Author of a

Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/. Author of a study manual for exam FM available at: http://smartURL.it/krzysioFM (paper) or http://smartURL.it/krzysioFMe (electronic) Instructor for online seminar for exam FM: http://smartURL.it/onlineactuary If you find these exercises valuable, please consider buying the manual or attending our seminar, and if you can’t, please consider making a donation to the Actuarial Program at Illinois State University: https://www.math.ilstu.edu/actuary/giving/. Donations will be used for scholarships for actuarial students. Donations are tax-deductible to the extent allowed by law. Questions about these exercises? E-mail: [email protected] Spring 2000 Casualty Actuarial Society Course 8 Examination, Problem No. 26 (multiple choice answers added), and Dr. Ostaszewski’s online exercise No. 81 posted December 2, 2006 You are given the following: • Stock price = $50. • The risk-free interest rate is a constant annual 8%, compounded continuously. • The price of a 6-month European call option with an exercise price of $48 is $5. • The price of a 6-month European put option with an exercise price of $48 is $3. • Assume no transaction costs, no margin requirements, and that the stock pays no dividend. If there is an arbitrage opportunity, calculate the payoff upon expiration of the options considered to an investor who utilizes such an opportunity for only one share of the stock, as well as a call and/or put on only one share of the stock. Assume that the investor can invest and borrow at the risk-free rate given, and uses that for any free cash flows. A. $1.88

B. $1.96

C. $2.00

D. $2.08

E. There is no arbitrage

Solution. The put-call parity relationship says that C + PV ( X ) = S + P. Using the values given in the problem, we get the following left-hand side of the above formula $5 + $48 ⋅ e−0.08⋅0.5 ≈ $51.12. The right-hand side of the put-call parity formula equals $50 + $3 = $53, and the two are not equal, even though they must be in the absence of arbitrage. Thus there is an arbitrage opportunity: one must be long the cheap assets and short the expensive ones, i.e., long call plus long PV(X), and short S plus short put. If the investor buys the call for $5 and invests $48 ⋅ e−0.08⋅0.5 ≈ $46.12 in a risk-free bond, this will result in a cash outlay of $51.12. By shorting the stock the investor obtains $50 and by writing a put the investor obtains $3, for a total cash inflow of $53. There is a net free cash flow of $1.88, which in six months will accumulate to approximately $1.96. At options expiration, the investor gets $48 from the bond purchased, and uses that and a call held to purchase a share of the stock to cover the short, if the price of a share is above $48 (in this case the put expires worthless and is irrelevant). If the price of a share is below $48, the call expires worthless

and is irrelevant, but the investor must use $48 bond proceeds to buy a share for $48 from the party long put, and then can use that share to cover the short. Net cash flow is $1.96. Answer B. © Copyright 2006 by Krzysztof Ostaszewski. All rights reserved. Reproduction in whole or in part without express written permission from the author is strictly prohibited. Exercises from the past actuarial examinations are copyrighted by the Society of Actuaries and/or Casualty Actuarial Society and are used here with permission.