Figure 11.8: Histogram of Portfolio Value Changes Using Full Valuation
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Pitfall in Delta and Gamma Approaches
• Here we use an example to illustrate that the gamma
approximation can sometimes be highly misleading
• Consider an option portfolio that consists of 3 types of
options all on the same asset, and that has a price of St =
100, all with
calendar days to maturity
• The risk-free rate is 0.02/365 and the volatility is 0.015 per
calendar day
• We take a short position in 1 put with a strike of 95, a short
position in 1.5 calls with a strike of 95, and a long position
in 2.5 calls with a strike of 105
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Pitfall in Delta and Gamma Approaches
• Using the BSM model to calculate the delta and gamma of the
individual options, we get
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Pitfall in Delta and Gamma Approaches
• Now we try to assess the accuracy of the delta and gamma
approximation for portfolio over a five trading day or
equivalently seven calendar day horizon
• Rather than computing VaRs, we will take a closer look at
the complete payoff profile of the portfolio for different
future values of underlying asset price, St+5
• We refer to the value of the portfolio today as VPFt and to
the hypothetical future value as VPFt+5(St+5)
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Pitfall in Delta and Gamma Approaches
• We first calculate the value of the portfolio today as
• The delta of the portfolio is similarly
• Now, the delta approximation to the portfolio value in five
trading days is easily calculated as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Pitfall in Delta and Gamma Approaches
• The gamma of the portfolio is
• and the gamma approximation to the portfolio value in five
trading days is
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
78
Pitfall in Delta and Gamma Approaches
• Finally, relying on full valuation, we must calculate the future
hypothetical portfolio values as
• where we subtract seven calendar days from the time to
maturity corresponding to the risk management horizon of
five trading days
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Figure 11.9 Future portfolio values for option portfolio
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Pitfall in Delta and Gamma Approaches
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• The important lesson of this three-option example is as
follows:
• The different strike prices and the different exposures to
the underlying asset price around the different strikes
create higher order nonlinearities, which are not well
captured by the simple linear and quadratic approximations
• In realistic option portfolios consisting of thousands of
contracts, there may be no alternative to using the full
valuation method
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Summary
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• This chapter has presented three methods for incorporating
options into the risk management model
– Delta-based (linear) risk models
– Gamma-based (quadratic) risk models
– Full Valuation
• A simple example
• Pitfalls in Delta and Gamma-based risk models
• The main lesson from the chapter is that for nontrivial
options portfolios and for risk management horizons
beyond just a few days, the full valuation approach may be
the only reliable choice
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen