The Delta-Gamma-Theta 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 48 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. See Section 47 here.

When considerable periods of time pass between the moments at which the original price of an option occurs and the new option price occurs, time decay - whose effects are represented by the option Greek theta - must be taken into account. Hence, the delta-gamma approximation from Section 47 can be amplified into the delta-gamma-theta approximation for the new option price and the market-maker's profit on the option when the underlying stock price changes by є and a time interval of duration h has passed.

C(S_t+h, T - t - h) = C(S_t, T - t) + єΔ(S_t, T - t) + (1/2)є²Γ(S_t, T - t) + hθ(S_t, T - t)

When a market-maker has purchased Δ shares and short-sold the call, his profit is

Profit = -(0.5є²Γ_t + θ_th + rh[Δ_tS_t - C(S_t)])

We can make a substitution for є²:

є² = σ²S_t²h

Thus, Profit = -(0.5σ²S_t²Γ_t + θ_t + r[Δ_tS_t - C(S_t)])h

Meaning of variables:

S_t = stock price at time t.

C = call option price.