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Table 11.2: Gamma of American Put Option

Table 11.2: Gamma of American Put Option

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Portfolio Risk Using Gamma



41



• When incorporating the second derivative, gamma, we

instead rely on the quadratic approximation

• where the portfolio  and  are calculated as



• where again mj denotes the number of option contract j in

the portfolio

Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



Cornish-Fisher Approximation



42



If we assume that the underlying asset return, Rt+1, is

normally distributed with mean zero and constant variance

2, and rely on the preceding quadratic approximation, then

the first three moments of the distribution of changes in the

value of a portfolio of options can be written as



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Cornish-Fisher Approximation

• For example, we can derive the expected value as



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



43



Cornish-Fisher Approximation



44



• In particular, we note that even if underlying return has mean

zero, the portfolio mean is no longer zero

• More important, the variance formula changes and the

portfolio skewness is no longer zero, even if the underlying

asset has no skewness

• The asymmetry of the options payoff itself creates

asymmetry in the portfolio distribution

• The linear-normal model presented earlier fails to capture the

skewness, but quadratic model considered here captures the

skewness at least approximately

• In this way, the quadratic model can offer a distinct

improvement over the linear model

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Cornish-Fisher Approximation



45



• The approximate Value-at-Risk of the portfolio can be

calculated using the Cornish-Fisher approach

• The Cornish-Fisher VaR allowing for skewness is



• Unfortunately, the analytical formulas for the moments of

options portfolios with many underlying assets are quite

cumbersome, and they rely on the unrealistic assumption of

normality and constant variance

• We will therefore now consider a much more general but

simulation-based technique that builds on the Monte Carlo

method

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



46



Simulation Based Gamma Approximation

• Consider again the simple case where the portfolio consists

of options on only one underlying asset and we are interested

in the K-day $VaR

• We have



• Using the assumed model for the physical distribution of

the underlying asset return, we can simulate MC pseudo Kday returns on the underlying asset



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



47



Simulation Based Gamma Approximation

• and calculate the hypothetical changes in the portfolio value as



• from which we can calculate the Value-at-Risk as



• In the general case of options on n underlying assets, we

have



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



48



Simulation Based Gamma Approximation

• where i and i are the aggregate delta and gamma of the

portfolio with respect to the ith return

• If we in addition allow for derivatives that depend on several

underlying assets, then we write



• which includes the so-called cross-gammas, ij

• For a call option, for example, we have



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



49



Simulation Based Gamma Approximation

• Cross-gammas are relevant for options with multiple sources

of uncertainty

• An option written on the US dollar value of the Tokyo stock

index is an example of such an option

• We now simulate a vector of underlying returns from the

multivariate distribution



• and we calculate

using



by summing over the different assets



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



50



Simulation Based Gamma Approximation

• The great benefit of this approach is that we are

aggregating all the options on one particular asset into a

delta and a gamma for that asset

• Thus, if the portfolio consists of a thousand different types

of option contracts, but only written on 100 different

underlying assets, then the dimension of the approximated

portfolio distribution is only 100

• As these formulas suggest, we could, in principle, simulate

the distribution of the future asset returns at any horizon

and calculate the portfolio Value-at-Risk for that horizon

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



51



Simulation Based Gamma Approximation

• However, a key problem with the delta and the deltagamma approaches is that if we calculate the VaR for a

horizon longer than one day, the delta and gamma numbers

may not be reliable approximations to the risk of option

position because they are assumed to be constant through

time when in reality they are not

• We therefore next consider an approach that is

computationally intensive, but does not suffer from the

problems arising from approximating the options by delta

and gamma

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Table 11.2: Gamma of American Put Option

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