The Delta-Gamma-Theta Approximation: Practice Problems and Solutions

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.

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

∆ = option delta.

Γ = option gamma.

θ = option theta

h = time interval under consideration.

r = annual continuously compounded risk-free interest rate

σ = annual standard deviation of the stock price movement.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 13, pp. 425-427.

Problem DGTA1. 22 days ago, the stock of Vindictive Co. traded for $511 per share. A certain call option on the stock had a delta of 0.66, a gamma of 0.001, and a daily theta of -0.03. The option used to trade for $59. Now the stock trades for $556. The annual continuously compounded risk-free interest rate is 0.08. Find the new option price using the delta-gamma-theta approximation.

Solution DGTA1. We use the equation

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).

Here, є = 45, h = 22 days, θ = -0.03, Δ = 0.66, Γ = 0.001, and C(S_t, T - t) = 59.

Thus, C(S_t+h, T - t - h) = 59 + 45*0.66 + (1/2)(45)²*0.001 + 22(-0.03) =

C(S_t+h, T - t - h) = $89.0525

Problem DGTA2. 22 days ago, the stock of Vindictive Co. traded for $511 per share. A certain call option on the stock had a delta of 0.66, a gamma of 0.001, and a daily theta of -0.03. The option used to trade for $59. Now the stock trades for $556. The annual continuously compounded risk-free interest rate is 0.08. A hypothetical market maker is has purchased delta shares and short-sold the call. Find what a market-maker's profit on one such option would be using the delta-gamma-theta approximation.

Solution DGTA2. We use the equation Profit = -(0.5є²Γ_t + θ_th + rh[Δ_tS_t - C(S_t)]). We note that we can use h = 22 days when we multiply h by a daily theta, but we must use h = 22/365 years when multiplying h by an annual interest rate. Thus,

Profit = -(0.5(45)²0.001 + (-0.03)22 + 0.08(22/365)[0.66*511 - 59]) = -$1.694246849

Problem DGTA3. 22 days ago, the stock of Vindictive Co. traded for $511 per share. A certain call option on the stock had a delta of 0.66, a gamma of 0.001, and a daily theta of -0.03. The option used to trade for $59. Now the stock trades for $556. The annual continuously compounded risk-free interest rate is 0.08. A hypothetical market maker is has purchased delta shares and short-sold the call. What is the annual standard deviation of the stock price movement?

Solution DGTA3. We use the equation є² = σ²S_t²h. Here, h = 22/365, є = 45, S_t = 511. We rearrange the equation thus: σ² = (є²)/(S_t²h) and σ = √[(є²)/(S_t²h)] = √[(45²)/(511²22/365)] = σ = 0.3586961419

Problem DGTA4. Active Co. stock has a price volatility of 0.55. A certain call option on Active Co. stock today costs $71.80. The option has a delta of 0.32, a gamma of 0.001, and a daily theta of -0.06. The stock price today is $3000 per share, and the annual continuously compounded risk-free interest rate is 0.1. Find what a market-maker's profit on one such option would be after 1 year using the delta-gamma-theta approximation.

Solution DGTA4. Here, h = 1, σ = 0.55, S_t = 3000, C(S_t) = 71.80, Δ_t = 0.32, Γ_t = 0.01, θ_t = -0.06. Thus, Profit = -(0.5σ²S_t²Γ_t + θ_t + r[Δ_tS_t - C(S_t)])h =

-(0.5*1*0.55²*3000²*0.001 + -0.06*365 + 0.1*1[0.32*3000 - 71.80]) =

Profit =-$1428.17

Problem DGTA5. The stock of Reactive Co. currently trades for $678 per share. 98 days ago, it traded for $450 per share. Imhotep owns a call option on Reactive Co. stock that cost him $56 when he bought it 98 days ago. Now the call option trades for $100. The option has a delta of 0.33 and a gamma of 0.006. What is the daily option theta? Use the delta-gamma-theta approximation.

Solution DGTA5. Here, є = 678 - 450 = 228 and h = 98 days

We use the equation

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), which we rearrange thus:
[C(S_t+h, T - t - h) - C(S_t, T - t) - єΔ(S_t, T - t) - (1/2)є²Γ(S_t, T - t)]/h = θ(S_t, T - t) =

(100 - 56 - 228*0.33 - (1/2)(228)²0.006)/98 = θ(S_t, T - t) = -1.910122449

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.