The Risk Premium and Sharpe Ratio of an Option: Practice Problems and Solutions

This is Section 43 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.

The risk premium of an option can be expressed as

γ - r = (α - r)Ω

The Sharpe ratio of any asset is (α - r)/σ. The Sharpe ratio for a call is the same as the Sharpe ratio for the underlying asset. If the option is a put, the sign of the Sharpe ratio is reversed, so the Sharpe ratio for a put becomes (r - α)/σ.

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.

σ = annual asset price volatility.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12, pp. 394-395.

Problem RPSRO1. You expect to get an annual continuously compounded return of 0.3 on the stock of Curious Co. The stock has annual price volatility of 0.22. The annual continuously compounded risk-free interest rate is 0.02. A certain call option on Curious Co. stock has elasticity of 2.3. Find the expected annual continuously compounded return on the call option.

Solution RPSRO1. We use the formula γ - r = (α - r)Ω, which we rearrange as

γ = (α - r)Ω + r to find γ = (0.3 - 0.02)2.3 + 0.02 = γ = 0.664

Problem RPSRO2. You expect to get an annual continuously compounded return of 0.3 on the stock of Curious Co. The stock has annual price volatility of 0.22. The annual continuously compounded risk-free interest rate is 0.02. A certain call option on Curious Co. stock has elasticity of 2.3. Find the Sharpe ratio of the call option.

Solution RPSRO2. We use the formula (α - r)/σ, which applies to the call option as well as the underlying stock. Here, (α - r)/σ = (0.3 - 0.02)/0.22 = Sharpe ratio = 1.272727272727

Problem RPSRO3. The stock of Delirious LLC has a Sharpe ratio of 0.77 and annual price volatility of 0.11. The annual continuously compounded risk-free interest rate is 0.033. For a certain call option on Delirious LLC stock, the elasticity is 3.4. Find the expected annual continuously compounded return on the option.

Solution RPSRO3. First, we find α. We know that (α - r)/σ = 0.77, so α = 0.77σ + r =

0.77*0.11 + 0.033 = α = 0.1177. Now we use the formula γ = (α - r)Ω + r to find

γ = (0.1177 - 0.033)3.4 + 0.033 = γ = 0.32098

Problem RPSRO4. You expect an annual continuously compounded return of 0.145 on the stock of Deleterious, Inc., and an annual continuously compounded return of 0.33 on a certain call option on that stock. The option elasticity is 4.44. Find the annual continuously compounded risk-free interest rate.

Solution RPSRO4. The formula γ = (α - r)Ω + r can be expressed as γ = Ωα + (1 - Ω)r, which we can rearrange as follows: r = (γ - Ωα)/(1 - Ω) = (0.33 - 4.44*0.145)/(1 - 4.44) =

r = 0.0912209302

Problem RPSRO5. You know that the Sharpe Ratio for a certain call option on Insidious Co. stock is 0.55 and the annual continuously compounded expected return on the option is 0.4. The option elasticity is 2.23. The annual continuously compounded risk-free interest rate is 0.05. Find the annual stock price volatility.

Solution RPSRO5. First, we find α. The formula γ = (α - r)Ω + r can be expressed as γ = Ωα + (1 - Ω)r, which we can rearrange as follows: α = [γ - (1 - Ω)r]/Ω = [0.4 - (1- 2.23)0.05]/2.23 =

α = 0.2069506726. We know that 0.55 = (α - r)/σ, so σ = (α - r)/0.55 =

(0.2069506726 - 0.05)/0.55 = σ = 0.2853648594

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