CFA 58: Basics of Derivative Pricing and Valuation

An arbitrage opportunity is least likely to be exploited when: one position is illiquid. the price differential between assets is large. the investor can execute a transaction in large volumes. Correct answer- A is correct. An illiquid position is a limit to arbitrage because it may be difficult to realize gains of an illiquid offsetting position. A significant opportunity arises from a sufficiently large price differential or a small price differential that can be employed on a very large scale. An arbitrageur will most likely execute a trade when: transaction costs are low. costs of short-selling are high. prices are consistent with the law of one price. Correct answer- A is correct. Some arbitrage opportunities represent such small price discrepancies that they are only worth exploiting if the transaction costs are low. An arbitrage opportunity may require short- selling assets at costs that eliminate any profit potential. If the law of one price holds, there is no arbitrage opportunity. An arbitrage transaction generates a net inflow of funds: throughout the holding period. at the end of the holding period. at the start of the holding period. Correct answer- C is correct. Arbitrage is a type of transaction undertaken when two assets or portfolios produce identical results but sell for different prices. A trader buys the asset or portfolio with the lower price and sells the asset or portfolio with the higher price, generating a net inflow of funds at the start of the holding period. Because the two assets or portfolios produce identical results, a long position in one and short position in the other means that at the end of the holding period, the payoffs offset. Therefore, there is no money gained or lost at the end of the holding period, so there is no risk. The price of a forward contract: is the amount paid at initiation. is the amount paid at expiration. fluctuates over the term of the contract. Correct answer- B is correct. The forward price is agreed upon at the start of the contract and is the fixed price at which the underlying will be purchased (or sold) at expiration. Payment is made at expiration. The value of the forward contract may change over time, but the forward price does not change. 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: lower than the value of the contract. equal to the value of the contract. greater than the value of the contract. Correct answer- C is correct. The price of a forward contract is a contractually fixed price, established at initiation, at which the underlying will be purchased (or sold) at expiration. The value of a forward contract at initiation is zero; therefore, the forward price is greater than the value of the forward contract at initiation. With respect to a forward contract, as market conditions change: only the price fluctuates. only the value fluctuates. both the price and the value fluctuate. Correct answer- B is correct. The value of the forward contract, unlike its price, will adjust as market conditions change. The forward price is fixed at initiation. The value of a forward contract at expiration is: positive to the long party if the spot price is higher than the forward price. negative to the short party if the forward price is higher than the spot price. positive to the short party if the spot price is higher than the forward price. Correct answer- A is correct. When a forward contract expires, if the spot price is higher than the forward price, the long party profits from paying the lower forward price for the underlying. Therefore, the forward contract has a positive value to the long party and a negative value to the short party. However, if the forward price is higher than the spot price, the short party profits from receiving the higher forward price (the contract value is positive to the short party and negative to the long party). 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: spot price. future value of the spot price. present value of the spot price. Correct answer- B is correct. At initiation, the forward price is the future value of the spot price (spot price compounded at the risk-free rate over the life of the contract). If the forward price were set to the spot price or the present value of the spot price, it would be possible for one side to earn an arbitrage profit by selling the asset and investing the proceeds until contract expiration. 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: lower. the same. higher. Correct answer- A is correct. The forward price of each stock is found by compounding the spot price by the risk-free rate for the period and then subtracting the future value of any benefits and adding the future value of any costs. In the absence of any benefits or costs, the one-year forward prices of BWQ and ZER should be equal. After subtracting the benefits related to BWQ, the one-year forward price of BWQ is lower than the one-year forward price of ZER. If the net cost of carry of an asset is positive, then the price of a forward contract on that asset is most likely: lower than if the net cost of carry was zero. the same as if the net cost of carry was zero. higher than if the net cost of carry was zero. Correct answer- A is correct. An asset's forward price is increased by the future value of any costs and decreased by the future value of any benefits: F0(T)=S0(1+r)T−(γ−θ)(1+r)T If the net cost of carry (benefits less costs) is positive, the forward price is lower than if the net cost of carry was zero. If the present value of storage costs exceeds the present value of its convenience yield, then the commodity's forward price is most likely: less than the spot price compounded at the risk-free rate. the same as the spot price compounded at the risk-free rate. higher than the spot price compounded at the risk-free rate. Correct answer- C is correct. When a commodity's storage costs exceed its convenience yield benefits, the net cost of carry (benefits less costs) is negative. Subtracting this negative amount from the spot price compounded at the risk-free rate results in an addition to the compounded spot price. The result is a commodity forward price which is higher than the spot price compounded. The commodity's forward price is less than the spot price compounded when the convenience yield benefits exceed the storage costs and the commodity's forward price is the same as the spot price compounded when the costs equal the benefits. Which of the following factors most likely explains why the spot price of a commodity in short supply can be greater than its forward price? Opportunity cost Lack of dividends Convenience yield Correct answer- C is correct. The convenience yield is a benefit of holding the asset and generally exists when a commodity is in short supply. The future value of the convenience yield is subtracted from the compounded spot price and reduces the commodity's forward price relative to it spot price. The opportunity cost is the risk-free rate. In the absence of carry costs, the forward price is the spot price compounded at the risk-free rate and will exceed the spot price. Dividends are benefits that reduce the forward price but the lack of dividends has no effect on the spot price relative to the forward price of a commodity in short supply. When interest rates are constant, futures prices are most likely: less than forward prices. equal to forward prices. greater than forward prices. Correct answer- B is correct. When interest rates are constant, forwards and futures will likely have the same prices. The price differential will vary with the volatility of interest rates. In addition, if futures prices and interest rates are uncorrelated, forward and futures prices will be the same. If futures prices are positively correlated with interest rates, futures contracts are more desirable to holders of long positions than are forwards. This is because rising prices lead to future profits that are reinvested in periods of rising interest rates, and falling prices lead to losses that occur in periods of falling interest rates. If futures prices are negatively correlated with interest rates, futures contracts are less desirable to holders of long positions than are forwards. The more desirable contract will tend to have the higher price. In contrast to a forward contract, a futures contract: trades over-the-counter. is initiated at a zero value. is marked-to-market daily. Correct answer- C is correct. Futures contracts are marked- to-market on a daily basis. The accumulated gains and losses from the previous day's trading session are deducted from the accounts of those holding losing positions and transferred to the accounts of those holding winning positions. Futures contracts trade on an exchange, forward contracts are over-the-counter transactions. Typically both forward and futures contracts are initiated at a zero value. To the holder of a long position, it is more desirable to own a forward contract than a futures contract when interest rates and futures prices are: negatively correlated. uncorrelated. positively correlated. Correct answer- A is correct. If futures prices and interest rates are negatively correlated, forwards are more desirable to holders of long positions than are futures. This is because rising prices lead to futures profits that are reinvested in periods of falling interest rates. It is better to receive all of the cash at expiration under such conditions. If futures prices and interest rates are uncorrelated, forward and futures prices will be the same. If futures prices are positively correlated with interest rates, futures contracts are more desirable to holders of long positions than are forwards. The value of a swap typically: is non-zero at initiation. is obtained through replication. does not fluctuate over the life of the contract. Correct answer- B is correct. Valuation of the swap during its life appeals to replication and the principle of arbitrage. Valuation consists of reproducing the remaining payments on the swap with other transactions. The value of that replication strategy is the value of the swap. The swap price is typically set such that the swap contract has a value of zero at initiation. The value of a swap contract will change during the life of the contract as the value of the underlying changes in value. The price of a swap typically: is zero at initiation. fluctuates over the life of the contract. is obtained through a process of replication. Correct answer- C is correct. Replication is the key to pricing a swap. The swap price is determined at initiation by replication. The value (not the price) of the swap is typically zero at initiation and the fixed swap price is typically determined such that the value of the swap will be zero at initiation. The value of a swap is equal to the present value of the: fixed payments from the swap. net cash flow payments from the swap. underlying at the end of the contract. Correct answer- B is correct. The principal of replication articulates that the valuation of a swap is the present value of all the net cash flow payments from the swap, not simply the present value of the fixed payments of the swap or the present value of the underlying at the end of the contract. A European call option and a European put option are written on the same underlying, and both options have the same expiration date and exercise price. At expiration, it is possible that both options will have: negative values. the same value. positive values. Correct answer- B is correct. If the underlying has a value equal to the exercise price at expiration, both options will have zero value since they both have the same exercise price. For example, if the exercise price is $25 and at expiration the underlying price is $25, both the call option and the put option will have a value of zero. The value of an option cannot fall below zero. The holder of an option is not obligated to exercise the option; therefore, the options each have a minimum value of zero. If the call has a positive value, the put, by definition, must have a zero value and vice versa. Both cannot have a positive value. At expiration, a European put option will be valuable if the exercise price is: less than the underlying price. equal to the underlying price. greater than the underlying price. Correct answer- C is correct. A European put option will be valuable at expiration if the exercise price is greater than the underlying price. The holder can put (deliver) the underlying and receive the exercise price which is higher than the spot price. A European put option would be worthless if the exercise price was equal to or less than the underlying price. The value of a European call option at expiration is the greater of zero or the: value of the underlying. value of the underlying minus the exercise price. exercise price minus the value of the underlying. Correct answer- B is correct. The value of a European call option at expiration is the greater of zero or the value of the underlying minus the exercise price. For a European call option with two months until expiration, if the spot price is below the exercise price, the call option will most likely have: zero time value. positive time value. positive exercise value. Correct answer- B is correct. A European call option with two months until expiration will typically have positive time value, where time value reflects the value of the uncertainty that arises from the volatility in the underlying. The call option has a zero exercise value if the spot price is below the exercise price. The exercise value of a European call option is Max(0, St - X ), where St is the current spot price at time t and X is the exercise price. When the price of the underlying is below the exercise price, a put option is: in-the-money. at-the-money. out-of-the-money. Correct answer- A is correct. When the price of the underlying is below the exercise price for a put, the option is said to be in-the-money. If the price of the underlying is the same as the exercise price, the put is at-the-money and if it is above the exercise price, the put is out-of-the-money. If the risk-free rate increases, the value of an in-the-money European put option will most likely: decrease. remain the same. increase. Correct answer- A is correct. An in-the-money European put option decreases in value with an increase in the risk-free rate. A higher risk-free rate reduces the present value of any proceeds received on exercise. The value of a European call option is inversely related to the: exercise price. time to expiration. volatility of the underlying. Correct answer- A is correct. The value of a European call option is inversely related to the exercise price. A lower exercise price means there are more potential outcomes at which the call expires in-the-money. The option value will be greater the lower the exercise price. For a higher exercise price, the opposite is true. Both the time to expiration and the volatility of the underlying are directly (positively) related to the value of a European call option. The table below shows three European call options on the same underlying: Time to Expiration Exercise Price Option 1 3 months $100 Option 2 6 months $100 Option 3 6 months $105 The option with the highest value is most likely: Option 1. Option 2. Option 3. Correct answer- B is correct. The value of a European call option is inversely related to the exercise price and directly related to the time to expiration. Option 1 and Option 2 have the same exercise price; however, Option 2 has a longer time to expiration. Consequently, Option 2 would likely have a higher value than Option 1. Option 2 and Option 3 have the same time to expiration; however, Option 2 has a lower exercise price. Thus, Option 2 would likely have a higher value than Option 3. The value of a European put option can be either directly or inversely related to the: exercise price. time to expiration. volatility of the underlying. Correct answer- B is correct. The value of a European put option can be either directly or indirectly related to time to expiration. The direct effect is more common, but the inverse effect can prevail the longer the time to expiration, the higher the risk-free rate, and the deeper in-the-money is the put. The value of a European put option is directly related to the exercise price and the volatility of the underlying. Prior to expiration, the lowest value of a European put option is the greater of zero or the: exercise price minus the value of the underlying. present value of the exercise price minus the value of the underlying. value of the underlying minus the present value of the exercise price. Correct answer- B is correct. Prior to expiration, the lowest value of a European put is the greater of zero or the present value of the exercise price minus the value of the underlying. A European put option on a dividend-paying stock is most likely to increase if there is an increase in: carrying costs. the risk-free rate. dividend payments. Correct answer- C is correct. Payments, such as dividends, reduce the value of the underlying which increases the value of a European put option. Carrying costs reduce the value of a European put option. An increase in the risk-free interest rate may decrease the value of a European put option. Based on put-call parity, a trader who combines a long asset, a long put, and a short call will create a synthetic: long bond. fiduciary call. protective put. Correct answer- A is correct. A long bond can be synthetically created by combining a long asset, a long put, and a short call. A fiduciary call is created by combining a long call with a risk free bond. A protective put is created by combining a long asset with a long put. Which of the following transactions is the equivalent of a synthetic long call position? Long asset, long put, short call Long asset, long put, short bond Short asset, long call, long bond Correct answer- B is correct. According to put−call parity, a synthetic call can be constructed by combining a long asset, long put, and short bond positions. Which of the following is least likely to be required by the binomial option pricing model? Spot price Two possible prices one period later Actual probabilities of the up and down moves Correct answer- C is correct. The actual probabilities of the up and down moves in the underlying do not appear in the binomial option pricing model, only the pseudo or "risk-neutral" probabilities. Both the spot price of the underlying and two possible prices one period later are required by the binomial option pricing model. An at-the-money American call option on a stock that pays no dividends has three months remaining until expiration. The market value of the option will most likely be: less than its exercise value. equal to its exercise value. greater than its exercise value. Correct answer- C is correct. Prior to expiration, an American call option will typically have a value in the market that is greater than its exercise value. Although the American option is at-the-money and therefore has an exercise value of zero, the time value of the call option would likely lead to the option having a positive market value. At expiration, American call options are worth: less than European call options. the same as European call options. more than European call options. Correct answer- B is correct. At expiration, the values of American and European call options are effectively the same; both are worth the greater of zero and the exercise value. Which of the following circumstances will most likely affect the value of an American call option relative to a European call option? Dividends are declared Expiration date occurs The risk-free rate changes Correct answer- A is correct. When a dividend is declared, an American call option will have a higher value than a European call option because an American call option holder can exercise early to capture the value of the dividend. At expiration, both types of call options are worth the greater of zero and the exercise value. A change in the risk-free rate does not affect the relative values of American and European call options. Combining a protective put with a forward contract generates equivalent outcomes at expiration to those of a: fiduciary call. long call combined with a short asset. forward contract combined with a risk-free bond. Correct answer- A is correct. Put−call forward parity demonstrates that the outcome of a protective put with a forward contract (long put, long risk-free bond, long forward contract) equals the outcome of a fiduciary call (long call, long risk-free bond). The outcome of a protective put with a forward contract is also equal to the outcome of a protective put with asset (long put, long asset).