The Elasticity and Risk Premium of an Option Portfolio: Practice Problems and Solutions

This is Section 44 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.

The elasticity of a portfolio of call options can be expressed as

Ω_portfolio = _i=1ⁿΣω_iΩ _i where Ω _i is the elasticity of the ith call option and ω_i is the percentage of the portfolio comprised of the ith call option.

The risk premium on the portfolio - where all call options are based on the same underlying asset - is

γ - r = Ω_portfolio(α - r)

Meaning of variables:

γ = expected annual continuously compounded return on the option.

α = expected annual continuously compounded return on the underlying asset (most often a stock).

Ω = option elasticity.

r = annual continuously compounded risk-free interest rate.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12,

p. 395 and this erratum.

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

Problem ERPOP1. You own a portfolio of calls options on Tenacious Co. stock. The portfolio consists of 444 Options A, 334 Options B, and 3434 Options C. Options A have elasticity of 4.4. Options B have elasticity of 5.5. Options C have elasticity of 1.22. Find the elasticity of this portfolio.

Solution ERPOP1. There are a total of 444 + 334 + 3434 = 4212 options in the portfolio. Options A comprise (444/4212) = 0.1054131054 of the portfolio; Options B comprise (334/4212) = 0.079297246 of the portfolio; Options C comprise (3434/4212) = 0.8152896486 of the portfolio. We use the formula Ω_portfolio = _i=1ⁿΣω_iΩ _i = 0.1054131054*4.4 + 0.079297246*5.5 + 0.8152896486*1.22 = Ω_portfolio = 1.894605888

Problem ERPOP2. You own a portfolio of calls options on Tenacious Co. stock. The portfolio consists of 444 Options A, 334 Options B, and 3434 Options C. Options A have elasticity of 4.4. Options B have elasticity of 5.5. Options C have elasticity of 1.22. The expected annual continuously compounded return on the stock is 0.24, and the annual continuously compounded risk-free interest rate is 0.05. Find the risk premium on this option portfolio.

Solution ERPOP2. We use the formula γ - r = Ω_portfolio(α - r). We know from Solution ERPOP1 that Ω_portfolio = 1.894605888. Thus, γ - r = 1.894605888(0.24 - 0.05) = γ - r = 0.3599751187

Problem ERPOP3. Your option portfolio consists of two distinct types of call options on Imperious LLC stock - Option D and Option E. Option D has elasticity 3.45, and Option E has elasticity 5.55. You have 100 options in the portfolio, and you know that the portfolio has elasticity of 4.374. How many Options D are in the portfolio?

Solution ERPOP3. Let D be the number of Options D in the portfolio.

By the formula Ω_portfolio = _i=1ⁿΣω_iΩ _i, we know that 4.374 = (D/100)3.45 + [(100 - D)/100]*5.55

Thus, 4.374 = 0.0345D + 5.55 - 0.0555D

-1.176 = -0.021D and D = -1.176/-0.021 = D = 56

Problem ERPOP4. Your option portfolio consists of three distinct types of call options on Imperious LLC stock - Option F, Option G, and Option H. You own 34 Options F with elasticity 2.4 and 45 Options G with elasticity 4.7. Option H has elasticity 5.9. How many options in total must be in the portfolio in order for portfolio elasticity to be 4.17?

Solution ERPOP4. Let H be the number of Options H in the portfolio.

By the formula Ω_portfolio = _i=1ⁿΣω_iΩ _i, we know that

4.17 = [34/(79 + H)]2.4 + [45/(79 + H)]4.7 + [H/(79 + H)]5.9

4.17(79 + H) = 293.1 + 5.9H

329.43 + 4.17H = 293.1 + 5.9H

36.33 = 1.73 H, so H = 21 and the total number of options in the portfolio is 34 + 45 + 21 = 100 options

Problem ERPOP5. You know that the risk premium on your call option portfolio is 0.56 and the expected annual continuously compounded return on the underlying asset - superwidgets - is 0.43. The annual continuously compounded risk-free interest rate is 0.09. The option portfolio consists of 44 Options I with elasticity 1.64 and 56 Options J with elasticity X. Find X.

Solution ERPOP5. First, we find portfolio elasticity using the formula γ - r = Ω_portfolio(α - r), which we rearrange to Ω_portfolio = (γ - r)/(α - r) = (0.56 - 0.09)/(0.43 - 0.09) =

Ω_portfolio = 1.382352941

Now we use the formula Ω_portfolio = _i=1ⁿΣω_iΩ _i. Since we conveniently have 100 options in the portfolio, the fraction consisting of I is 0.44 and the fraction consisting of J is 0.56. Thus,

1.382352941 = 0.44*1.64 + 0.56X

0.6607529412 = 0.56X.

Thus, X = 1.179915966

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