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Figure 11.8: Histogram of Portfolio Value Changes Using Full Valuation

73

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

74

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

75

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

76

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

77

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

79

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

80

• 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

81

• 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

Figure 11.8: Histogram of Portfolio Value Changes Using Full Valuation