Pricing and Valuation of Options — MCQs

Hawthorne Equity Portfolio holds a diversified portfolio of non-dividend-paying shares and uses European options to modify its payoff distribution. The portfolio manager observes the market data in Exhibit 1 for a matched pair of European call and put options. Both options have the same underlying share, exercise price, and expiration date. The call is actively traded, but the put is not, so the manager estimates the put's no-arbitrage value using put–call parity. There are no transaction costs, taxes, short-sale restrictions, or interim cash flows from the underlying share. For a different share, the manager constructs the one-period binomial model in Exhibit 2. The share will have exactly one of the two stated values at the end of the period. A European call expires at that time and pays the greater of zero or the share price less the exercise price. The manager plans to value the call by forming a portfolio of the underlying share and risk-free borrowing that exactly replicates the call's state-contingent payoffs. The risk-free rate is effective for the single period, and borrowing and lending are available at that rate. A client represented by Hawthorne already owns a concentrated share position and expects the shares to appreciate over the next year. The client cannot tolerate an exit value below a specified floor but is willing to pay an option premium to retain participation in price increases. The client does not want a strategy that requires selling the shares today, creates an uncovered option exposure, or completely fixes the future sale price. The manager also reviews how a change in expected volatility affects otherwise identical European calls and puts. The options' exercise prices, time to expiration, risk-free rate, and underlying share price are held constant. Neither option can be exercised before expiration, and both option holders have limited downside equal to the premiums paid. The analysis concerns option value before expiration rather than the options' expiration payoffs or the profit earned after deducting premiums.

Meridian Capital manages an equity portfolio containing a concentrated position in a non-dividend-paying company. The derivatives team is evaluating European calls and puts on the shares using a one-period binomial model. The model assumes that by the end of the period the share price will take one of the two values shown in Exhibit 1. The one-period risk-free rate is 5.00%, and borrowing and lending are available at that rate. There are no transaction costs, taxes, dividends, or arbitrage opportunities. A junior analyst proposes valuing the call by assigning equal probabilities to the up and down states. The senior analyst rejects that approach, noting that no real-world state probabilities have been supplied and are not required for no-arbitrage valuation. Instead, the team can determine a risk-neutral probability or construct a portfolio of the underlying share and risk-free borrowing that exactly reproduces the call's state-contingent payoffs. The call is European and has an exercise price of USD 100.00. Therefore, its expiration payoff is USD 20.00 in the up state and zero in the down state. The team also considers a European put with the same underlying, exercise price, and expiration date. The senior analyst instructs the team to value this put through European put–call parity using the calculated call value of USD 10.88. Because the stock pays no dividends, the applicable relationship is between the call, the put, the current stock price, and the present value of the exercise price. The risk-free asset used in the parity relationship pays USD 100.00 at expiration. For risk management, Meridian wants to understand the number of shares represented by the call's replicating portfolio. The senior analyst explains that the option's hedge ratio must be based on the change in option value across the two terminal states divided by the corresponding change in the underlying share price. It is not based on the option's unconditional expected payoff. A portfolio manager separately asks whether changing the put from European to American exercise would affect its value. The stock still pays no dividends, but the exercise price is received when a put is exercised. The senior analyst notes that the American holder may exercise at any time through expiration but is not required to do so. All other contract terms would remain unchanged.

Northbridge Options Fund uses listed European options to modify portfolio payoffs and obtain synthetic exposures. Elena Rossi, the fund's option strategist, is reviewing four potential analyses. Unless otherwise stated, the underlying stocks pay no dividends, options can be exercised only at expiration, and markets are free of taxes, transaction costs, and arbitrage opportunities. For the first analysis, Rossi observes a stock, a call option, and a risk-free bond and is asked to infer the value of a put with the same exercise price and expiration date as the call. She plans to use European put–call parity. A colleague proposes using the exercise price without discounting it, but Rossi notes that the parity relationship requires the present value of the exercise price. For the second analysis, Rossi applies a one-period binomial model. The stock can reach only the specified up-state or down-state price at the end of the period. She intends to calculate the risk-neutral probability using the gross risk-free return and then discount the probability-weighted option payoff. The actual probability of an upward stock movement is not needed for no-arbitrage valuation. A portfolio manager then asks Rossi to protect an existing long stock position. The manager requires a minimum expiration value but wants to retain the benefit if the stock appreciates. Premium cost is acceptable, and the manager does not want to sell the shares. Rossi compares buying a put, selling a call, and selling a put on the same stock. Finally, Rossi reviews a covered-call position established by buying a share and simultaneously selling a call. Both positions will be held to option expiration. She explains that the call premium reduces the net initial investment but that the short call limits the stock price at which further gains accrue. Exhibit 1 presents the complete inputs for the four analyses.

Northbridge Options Advisory assists portfolio managers with option valuation and strategy design. All options considered are European unless explicitly stated otherwise, and all underlying shares are free of default risk. Transaction costs and taxes are ignored. Options with the same underlying, exercise price, and expiration date are subject to no-arbitrage relationships. Analyst Priya Shah first reviews one-year options on a non-dividend-paying stock. She intends to calculate the put price implied by the observed call price using put–call parity. A colleague suggests discounting the option premium because it is paid before expiration. Shah notes instead that parity compares current option prices with the present value of the exercise price. The relevant market information appears in Exhibit 1. Shah then considers synthetic positions. One institutional client wants the same expiration payoff as owning a stock while purchasing a put with exercise price X. The client is willing to use a call and a default-free zero-coupon bond rather than trade the stock and put directly. The bond can be purchased today for the present value of X and pays X at expiration. For another client, Shah values a one-year European call in a single-period binomial framework. The stock has only the two possible expiration values shown in Exhibit 2. The effective risk-free return lies between the down-state and up-state stock returns, so the model has no immediate arbitrage. Shah may value the call either with a replicating portfolio or by discounting its risk-neutral expected payoff. Finally, client Elena Ortega owns a concentrated position in technology shares that she cannot sell for six months because of an employer-imposed trading restriction. She maintains the shares as a long-term holding but has a separate moderately bullish six-month view and wants an option overlay whose maximum loss is limited to the initial net premium. She does not want to pay the relatively high premium required for an at-the-money protective put. She is willing to give up gains above a specified upper price in exchange for reducing the option premium, but she does not want to write an uncovered option. Shah considers a covered call, a protective put, and a bull call spread using options that expire when the restriction ends.

Northbridge Asset Management is reviewing option positions for an equity portfolio. All options in the review are European, the underlying stocks pay no dividends during the relevant periods, and there are no transaction costs or arbitrage restrictions. Cash flows occurring at expiration are discounted using the risk-free rates shown in Exhibit 1. For the first analysis, an options specialist checks whether a one-year put is priced consistently with an observed call, the underlying stock, and a risk-free zero-coupon bond. The call and put have the same exercise price and expiration date. The specialist uses the annual effective risk-free rate and notes that the exercise price must be discounted for put–call parity. A junior analyst instead uses the undiscounted exercise price, arguing that exercise occurs at expiration; the specialist explains that parity compares present values at the valuation date. The team next applies a one-period binomial model to a different stock. At the end of the period, the stock will have one of the two values stated in Exhibit 1. The team is valuing a European call with an exercise price of USD 85. The replicating portfolio consists of shares of stock and borrowing or lending at the risk-free rate. For the hedge ratio, Northbridge uses the difference between the option's up-state and down-state payoffs divided by the difference between the corresponding stock prices. The team does not use the actual probability of either stock-price outcome. A portfolio manager also owns a concentrated position in a listed stock. She wants to establish a minimum value at which the position can effectively be sold during the next six months, but she wants to retain gains if the stock appreciates. She is willing to pay an option premium to achieve this asymmetric payoff. A covered call and a protective put are both liquid, but only one directly meets both objectives. She is also weighing whether adding a short call to the protective put, forming a collar, would better suit her cost and upside preferences. Finally, the derivatives committee assesses a long at-the-money straddle consisting of a call and a put with identical exercise prices and expiration dates. The committee expects a material increase in volatility but has no directional view on the stock price. Interest rates, the stock price, time to expiration, and expected dividends are assumed unchanged during this assessment.