7.6. SENSITIVITY, HEDGING AND THE “GREEKS” 263
formulas presented above. In practice, transaction costs make frequent bal-
ancing expensive. Rather than trying to eliminate all risks, an option trader
usually concentrates on assessing risks and deciding whether they are accept-
able. Traders tend to use Delta, Gamma, and Vega measures to quantify the
different aspects of risk in their portfolios.
Sources
The material in this section is adapted from ‘Quantitative modeling of Deriva-
tive Securitiesby Marco Avellaneda, and Peter Laurence, Chapman and Hall,
Boca Raton, 2000, pages 44–56,; andOptions, Futures, and other Derivative
Securitiessecond edition, by John C. Hull, Prentice Hall, 1993, pages 298–
318.
Problems to Work for Understanding
- How can a short position in 1,000 call options be made Delta neutral
when the Delta of each option is 0.7? - Calculate the Delta of an at-the-money 6-month European call option
on a non-dividend paying stock, when the risk-free interest rate is 10%
per year (compound continuously) and the stock price volatility is 25%
per year. - Use the put-call parity relationship to derive the relationship between
(a) The Delta of European call and the Delta of European put.
(b) The Gamma of European call and the Gamma of European put.
(c) The Vega of a European call and a European put.
(d) The Theta of European call and a European put.- (a) Derive the expression for Γ for a European call option as given in
the notes.
(b) Draw a graph of Γ versus S for K = 50,r = 0.10, σ = 0.25,
T−t= 0.25.
(c) Draw a graph of Γ versustfor a call option on an at-the-money
stock, withK= 50,r= 0.10,σ= 0.25,T−t= 0.25.