Elements of Financial Risk Management
Focusing on implementation, this book is for practitioners in the financial services and investment industries, as well as graduate students and advanced undergraduates who want exposure to these techniques.
TABLE OF CONTENTS Elements of Financial Risk Management
6.9. APPENDIX: THE CFG OPTION PRICING FORMULA
6.9. APPENDIX: THE CFG OPTION PRICING FORMULA
The probabilities P 1 and P 2 in the closed-form GARCH ( CFG) formula are derived by first solving for the conditional moment generating function. The conditional, time- t, moment generating function of the log asset prices as time t + 
is
In the CFG model, this function takes a log-linear form (omitting the time subscripts on f( ?))
where
and
These functions can be solved by recursing backward one period at a time from the maturity date using the terminal conditions,
A fundamental result in probability theory establishes the following relationship between the characteristic function, f(i ?), and the probability density function p(x):
where the Re(*) functions take the real value of the argument.
Using these results, we can calculate the conditional expected payoff as
To price the call option, we use the risk-neutral distribution to get
Where we have used the fact that 
. Note that under the risk-neutral distribution, ? is set to ? 1/2, and ? is replaced by ?*. Finally, we note that the previous integrals must be solved numerically.
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