Chp09 Hedging and Volatility

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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Topics

Dynamic (Delta) Hedging

The Greeks

BOPM and the Greeks


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Dynamic Hedging


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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



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



THE GREEKS



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)



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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K.Cuthbertson and D.Nitzsche 12

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

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Chp09 Hedging and Volatility