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

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 50 (Version 2.0)

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 50 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. See Section 48 here. See Section 49 here.

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: