**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 46 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.**

The problems in this section were designed to be similar to problems from past versions of Exam 3F / Exam MFE. They use original exam questions as their inspiration – and the specific inspiration for each problem is cited so as to give students a chance to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

**Problem ESQOEOVBSF1.**

**Similar to Question 22 from the Casualty Actuarial Society’s Fall 2007 Exam 3:**

Evasive Co. stock currently trades for $77 per share. The stock pays no dividends, and the annual continuously-compounded risk-free interest rate is 0.05. The stock price volatility is 0.44. A call option on Evasive Co. stock has a strike price of $74 and time to expiration of 2 years. Calculate Ω, the elasticity of this call option within the Black-Scholes framework.

**Solution ESQOEOVBSF1.**

In order to use the formula Ω = S∆/C, we first need to find C and ∆.

To find C, we use the Black-Scholes formula, where

d1 = [ln(S/K) + (r – ∂ + 0.5σ2)T]/[σ√(T)] = [ln(77/74) + (0.05 – 0 + 0.5*0.442)2]/[0.44√(2)] =

d1 = 0.5356981973

d2 = d1 – σ√(T) = 0.5356981973 – 0.44√(2) = d2 = -0.0865557701

In MS Excel, using the input “=NormSDist(0.5356981973)”, we find that N(d1) = 0.70391645

In MS Excel, using the input “=NormSDist(-0.0865557701)”, we find that N(d2) = 0.465512247

Now we use the Black-Scholes formula:

C(S, K, σ, r, T, ∂) = Se-∂TN(d1) – Ke-rTN(d2) = 77*0.70391645 – 74*e-2*0.050.465512247 =

C = 23.03181208

Also, ∆call = e-∂TN(d1), which in this case is just N(d1) = 0.70391645. Thus,

Ω = S∆/C = 77*0.70391645/23.03181208 = **Ω = 2.353334877**

**Problem ESQOEOVBSF2.**

**Similar to Question 29 from the Casualty Actuarial Society’s Fall 2007 Exam 3:**

You know the following about the stock of Selective LLC and a certain call option on it. The stock price is $55, and the option’s strike price is $65. The option expires in 1 year. The stock pays no dividends, and the annual continuously-compounded risk-free interest rate is 0.09. You expect an annual continuously-compounded return of 0.18 on the stock, and you estimate that stock volatility is 0.33. Find the Sharpe ratio of the call option.

**Solution ESQOEOVBSF2.**

The Sharpe ratio for a call option is the same as the Sharpe ratio for the underlying stock, i.e.,

(α – r)/σ = (0.18 – 0.09)/0.33 = **0.272727272727**

**Problem ESQOEOVBSF3.**

**Similar to Question 5 from the Society of Actuaries’ May 2007 Exam MFE:**

The stock of Methodical, Inc., currently trades for $500 per share. The stock pays no dividends, and the annual continuously-compounded risk-free interest rate is 0.13. The stock price volatility is 0.39. A certain call option on Methodical, Inc., stock has a strike price of $433 and time to expiration of 1 year. Find the volatility of this call option within the Black-Scholes framework.

**Solution ESQOEOVBSF3.**

The volatility of an option can be expressed as

σoption = σstock*│Ω│. We need to find Ω before using this formula.

In order to use the formula Ω = S∆/C, we first need to find C and ∆.

To find C, we use the Black-Scholes formula, where

d1 = [ln(S/K) + (r – ∂ + 0.5σ2)T]/[σ√(T)] = [ln(500/433) + 1(0.13 – 0 + 0.5*0.392)]/ [0.39√(1)] =

d1 = 0.3499203704 and d2 = d1 – σ√(T) = 0.3499203704 – 0.39 = d2 = -0.0400796296

In MS Excel, using the input “=NormSDist(0.3499203704)”, we find that N(d1) = 0.63680079. Since the stock pays no dividends, this is also ∆.

In MS Excel, using the input “=NormSDist(-0.0400796296)”, we find that N(d2) = 0.484014759.

Now we use the Black-Scholes formula:

C(S, K, σ, r, T, ∂) = Se-∂TN(d1) – Ke-rTN(d2) = 500*0.63680079 – 433e-0.130.484014759 = C = 134.3705678 and Ω = S∆/C = 500*0.63680079/134.3705678 = Ω = 2.36956947

σoption = σstock*│Ω│ = 0.39*2.36956947 = **σoption = 0.9241320933**

**Problem ESQOEOVBSF4.**

**Similar to Question 15 from the Society of Actuaries’ May 2007 Exam MFE:**

The stock of Insidious LLC currently trades for $10 per share. In 4 years, the stock will pay a dividend of $0.50. It pays no other dividends. The volatility relevant for the Black-Scholes formula is 0.37. The annual continuously-compounded risk-free interest rate is 0.07. A certain put option on Insidious LLC stock has a strike price of $10 and time to expiration of 8 years. Find the price of such an option using the Black-Scholes formula.

**Solution ESQOEOVBSF4.**

We can use the Black-Scholes formula with prepaid forwards:

FP0,T(S) = 10 – 0.50e-0.07*4 = 9.622108129

FP0,T(K) = 10e-0.07*8 = 5.712090638

d1 = [ln(FP0,T(S)/FP0,T(K)) + 0.5σ2T]/[σ√(T)] =

[ln(9.622108129/5.712090638) + 0.5*0.372*8]/[0.37√(8)] = d1 = 1.021557442

d2 = d1 – σ√(T) = 1.021557442 – 0.37√(8) = d2 = -0.0249605942

In MS Excel, using the input “=NormSDist(-1.021557442)”, we find that N(-d1) = 0.153495219.

In MS Excel, using the input “=NormSDist(0.0249605942)”, we find that N(-d2) = 0.50995685.

The Black-Scholes formula for the put price is

P(FP0,T(S), FP0,T(K), σ, T) = FP0,T(K)N(-d2)- FP0,T(S)N(-d1) =

5.712090638*0.50995685 – 9.622108129*0.153495219 = **P = $1.435972154**

**Problem ESQOEOVBSF5.**

**Similar to Question 8 from the Society of Actuaries’ Sample MFE Questions and Solutions:**

Treacherous Co. stock currently sells for $500 per share. The stock pays no dividends, and its price volatility is 0.34. A certain call option on Treacherous Co. stock has a strike price of $700 and time to expiration of 4 years. The call option has a delta of 0.5. Which of these expressions represents the price of this option?

(a) 251.3558251-∞0.68∫(e-(x^2)/2)dx – 380.0556182

(b) 250 – 251.3558251-∞0.68∫(e-(x^2)/2)dx

(c) 250 – 630.0556182-∞0.68∫(e-(x^2)/2)dx

(d) 250 – 380.0556182-∞0.68∫(e-(x^2)/2)dx

(e) 630.0556182-∞0.68∫(e-(x^2)/2)dx – 380.0556182

**Solution ESQOEOVBSF5.**

This is indeed a treacherous problem. The annual continuously compounded risk-free interest rate (r) is unknown and needs to be found. Fortunately, it is possible to do so. The stock pays no dividends, so ∆call = N(d1), which is given to be 0.5. Since N(d1) = 0.5, d1 = 0.

d1 = [ln(S/K) + (r – ∂ + 0.5σ2)T]/[σ√(T)] = 0 in this case.

Thus, [ln(500/700) + 4r + 4*0.5*0.342]/[0.34√(4)] = 0

and ln(500/700) + 4r + 4*0.5*0.342 = 0

and 4r = 0.1052722366. Thus, r = 0.0263180592

d2 = d1 – σ√(T) = 0 – 0.34√(4) = d2 = -0.68

Now the function N(t) can be represented as the following integral:

[1/√(2π)]-∞t∫(e-(x^2)/2)dx.

However, we note that the expression in the integral in all of our possible answers is *not* N(d2); rather, it is N(-d2) = 1 – N(d2). However, we can express N(d2) as 1 – N(-d2).

Now we are ready to use the Black-Scholes formula:

C(S, K, σ, r, T, ∂) = Se-∂TN(d1) – Ke-rTN(d2) = S∆call – Ke-rT[1 – N(-d2)] =

500*0.5 – 700e-4*0.0263180592[1 – [1/√(2π)]-∞0.68∫(e-(x^2)/2)dx] =

250 – 630.0556182 + 251.3558251-∞0.68∫(e-(x^2)/2)dx =

**251.3558251-∞0.68∫(e-(x^2)/2)dx – 380.0556182**. So **(a)** is the correct answer.

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