prices.
15. Arbitrage opportunities exist when:
A. two identical assets or derivatives sell for different prices.
B. combinations of the underlying asset and a derivative earn the risk-free rate.
C. arbitrageurs simultaneously buy takeover targets and sell takeover acquirers.
CHAPTER 2
BASICS OF DERIVATIVE PRICING AND VALUATION
LEARNING OUTCOMES
After completing this chapter, you will be able to do the following:
● explain how the concepts of arbitrage, replication, and risk neutrality are used in pricing derivatives;
● distinguish between value and price of forward and futures contracts;
● explain how the value and price of a forward contract are determined at expiration, during the life of the contract, and at initiation;
● describe monetary and nonmonetary benefits and costs associated with holding the underlying asset and explain how they affect the value and price of a forward contract;
● define a forward rate agreement and describe its uses;
● explain why forward and futures prices differ;
● explain how swap contracts are similar to but different from a series of forward contracts;
● distinguish between the value and price of swaps;
● explain how the value of a European option is determined at expiration;
● explain the exercise value, time value, and moneyness of an option;
● identify the factors that determine the value of an option and explain how each factor affects the value of an option;
● explain put–call parity for European options;
● explain put–call–forward parity for European options;
● explain how the value of an option is determined using a one-period binomial model;
● explain under which circumstances the values of European and American options differ.
SUMMARY OVERVIEW
This reading on derivative pricing provides a foundation for understanding how derivatives are valued and traded. Key points include the following:
● The price of the underlying asset is equal to the expected future price discounted at the risk-free rate, plus a risk premium, plus the present value of any benefits, minus the present value of any costs associated with holding the asset.
● An arbitrage opportunity occurs when two identical assets or combinations of assets sell at different prices, leading to the possibility of buying the cheaper asset and selling the more expensive asset to produce a risk-free return without investing any capital.
● In well-functioning markets, arbitrage opportunities are quickly exploited, and the resulting increased buying of underpriced assets and increased selling of overpriced assets returns prices to equivalence.
● Derivatives are priced by creating a risk-free combination of the underlying and a derivative, leading to a unique derivative price that eliminates any possibility of arbitrage.
● Derivative pricing through arbitrage precludes any need for determining risk premiums or the risk aversion of the party trading the option and is referred to as risk-neutral pricing.
● The value of a forward contract at expiration is the value of the asset minus the forward price.
● The value of a forward contract prior to expiration is the value of the asset minus the present value of the forward price.
● The forward price, established when the contract is initiated, is the price agreed to by the two parties that produces a zero value at the start.
● Costs incurred and benefits received by holding the underlying affect the forward price by raising and lowering it, respectively.
● Futures prices can differ from forward prices because of the effect of interest rates on the interim cash flows from the daily settlement.
● Swaps can be priced as an implicit series of off-market forward contracts, whereby each contract is priced the same, resulting in some contracts being positively valued and some negatively valued but with their combined value equaling zero.
● At expiration, a European call or put is worth its exercise value, which for calls is the greater of zero or the underlying price minus the exercise price and for puts is the greater of zero and the exercise price minus the underlying price.
● European calls and puts are affected by the value of the underlying, the exercise price, the risk-free rate, the time to expiration, the volatility of the underlying, and any costs incurred or benefits received while holding the underlying.
● Option values experience time value decay, which is the loss in value due to the passage of time and the approach of expiration, plus the moneyness and the volatility.
● The minimum value of a European call is the maximum of zero and the underlying price minus the present value of the exercise price.
● The minimum value of a European put is the maximum of zero and the present value of the exercise price minus the price of the underlying.
● European put and call prices are related through put–call parity, which specifies that the put price plus the price of the underlying equals the call price plus the present value of the exercise price.
● European put and call prices are related through put–call–forward parity, which shows that the put price plus the value of a risk-free bond with face value equal to the forward price equals the call price plus the value of a risk-free bond with face value equal to the exercise price.
● The values of European options can be obtained using the binomial model, which specifies two possible prices of the asset one period later and enables the construction of a risk-free hedge consisting of the option and the underlying.
● American call prices can differ from European call prices only if there are cash flows on the underlying, such as dividends or interest; these cash flows are the only reason for early exercise of a call.
● American put prices can differ from European put prices, because the right to exercise early always has value for a put, which is because of a lower limit on the value of the underlying.
PROBLEMS
1. An arbitrage opportunity is least likely to be exploited when:
A. one position is illiquid.
B. the price differential between assets is large.
C. the investor can execute a transaction in large volumes.
2. An arbitrageur will most likely execute a trade when:
A.
