Assorted Exam-Style Questions and Solutions for Exam 3F / Exam MFE

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 84 of the Study Guide. See an index of all sections by following the link in this paragraph.

To provide additional practice, I offer five exam-style questions relating to material covered throughout the previous sections. All material from the study guide is fair game in this section.

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 AESQ1.

Similar to Question 18 from the Society of Actuaries' Sample MFE Questions and Solutions:

Okonkwo is a market-maker who has sold 7000 European 2-year gap options, each on one share of the stock of Yams, Inc., which does not pay any dividends. The stock price volatility is 0.55. Each gap option has a payment trigger price of $555 and a strike price of $666. The stock currently trades for $545. The annual continuously-compounded risk-free interest rate is 0. Okonkwo has decided to delta-hedge his position. How many shares are initially in the delta-hedge?

Solution AESQ1. A modified Black-Scholes formula can be used to price a gap call:
C_gap(S, K₁, K₂, σ, r, T, δ) = Se^-δTN(d₁) - K₁e^-rTN(d₂), where
d₁ = (ln[(Se^-δT)/(K₂e^-rT)] + 0.5σ²T)/(σ√(T)) and d₂ = d₁ - σ√(T)

Here, S is the stock price, K₁ is the strike price, and K₂is the trigger price.

We also know an expression for delta for a regular call: ∆_{regular call} = e^-δTN(d₁)

If this gap option were a regular call option, then the strike price would be equal to 100, or the trigger price. So C(S, K₁, K₂, σ, r, T, δ) = Se^-δTN(d₁) - K₂e^-rTN(d₂).