Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities — MCQs

Meridian Options Advisory is reviewing several option analyses for an institutional client. The client permits only fully funded or defined-risk positions and requires valuations to be based on no-arbitrage relationships. There are no transaction costs, taxes, or restrictions on short sales. Unless otherwise stated, the underlying shares pay no dividends during the relevant option lives. The first analysis uses one-year European calls and puts with identical exercise prices and expiration dates. An associate observes that buying a call and investing the present value of the exercise price must produce the same expiration payoff as buying the stock and a put. The associate will use this relationship and the market data in Exhibit 1 to infer the put price. All cash flows occur exactly one year from today, and the stated risk-free rate is an effective annual rate. The second analysis values a European call with a one-period binomial model. The stock can move only to the two terminal prices supplied in Exhibit 1. The risk-free asset grows by 5% during the period. The analyst is instructed to derive the risk-neutral probability from the stock's up and down gross returns, calculate the expected option payoff under that probability, and discount the result for one period. The actual probability of the stock rising is not required for no-arbitrage valuation. The third analysis considers the two-call position in Exhibit 2. The calls have the same expiration date and are written on the same share. The client expects a moderate increase in the share price but is willing to cap gains in exchange for reducing the initial premium. Meridian reports option profits after incorporating all premiums paid and received. Premiums are measured at initiation and are not carried forward with interest for this analysis. Finally, an analyst considers whether the client should exercise an American call on a non-dividend-paying share before expiration. The call is currently in the money and has remaining time value. The client can sell the option at its fair market value, and there are no market frictions preventing a sale. The analyst must distinguish intrinsic value from total option value and consider the financing benefit of delaying payment of the exercise price.

Northbridge Options Fund uses listed options both to alter portfolio payoffs and to identify relative-value opportunities. Portfolio manager Anika Patel asks analyst David Lin to review option prices on non-dividend-paying shares. All options described in the case have the same contract multiplier, and transaction costs, taxes, and counterparty default are ignored. Lin first evaluates a European call and put written on the same share. Both options have the same exercise price and expiration date. The share cannot be sold short only through the option market; ordinary share borrowing is available, so the standard no-arbitrage put–call parity relationship applies. Exhibit 1 provides the current share price, call value, and present value of the exercise price. Lin must infer the put value that prevents an arbitrage among the share, call, put, and risk-free bond. Lin next values a different European call using a one-period binomial model. Over the single period, the underlying share will finish at either the specified up-state price or down-state price. The exercise price equals the current share price. The risk-free growth factor is positive and lies between the share's up and down gross returns, so a valid risk-neutral probability exists. The option cannot be exercised before the end of the period. All binomial inputs needed for a direct valuation are included in Exhibit 1. Patel also owns a concentrated position in another non-dividend-paying company. She expects its share price to remain approximately unchanged or rise only modestly over the next six months. She is willing to relinquish gains above a specified exercise price in exchange for immediate option premium income, but she does not want to sell the shares. She asks Lin to identify the option strategy most directly aligned with that objective. During the review, Lin compares American and European puts with otherwise identical terms. He recognizes that American exercise rights cannot make the holder worse off because the holder may choose to wait until expiration. For some sufficiently deep-in-the-money puts, receiving the exercise proceeds before expiration can be advantageous. Patel's statements about the covered-call strategy and the American put are reproduced in Exhibit 2. Each conclusion should be assessed independently using the facts applicable to that instrument.

Meridian Wealth Advisors is reviewing option positions for a client and testing the consistency of several option valuation conclusions. The analysis assumes frictionless markets, no taxes or transaction costs, unrestricted borrowing and lending at the risk-free rate, and no counterparty default. All options are on non-dividend-paying underlying securities unless otherwise stated. The first valuation involves European calls and puts with the same underlying security, exercise price, and expiration date. Analyst Priya Shah applies put–call parity and notes that the exercise price represents a payment at expiration. Accordingly, the present value of the exercise price, rather than its undiscounted amount, enters the parity relationship. A colleague proposes using the undiscounted exercise price because the options can only be exercised at expiration. The second valuation uses a one-period binomial model. The underlying can move to only the stated up-state or down-state value. Shah will calculate the risk-neutral probability from the gross risk-free return and the up and down price multipliers. The model does not require an estimate of the underlying security's actual expected return or the investor's risk aversion. The call's terminal payoff equals the greater of zero and the terminal underlying price less the exercise price. Meridian also advises a client who already owns a diversified equity portfolio. The client is concerned about a material market decline during the option horizon but does not want to sell the portfolio or surrender gains if the market rises. The client accepts that obtaining this asymmetric protection can require payment of a net option premium. The strategy must therefore create a floor while preserving unlimited appreciation above that floor. Finally, Shah reviews the distinction between American and European exercise rights. She focuses on a call written on a non-dividend-paying stock. Although the American contract permits exercise before expiration, exercising early would require paying the exercise price sooner and would eliminate the option's remaining time value. The absence of dividends removes a potential benefit from obtaining the stock before expiration. The option data, client objective, and analyst statements are presented in Exhibits 1 and 2.

Halcyon Capital is evaluating one-year European options on a non-dividend-paying stock using a one-period binomial framework. The stock currently trades at USD 50.00. At option expiration, the stock price will be either USD 60.00 or USD 40.00. Both the call and put have a strike price of USD 52.00 and can be exercised only at expiration. The annual effective risk-free rate is 5.00%, borrowing and lending are available at that rate, and markets are free of taxes and transaction costs. The two stock-price outcomes permit complete replication of each contingent claim. An analyst first proposes estimating the call's value by assigning an equal probability to each terminal stock price. The chief risk officer responds that no-arbitrage valuation requires the risk-neutral probability implied by the stock-price movements and the risk-free rate. The desk must therefore calculate the option's state-contingent payoffs, derive the risk-neutral probability, and discount the resulting expected payoff. No estimate of the stock's actual expected return is needed. A second analyst evaluates the European put. Because the put and call have identical underlying securities, strike prices, and expiration dates, the analyst can apply European put–call parity. The present value of the strike must be calculated using the stated annual effective rate. The analyst is cautioned not to substitute the strike price itself for its present value and not to use an American-option relationship. The desk also considers replicating the call directly. A portfolio containing a fractional number of shares and a borrowing or lending position must produce the call's payoff in both terminal states. The stock position is determined from the change in option payoff divided by the change in stock price, not from either the physical or risk-neutral probability of an up move. Separately, a private client already owns shares of the same stock. The client expects long-term appreciation but is concerned about a substantial decline over the next year. The client wants a contractual minimum value at the end of the year and does not want to surrender gains if the stock rises sharply. The client accepts paying an upfront premium and has no objective to generate current income from writing options.

Summit Capital advises institutional and private-wealth clients on the use of exchange-traded equity options. Daniel Ortega, an options strategist, is reviewing contracts on the non-dividend-paying common stock of Horizon Systems. All options discussed are free of default risk, and transaction costs and taxes are ignored. Ortega first examines a European call and put with identical exercise prices and expiration dates. The call's observed market price and the other inputs required to apply put–call parity are shown in Exhibit 1. Ortega notes that the present value of the exercise price, rather than the undiscounted exercise price, must be used in the parity relationship. The put price requested is the no-arbitrage price implied by the observed call. He next applies a one-period binomial model to a different European call on the same stock. At the end of the period, the stock will be worth either USD 120 or USD 90. The exercise price is USD 105, and the effective risk-free rate is 2% for the period. The model assumes the stock's up and down factors span the risk-free gross return, making a valid risk-neutral probability available. The option has no interim cash flows, and its value is obtained by discounting its risk-neutral expected payoff. A private client of Summit owns a concentrated position in Horizon Systems. The client expects the stock to appreciate over the coming year but cannot tolerate a terminal stock value below a specified floor. The client is willing to pay an option premium to establish that floor and explicitly wants to retain the benefit of stock-price appreciation above the current market price. Ortega is considering a covered call, a protective call, and a protective put. Finally, Ortega compares otherwise identical American and European puts, and separately American and European calls, on the stock. Each pair has the same exercise price and expiration date. The American contracts may be exercised at any time through expiration, whereas the European contracts may be exercised only at expiration. Ortega is not asked to determine whether early exercise is optimal for the put under current conditions; instead, he must determine the general no-arbitrage value relationship created by the American put's additional exercise flexibility, as well as the well-known result governing early exercise of calls on non-dividend-paying stocks.