The Return and Variance of the Return to a Delta-Hedged Market-Maker: Practice Problems and Solutions

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The period i return to a delta-hedged market-maker who has purchased a call - R_h,i - can be written as R_h,i = (1/2)S²σ²Γ(x_i²-1)h

For a delta-hedged market-maker who has written a call, the period i return is the negative of the previous expression: R_h,i = -(1/2)S²σ²Γ(x_i²-1)h

The variance of this return is

Var(R_h,i) = (1/2)(S²σ²Γh)²

It is assumed that x_i is uncorrelated across time.

Meaning of variables:

R_h,i = The period i return to a delta-hedged market-maker who has written a call.

S = stock price.

σ = standard deviation of the stock price movement.

Γ = option gamma.

h = time interval between hedge readjustments.

x_i = the number of standard deviations the stock price moves during period i.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 13, pp. 431.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem RVRDHMM1. Imhotep is a delta-hedged market-maker who readjusts his hedges every 5 months. He hedges using the stock of Tranquil Co., which has a price of $500 per share; the standard deviation of this price is 0.03. A certain call option on Tranquil Co. stock has a gamma of 0.001. Imhotep has a long position in this option. During a particular 5-month period, the stock price moves by 0.23 standard deviations. Find the return to Imhotep during this time period.

SolutionRVRDHMM1. Since Imhotep has a long position, we use the formula

R_h,i = (1/2)S²σ²Γ(x_i²-1)h = (1/2)*500²*0.03²*0.001(0.23²-1)(5/12) = R_h,i = - 0.0443953125

Problem RVRDHMM2. Imhotep is a delta-hedged market-maker who readjusts his hedges every 5 months. He hedges using the stock of Tranquil Co., which has a price of $500 per share; the standard deviation of this price is 0.03. A certain call option on Tranquil Co. stock has a gamma of 0.001. Imhotep has a long position in this option. During a particular 5-month period, the stock price moves by 0.23 standard deviations. Find the variance of the return to Imhotep during this time period.

Solution RVRDHMM2. We use the formula Var(R_h,i) = (1/2)(S²σ²Γh)² = (1/2)(500²0.03²0.001*(5/12))² = Var(R_h,i) = 0.0043945313

Problem RVRDHMM3. Cuauhtemoc is a delta-hedged market-maker who has a short position in a call option on the stock of Volatile Co. Cuauhtemoc readjusts his hedges every 2 months. The stock has a price of $45; the standard deviation of this price is 0.33. The gamma of the call option is 0.02. During a particular 2-month period, the stock price moves by 0.77 standard deviations. Find the return to Cuauhtemoc during this time period.

Solution RVRDHMM3. Since Cuauhtemoc has a short position, we use the formula

R_h,i = -(1/2)S²σ²Γ(x_i²-1)h = -(1/2)45²0.33²0.02(0.77²-1)(2/12) = R_h,i = 0.1496245163

Problem RVRDHMM4. Cuauhtemoc is a delta-hedged market-maker who has a short position in a call option on the stock of Volatile Co. Cuauhtemoc readjusts his hedges every 2 months. The stock has a price of $45; the standard deviation of this price is 0.33. The gamma of the call option is 0.02. During a particular 2-month period, the stock price moves by 0.77 standard deviations. Find the variance of the return to Cuauhtemoc during this time period.

Solution RVRDHMM4. We use the formula Var(R_h,i) = (1/2)(S²σ²Γh)² = (1/2)(45²0.33²0.02*2/12)² = Var(R_h,i) = 0.2701676278

Problem RVRDHMM5. Gilgamesh - a delta-hedged market-maker - has a short position in a call option on the stock of Mesopotamian Industries. The stock price has a standard deviation of 0.39. The call option has a gamma of 0.007. Gilgamesh readjusts his hedges daily. During a particular day, the stock price of Mesopotamian Industries changed by 0.03 standard deviations. Gilgamesh obtained a return of $1.23 as a result. What was the original stock price of Mesopotamian Industries on that day? Assume that there are 365 days in a year.

Solution RVRDHMM5. Since Gilgamesh has a short position, we use the formula

R_h,i = -(1/2)S2σ²Γ(x_i²-1)h, where we want to find S. Here, h = 1/365.

We rearrange the formula thus:

√[-2R_h,i/(σ²Γ(x_i²-1)h)] = S = √[-2*1.23/(0.03²0.007(0.03²-1)(1/365))] = S= $142,652,196.50

(This stock is something akin to Berkshire Hathaway - except on the scale of millennia. It might not have had any splits since 3000 BC!)

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