bm fi6051 wk5 lecture

8/14/2019 Bm Fi6051 Wk5 Lecture

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Financial DerivativesFI6051

Finbarr MurphyDept. Accounting & FinanceUniversity of LimerickAutumn 2009

Week 5 The Greeks


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The delta of an option is the rate of change of theoption price with respect to the underlying assetprice

That is,

Suppose the delta of an option is 0.6

As the underlying asset price changes by a smallamountthe option price changes by about60%of that amount

Delta Hedging

S

c

=


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The following is a graphical representation ofdelta

Delta Hedging

6.0==slope

Stock Price

Option Price

A

B


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B-S derived the following formula for the price ofa European call option on a non-dividend payingstock

where

and is the cumulative probability distributionfor the standardnormal distribution

The Black-Scholes Delta

( ) ( )2100 dNKedNScrT

=

( ) ( )

( ) ( )Td

T

TrKSd

T

TrKSd

=+

=

++=

1

2

0

2

2

0

1

2//ln

;2//ln

( )N


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From this equation, it can be shown that the deltaof such a European call stock option is

Delta hedging of a shortcall option position theninvolves buying in shares in the underlying

Delta hedging of a long call option position theninvolves shorting shares in the underlying


( )1dNS

cc

=

=

( )1

dN

( )1

dN


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B-S derived the following formula for the price ofa European put option on a non-dividend payingstock

where d1, d2, and are as before

From this equation, it can be shown that the deltaof such a European put stock option is


( )N

( ) ( )1020 dNSdNKeprT

=

( ) 11 =

= dN

S

pp


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Noting that by definition , and so thefollowing relation holds

Delta hedging of a shortput option position theninvolves shorting shares in the underlying

Delta hedging of a long put option position theninvolves going long shares in the underlying


( ) 101 < dN

01


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The following graphs shows how the delta of acalloption varies with the stock price


K

1

0

Stock Price

Call Delta


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The following graphs shows how the delta of aputoption varies with the stock price


K

-1

0

Stock Price

Put Delta


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It is important now to consider theimplementation of a delta hedging strategy

Consider a financial institution that sells aEuropean call option on 100,000 shares of a non-dividend paying stock

Assume the following Black-Scholes parametervalues

The Dynamics of Delta Hedging

3846.052/20;%20;%5;50;490 ====== TrKS


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MATLAB has inbuilt functions that allow you toeasily calculate delta

[CD,PD] = blsdelta(SO,X,R,T,SIG,Q)

Cut and paste the following

[CD,PD] = blsdelta(49,50,0.05,20/52,.2,0)

GivesCD = 0.5216PD = -0.4784


3846.052/20;%20;%5;50;490 ====== TrKS


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The link below leads to an Excel workbookdetailing the dynamic delta hedging of this shortoption position

The first table assumes that the option expiresITM at expiration

The second table assumes that the option expiresOTM at expiration

Excel link:FI6051_DynamicDeltaHedging_Example_HullTable14-2-3

https://staffexchange1.ul.ie/exchange/Finbarr.Murphy/Drafts/No%20Subject-2.EML/BM%20FI6051-WK5-LecturerNotes.ppt/C58EA28C-18C0-4a97-9AF2-036E93DDAFB3/FI6051_DynamicDeltaHedging_Example_HullTable14-2-3.xlshttps://staffexchange1.ul.ie/exchange/Finbarr.Murphy/Drafts/No%20Subject-2.EML/BM%20FI6051-WK5-LecturerNotes.ppt/C58EA28C-18C0-4a97-9AF2-036E93DDAFB3/FI6051_DynamicDeltaHedging_Example_HullTable14-2-3.xls


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The delta of a portfolio of options dependent on asingle asset whose price is S is given by

where is the value of the portfolio

The delta of the portfolio can be calculated fromthe deltas of the individual options

The Delta of a Portfolio

S


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The theta of an option is the rate of change of theoption price with respect to time

That is,

Assume the Black-Scholes equation for the priceof a European call option on a non-dividend

paying stock

In this case,

Theta

tc=

( )( )

2

10

2dNrKe

T

dNS rTc

=


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In the above equation for theta note that

Assume now the Black-Scholes equation for theprice of a European put option on a non-dividendpaying stock

In this case,

Theta

( ) 2/1

21

2

1 dedN

=

( )( )

2

10

2dNrKe

T

dNS rTp +

=


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To illustrate, consider a 4-month put option on anon-dividend paying stock

Assume the following details

[CD,PD] = blstheta(305,300,0.08,4/12,.25,0)

CD = -38.3713PD = -15.0028

Theta

25.0;08.0;300;3050 ==== rKS


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Consider again a portfolio of options the value ofwhich is denoted

Gamma is defined as the rate of change of theportfolios delta with respect to the underlyingasset price

That is,

Gamma

S

PP

=


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Recall that by definition

Therefore

That is, is the second derivative of theportfolios value with respect to the underlyingasset price

Gamma

2

2

SP

=

SP

=


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Note that if is small (in absolute terms) thenthis means that delta will change only slowly

In this case adjustments to a delta-neutralportfolio need only be made infrequently

If is large (in absolute terms) then this means

that delta will change quite quickly

In this case adjustments to a delta-neutralportfolio need to be made quite frequently

Gamma


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Note that the gamma of the underlying asset iszero

The gamma of a forward contract on theunderlying asset is also zero The reason for this is that a forward contract is linearly

dependent on the underlying asset

The gamma of an options portfolio can thereforebe changed with an instrument that is nonlinearlydependent on the asset

An example of such an instrument is another tradedoption

Creating a Gamma Neutral Portfolio


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Consider a delta-neutral portfolio with gamma

Consider a traded option with gamma andassume that is the number of options added tothe portfolio

The gamma of the newportfolio is

The appropriate position in the traded option toensure a neutral portfolio gamma is found by

solving


w

+w

0 =+w


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This clearly leads to

Note also that by adding the traded option to theportfolio, the portfolios delta is changes also

In order to ensure a neutral delta it is alsonecessary to adjust the position in the underlyingasset


=

w


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A delta- and gamma-neutral portfolio can beregarded as correcting for not being able toadjust delta continuously

Delta neutrality protects against relatively smallchanges in the underlying asset price betweenrebalancing

Gamma neutrality protects against largermovements in the underlying asset price betweenrebalancing



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Consider again the example used to illustrate thedynamics of delta hedging

That is, consider a financial institution that sells aEuropean call option on 100,000 shares of a non-dividend paying stock

Assume the following Black-Scholes parametervalues

The Dynamics of Gamma Hedging

3846.052/20;%20;%5;50;490 ====== TrKS


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For the gamma hedging of the short optionsposition assume another traded options contractis used

That is, consider an options contract with thesame details but where the strike price isdifferent

Assume the strike price for this option is K1 = 49,

(the option is ITM)


3846.052/20;%20;%5;49;49 10 ====== TrKS


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We can again use MATLAB to calculate gamma

[Gamma] = blsgamma(49,50,0.05,20/52,.2,0)

Gamma = 0.0655


3846.052/20;%20;%5;49;49 10 ====== TrKS


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The link below leads to an Excel workbookdetailing the dynamic delta and gamma hedgingof this short option position

The first table assumes that the option expiresITM at expiration

The second table assumes that the option expiresOTM at expiration

Excel link:FI6051_DynamicDeltaGammaHedging_Example_HullTable14-2-3

https://staffexchange1.ul.ie/exchange/Finbarr.Murphy/Drafts/No%20Subject-2.EML/BM%20FI6051-WK5-LecturerNotes.ppt/C58EA28C-18C0-4a97-9AF2-036E93DDAFB3/FI6051_07/FI6051_DynamicDeltaGammaHedging_Example_HullTable14-2-3.xlshttps://staffexchange1.ul.ie/exchange/Finbarr.Murphy/Drafts/No%20Subject-2.EML/BM%20FI6051-WK5-LecturerNotes.ppt/C58EA28C-18C0-4a97-9AF2-036E93DDAFB3/FI6051_07/FI6051_DynamicDeltaGammaHedging_Example_HullTable14-2-3.xls


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Recall that within the B-S framework volatility isassumed to be constant

In reality of course volatility will change over time

So the value of an option can change as a resultof changes in volatility As well as a result of changes in the underlying asset

and time

Vega


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Again let denote the value of a portfolio ofoptions

Vega is the rate of change of with respect tothe volatility of the underlying asset

That is,

Vega

=


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If vega is high in absolute terms, the portfoliosvalue is very sensitive to smallchanges involatility

The opposite is true in the case where vega is low inabsolute terms

Note that a position in the underlying asset has a

vega of zero, therefore, another traded optioncan be used to change a portfolios vega

[VEGA] = blsvega(49,50,0.05,20/52,.2,0)VEGA = 12.1055

Vega


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Let be the vega of the traded option

The position to be taken in the traded option to

vega neutrality is given by

Note that a portfolio with such a vega hedge inplace will not be in general gamma neutral

To ensure both gamma and vega neutrality atleast two traded o tions must be used see Hull

Vega

=w


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The rho of a portfolio - with value - is the rateof change of with respect to the interest rate

That is,

Rho measures the sensitivity of an portfoliosvalue to changes in the level of interest rates

[rho] = blsrho(49,50,0.05,20/52,.2,0)rho = 8.9070

Rho

r

r

=


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Hull, J.C, Options, Futures & Other Derivatives,2009, 7th Ed. Chapter 17

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