Table 11.2: Gamma of American Put Option
Tải bản đầy đủ - 0trang
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