DER 8 Pricing and Valuation of Options

An option is a contingent claim: one side, the buyer, decides whether a trade happens. That right is not free. At inception a forward commitment is worth zero to both parties, but an option buyer pays the seller a premium up front in exchange for the choice to transact the underlying later at a set price. The lessons here deal with European options, which may be exercised only at expiration, and they ignore any cost or benefit of holding the underlying beyond the opportunity cost captured by the risk-free rate.

Because the payoff is one-sided, buyers and sellers watch three related gauges over the life of the contract: the exercise value (also called the intrinsic value), the moneyness, and the time value. Together the first and third add up to the option price, so we begin with the exercise value.

Payoff at maturity

At expiration the buyer walks away unless the trade returns a positive amount. A call buyer transacts only when the underlying finishes above the exercise price; a put buyer transacts only when it finishes below. Writing the underlying price at maturity as and the exercise price as , the buyer receives:

Exercise value before maturity

At any earlier date , the exercise value asks a hypothetical question: what would the contract be worth if it could be settled right now? Answering it means comparing the current spot price with the exercise price, but the exercise price is not paid until maturity, so it is discounted back at the risk-free rate. The exercise value therefore relies on the discounted exercise price rather than its face amount:

There is a neat link to forward pricing. Set the exercise price to match the forward price, so that . When the spot price sits above PV(X), the call exercise value matches what a long forward commitment is worth at that same instant, namely . The two coincide only over the region where the option would be exercised; below it the option floors at zero while the forward keeps falling. This comparison also sets aside the premium the option buyer paid at inception.

Moneyness compares the option value with its exercise price and sorts every option into one of three states. A contract is in the money (ITM) when exercising it now would pay something, at the money (ATM) when the two prices are equal, and out of the money (OTM) when exercising now would pay nothing. Because a call and a put point in opposite directions, the same spot price places them on opposite sides of the ledger.

Moneyness does more than label a state; it also governs how sharply the option price reacts to the underlying. A deep-in-the-money option is almost certain to be exercised, so its price tracks the underlying nearly one for one. A deep-out-of-the-money option is almost certain to expire unused, so its price barely stirs when the underlying moves. An at-the-money option sits on the knife edge, where a small move in the underlying can decide whether exercise happens at all. Since moneyness works off the same underlying, it gives a clean way to compare options that differ only in exercise price or time to maturity.

Exercise value captures what an option would return if settled today. The rest of its price comes from what might still happen before expiration. Even though a European option cannot be exercised early, it can be bought or sold in the meantime at a price ( for a call, for a put) that reflects the expected payoff at maturity. That extra amount is the time value, and it is defined as the option price minus the exercise value:

A longer horizon usually widens the range of prices the underlying could reach for a given level of volatility, which raises the chance of a large favorable move. Because the buyer can ignore the outcomes where the option expires unused, that wider spread only helps. Time value is therefore always positive, and it erodes toward zero as expiration approaches, a process known as time value decay. At maturity nothing remains but exercise value.

Riskless arbitrage becomes possible whenever the law of one price fails, meaning the same asset trades at two prices at once. For derivatives, the no-arbitrage conditions come from the payoff at maturity. A forward commitment has a symmetric payoff, settling on the gap between the forward price agreed at inception and the underlying price at maturity, so a long forward earns . An option is different: its payoff is asymmetric because the buyer transacts only when doing so pays. Including the premium, the buyer profit is:

This conditional payoff forces price bounds on the option at every date . A call buyer exercises only if the underlying ends above the exercise price, so the call can never trade below its exercise value; an option quoted under its exercise value would itself be an arbitrage. At the same time no rational buyer pays more for the right to buy an asset than the asset itself costs, which caps the call at the spot price:

A put runs the other way. Its buyer exercises only when the underlying ends below the exercise price, and the most a put can ever deliver is the exercise price itself (received when the underlying is worthless), which sets the upper bound. Its lower bound mirrors the call, being the discounted exercise price minus the spot price, floored at zero:

Replication rebuilds a derivative payoff from positions in the underlying and cash borrowed or lent. The fact that traders can do this is what keeps the law of one price intact and closes off arbitrage. A call resembles a long forward in that both gain when the underlying rises, but the call settles only when exercise pays. To copy a call at inception, borrow at the risk-free rate and buy the underlying at its spot price. At expiration two branches are possible:

• Underlying above the exercise price: sell it for and use the proceeds to repay the exercise price X.
• Underlying below the exercise price: no settlement is needed.

If exercise were certain, the trader would borrow the full at inception, just as with a forward. Because exercise is uncertain, the trader instead borrows only a fraction of that amount, scaled to the likelihood of exercise, which is tied to the moneyness. As that likelihood shifts, the borrowing has to be adjusted; a forward, whose settlement is certain, needs no such rebalancing.

A put is the mirror image. A short put behaves like a long forward when the underlying ends below the forward price, so replicating a long put means selling the underlying short at its spot price and lending the proceeds at the risk-free rate. At expiration:

• Underlying below the exercise price: buy the underlying for using the loan proceeds.
• Underlying above the exercise price: no settlement is needed.

Six inputs drive an option value: the current price of the underlying asset, the exercise price, how much time remains, the risk-free rate, how volatile the underlying is, and any income or cost tied to holding it. Some of these also move forward commitments; others are specific to the one-sided payoff of a contingent claim. Taking each in turn clarifies both the direction and, for a couple of them, why the direction differs between calls and puts.

Value of the underlying

A rise in the underlying lifts a call and a long forward alike, while a fall lifts a put and a short forward. The difference is in the magnitude of the response. A forward moves one for one with the underlying, but an option moves by an amount that depends on how likely exercise is. The deeper in the money the option, the more its value tracks the underlying, because exercise is close to certain.

Exercise price

For a call, the exercise price is the floor on what the buyer must pay to acquire the underlying, so a lower exercise price raises both the chance of exercise and the settlement amount , lifting the call value. For a put, the exercise price is the amount the holder collects on sale, so a higher exercise price raises the value of the put. The two respond to exercise price in opposite directions.

Time to expiration

A longer horizon almost always helps a call, since the upside of the underlying is effectively unbounded and grows with time, while the downside is capped at the premium. A put usually benefits too, because more time means more room for the underlying to fall below the exercise price. There is one wrinkle for puts: waiting to collect pushes that receipt further into the future, lowering its present value. For a deep-in-the-money put when the risk-free rate is high, that drag can dominate, so occasionally a longer life lowers a put value. This is why time to expiration is the one factor whose put effect is not strictly positive.

Risk-free rate

The risk-free rate discounts the exercise price. Writing the exercise values as for a call and for a put makes the effect clear: a higher rate shrinks PV(X). For an in-the-money call that raises the exercise value, and for an in-the-money put it lowers the exercise value. The rate touches only the exercise value, not the time value.

Volatility of the underlying

Volatility measures the expected dispersion of the future underlying price. Wider dispersion raises the odds of a large positive exercise value while leaving the worthless-expiration outcome unchanged, so higher volatility lifts both calls and puts. Lower volatility trims the time value of both. This is one of the clearest cases where a call and a put move in the same direction.

Income or cost of owning the underlying

Income and other non-cash benefits, such as a convenience yield, accrue to the holder of the underlying but not to the holder of a derivative on it. Such benefits reduce a call value and raise a put value. Carry costs, such as storage and insurance on a commodity, do the reverse: they raise a call value and lower a put value.

Sign shows the impact of an increase in the factor. For the last row, the first sign refers to income and the second to carry cost.

Two patterns are worth committing to memory. Volatility and time to expiration usually push calls and puts the same way, while the exercise price, any income or carry cost, and the risk-free rate tend to push calls and puts in opposite directions.