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. transaction costs are low.
B. costs of short-selling are high.
C. prices are consistent with the law of one price.
3. An arbitrage transaction generates a net inflow of funds:
A. throughout the holding period.
B. at the end of the holding period.
C. at the start of the holding period.
4. The price of a forward contract:
A. is the amount paid at initiation.
B. is the amount paid at expiration.
C. fluctuates over the term of the contract.
5. Assume an asset pays no dividends or interest, and also assume that the asset does not yield any non-financial benefits or incur any carrying cost. At initiation, the price of a forward contract on that asset is:
A. lower than the value of the contract.
B. equal to the value of the contract.
C. greater than the value of the contract.
6. With respect to a forward contract, as market conditions change:
A. only the price fluctuates.
B. only the value fluctuates.
C. both the price and the value fluctuate.
7. The value of a forward contract at expiration is:
A. positive to the long party if the spot price is higher than the forward price.
B. negative to the short party if the forward price is higher than the spot price.
C. positive to the short party if the spot price is higher than the forward price.
8. At the initiation of a forward contract on an asset that neither receives benefits nor incurs carrying costs during the term of the contract, the forward price is equal to the:
A. spot price.
B. future value of the spot price.
C. present value of the spot price.
9. Stocks BWQ and ZER are each currently priced at $100 per share. Over the next year, stock BWQ is expected to generate significant benefits whereas stock ZER is not expected to generate any benefits. There are no carrying costs associated with holding either stock over the next year. Compared with ZER, the one-year forward price of BWQ is most likely:
A. lower.
B. the same.
C. higher.
10. If the net cost of carry of an asset is positive, then the price of a forward contract on that asset is most likely:
A. lower than if the net cost of carry was zero.
B. the same as if the net cost of carry was zero.
C. higher than if the net cost of carry was zero.
11. If the present value of storage costs exceeds the present value of its convenience yield, then the commodity's forward price is most likely:
A.