The Delta-Gamma Approximation: Practice Problems and Solutions

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 47 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here. See Section 45 here. See Section 46 here.

A market-maker sells assets or contracts to buyers and buys them from sellers. He is an intermediary between the buyers and sellers. A market-maker's function is in contrast to proprietary trading, which is "trading to express an investment strategy" (McDonald, p. 414).

A delta-hedged position is a position designed to earn the risk-free rate of interest and is used to offset the risk of an option position.

The delta-gamma approximation is used to estimate option price movements if the underlying stock price changes.

The delta-gamma approximation for call options can be expressed via the following formula:

C(S_t+h) = C(S_t) + є∆(S_t) + (1/2)є²Γ(S_t)

For a put option, the same formula holds, but delta is now negative - so the put price will decrease if the stock price increases.

Meaning of variables:

S_t = stock price at time t.

S_t+h = stock price at time t+h.

C = call option price.

є = stock price change from time t to time t + h.

∆ = option delta.

Γ = option gamma.