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 41 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.
Theta
The option Greek theta (θ) “measures the change in the option price when there is a decrease in the time to maturity of 1 day” (McDonald 2006, p. 383).
R. L. McDonald suggests the following mnemonic device to help memorize the definition of theta: “theta” and “time” begin with the same letter.
In most cases, theta is negative – that is, an option loses value as expiration time approaches. The fastest time decay at expiration occurs for at-the-money options.
There are a few exceptions to this rule. Theta can be positive for deep in-the-money European puts and deep in-the-money European calls when the underlying asset has a high dividend yield.
The option Greek rho (ρ) “measures the change in the option price when there is an increase in the interest rate of 1 percentage point (100 basis points)” (McDonald 2006, p. 383).
R. L. McDonald suggests the following mnemonic device to help memorize the definition of rho: “rho” begins with the letter r, which is commonly used to denote the annual continuously-compounded risk-free interest rate.
For an ordinary call option, rho is positive; for a put, rho is negative.
Psi
The option Greek psi (Ψ) “measures the change in the option price when there is an increase in the continuous dividend yield of 1 percentage point (100 basis points)” (McDonald 2006, p. 383).
For call options, psi is negative. For put options, psi is positive. “The absolute value of psi increases with time to expiration” (McDonald 2006, p. 388).
Greek Measures for Portfolios
“The Greek measure of a portfolio is the sum of the Greeks of the individual portfolio components” (McDonald 2006, p. 388). For delta, this can be expressed by the formula
∆portfolio = i=1nΣωi∆i
where ωi is the quantity of each option i and ∆i is the delta of each option i and the portfolio contains n options.
Original Practice Problems and Solutions from the Actuary’s Free Study Guide:
Problem OGTRPGMP1. The stock of Theta-Rho-Psi Co. pays dividends at an annual continuously compounded yield of 0.12. The annual continuously compounded risk-free interest rate is 0.34. Certain call options on the stock of Theta-Rho-Psi Co. have time to expiration of 99 days, θ = -0.03, ρ = 0.11, Ψ = -0.04. One call option currently trades for $56. Find the price of the call option 65 days from expiration, all other things equal.
Solution OGTRPGMP1. We want to calculate the option price once 99-65 = 34 days have passed. The price change is 34θ = -0.03*34 = -1.02. Thus, the new option price is 56 – 1.02 = $54.98
Problem OGTRPGMP2. The stock of Theta-Rho-Psi Co. pays dividends at an annual continuously compounded yield of 0.12. The annual continuously compounded risk-free interest rate is 0.34. Certain call options on the stock of Theta-Rho-Psi Co. have time to expiration of 99 days, θ = -0.03, ρ = 0.11, Ψ = -0.04. One call option currently trades for $56. Find the price of the call option if the interest rate suddenly increases to 0.66, all other things equal.
Solution OGTRPGMP2. The change in the interest rate is 0.66 – 0.34 = 0.32, so the change in the option price is 32ρ = 32*0.11 = 3.52 and the new option price is 56 + 3.52 = $59.52
Problem OGTRPGMP3. The stock of Theta-Rho-Psi Co. pays dividends at an annual continuously compounded yield of 0.12. The annual continuously compounded risk-free interest rate is 0.34. Certain call options on the stock of Theta-Rho-Psi Co. have time to expiration of 99 days, θ = -0.03, ρ = 0.11, Ψ = -0.04. One call option currently trades for $56. Find the price of the call option if the stock’s dividend yield suddenly decreases to 0.02, all other things equal.
Solution OGTRPGMP3. The change in the dividend yield is 0.02 – 0.12 = -0.1. So the change in the option price is -10Ψ = -10*-0.04 = 0.4 and the new option price is 56 + 0.4 = $56.40
Problem OGTRPGMP4. The stock of Theta-Rho-Psi Co. pays dividends at an annual continuously compounded yield of 0.12. The annual continuously compounded risk-free interest rate is 0.34. Certain call options on the stock of Theta-Rho-Psi Co. have time to expiration of 99 days, θ = -0.03, ρ = 0.11, Ψ = -0.04. One call option currently trades for $56. Find the price of the call option if the stock’s dividend yield increases to 0.45, the interest rate drops to 0.01, and time to expiration becomes 12 days, all other things equal.
Solution OGTRPGMP4. We consider the effect of the dividend yield increase on the option price: The change in the dividend yield is 0.45 – 0.12 = 0.33. So the change in the option price from this alone is 33Ψ = 33*-0.04 = -1.32
We consider the effect of the interest rate decrease on the option price: The change in the interest rate is 0.01 – 0.34 = -0.33. So the change in the option price from this alone is -33ρ = -33*0.11 = -3.63
Also, 99-12 = 87 days have passed, so from this alone the option price changed by 87θ =
87*-0.03 = -2.61.
Thus, the new option price is 56 – 1.32 – 3.63 – 2.61 = $48.44
Problem OGTRPGMP5. You own 45 call options on Asset A, 14 put options on Asset B, 44 put options on Asset C, and 784 call options on Asset D. Asset A call options have ∆ = 0.22. Asset B put options have ∆ = -0.82. Asset C put options have ∆ = -0.33. Asset D call options have ∆ = 0.01. Find the delta of your entire portfolio.
