvalue 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
111
SeDDo
tr
o
)01.0(05.0e
1111 CDS
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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: DTK = 111.29 100 = 11.29% Net cost at T: (DTK - C0) / C0 = 8.1%
% Cost of the delta hedge = risk free rate
%Hedge Performancer = sd( DTe-rT- C0) / C0
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THE GREEKS
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Figure 9.2 : Delta and gamma : long call
0
0.2
0.4
0.6
0.8
1
1.2
1 11 21 31 41 51 61 71 81 91
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Delta Gamma
Stock Price (K= 50)
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THE GREEKS: A RISK FREE HOLIDAY ON THE ISLANDS
2
2
Sf
f
Gamma and Lamda
df .dS +(1/2) (dS)2 + dt + rdr + d
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HEDGING WITH THE GREEKS:
Gamma Neutral Portfolio= gamma of existing portfolioT= gamma of new options
port= NTT+ = 0
therefore buy : NT= - /T new options
Vega Neutral PortfolioSimilarly : N= -/ T new options
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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 deltaneutral by trading only the underlying stocks (cant tradeoptions because this would break the gamma/vega
neutrality).
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Hedging With The Greeks: A Simple Example
PortfolioA: is delta neutral but = -300.
A Call option Z with the same underlying (e.g. stock) has adelta = 0.62and gamma of 1.5.How can you use Z to make the overall portfolio gamma anddelta neutral?
We require: nzz+ = Onz= - / z= -(-300)/1.5 = 200
implies 200 long contracts in ZThe delta of this new portfolio is now
= nz.z= 200(0.62) = 124Hence to maintain delta neutrality you must short 124units of the underlying.
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BOPM and the Greeks
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Figure 9.5 : BOPM lattice
Index,j
Time, t
1,0 2,0 3,0 4,00,0
1,1
2,2
3,3
4,4
2,1 3,1 4,1
3,2 4,2
4,3
10 2 3 4
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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 =
1011
101100
SS
ff
2122
212211
SS
ff
2021
202110
SS
ff
)()[( 20212122 SSSS 2/)( 2022 SS
101100
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End of Slides