The Delta-Gamma Approximation: Practice Problems and Solutions

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.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 13, pp. 413-425.

Problem DGA1. The stock of Delta-Gamma Co. currently trades for $657 per share. A certain call option on the stock of Delta-Gamma Co. has a price of $120, a delta of 0.47, and a gamma of 0.01. Use a delta-gamma approximation to find the price of the call option if, after 1 second, the stock of Delta-Gamma Co. suddenly begins trading at $699 per share.

Solution DGA1. 1 second is an infinitesimally small unit of time, so the delta-gamma approximation will be reasonably accurate here. Here, є = 699 - 657 = 42. Thus,

C(S_t+h) = C(S_t) + є∆(S_t) + (1/2)є²Γ(S_t) = 120 + 42*0.47 + (1/2)(42²)*0.01 =

C(S_t+h) = $148.56

Problem DGA2. The stock of Frivolous LLC currently trades for $1200 per share. A certain call option on the stock of Frivolous LLC has a price of $35, a delta of 0.72, and a certain value of gamma. When the stock price suddenly falls to $1178, the call price falls to $23. Using the delta-gamma approximation, what is the gamma of this call option?

Solution DGA2.

We use the formula C(S_t+h) = C(S_t) + є∆(S_t) + (1/2)є²Γ(S_t), where ∆(S_t) = 0.72, є = -22, C(S_t) = $35, and C(S_t+h) = $23. We rearrange the formula thus:

C(S_t+h) - C(S_t) - є∆(S_t) = (1/2)є²Γ(S_t) and Γ(S_t) = [C(S_t+h) - C(S_t) - є∆(S_t)]/[(1/2)є²]. Thus,

Γ(S_t) = [23 - 35 - (-22)(0.72)]/(1/2)(-22)²] = Γ(S_t) = 0.0158677686

Problem DGA3. The stock of Precarious LLC currently trades for $13 per share. A certain call option on the stock of Frivolous LLC has a price of $1.34, a gamma of 0.025, and a certain value of delta. When the stock price suddenly rises to $19 per share, the call option price increases to $5.67. Using the delta-gamma approximation, what is the original delta of this call option?

Solution DGA3. We use the formula C(S_t+h) = C(S_t) + є∆(S_t) + (1/2)є²Γ(S_t), where Γ(S_t) = 0.025, є = 6, C(S_t) = $1.34, and C(S_t+h) = $5.67.

We rearrange the formula thus:

C(S_t+h) - C(S_t) - (1/2)є²Γ(S_t) = є∆(S_t) and ∆(S_t) = [C(S_t+h) - C(S_t) - (1/2)є²Γ(S_t)]/є

Thus, ∆(S_t) = [5.67 - 1.34 - (1/2)6²*0.025]/6 = ∆(S_t) = 0.646666666667

Problem DGA4. The stock of Imperious LLC suddenly began to trade for $1440 per share. Previously, it traded for $X per share. When the stock traded for $X per share, a certain call option on the stock had a price of $200. Now the call option has a price of $250. The call option has a delta of 0.55 and a gamma of 0.003. Use the delta-gamma approximation to find X.

Solution DGA4. We use the formula C(S_t+h) = C(S_t) + є∆(S_t) + (1/2)є²Γ(S_t), where we desire to find є. We are given that C(S_t+h) = 250, C(S_t) = 200, ∆(S_t) = 0.55, and Γ(S_t) = 0.003. Thus, 250 = 200 + 0.55є + 0.0015є² and 0.0015є² + 0.55є - 50 = 0. By the quadratic formula, the relevant (positive) є = 75.4029116, and so X = 1440 - 75.4029116 = X = $1364.597088

Problem DGA5. When the stock of Odious Co. suddenly decreased in price by $6 per share, a certain put option on the stock of Odious Co. increased in price to $5.99. The put option had an original ∆ of -0.49 and a gamma of 0.002. Find the original put option price using the delta-gamma approximation.

Solution DGA5. Here, є = -6, ∆ = -0.49, Γ = 0.002, and P(S_t+h) = 5.99

We use the formula P(S_t+h) = P(S_t) + є∆(S_t) + (1/2)є²Γ(S_t), which we can rearrange thus:

P(S_t) = P(S_t+h) - є∆(S_t) - (1/2)є²Γ(S_t) = 5.99 - (-6)*(-0.49) + (1/2)(-6²)(0.002) =

P(S_t) = $3.086

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