Quant Finance Foundations, Part 5. Previously: Forwards, Futures and Swaps.
Option payoffs bend where everything priced so far ran straight. A forward moves one-for-one with its underlying, which is why a yield curve and arithmetic were enough. An option breaks that pattern.
Moreover, that bend is exactly what makes options both useful to clients and mathematically demanding to price. Before reaching for any formula, therefore, it is worth being precise about what the instrument actually is.
Right, not obligation
An option gives its holder the right but not the obligation to buy (a call) or sell (a put) an underlying asset at a fixed strike price, on or before a fixed date. The seller has the mirror image: an obligation with no right.
That asymmetry is the whole product. In a forward, both parties are committed and neither pays the other anything up front. In an option, one party holds all the choice, so that party pays a premium for it. Optionality is never free, and the size of the premium is the subject of everything that follows.
Exercise style matters for pricing though not for intuition. European options can be exercised only at expiry. American options can be exercised any time up to expiry, which adds an optimal-stopping problem on top of the valuation. Most of what a beginner needs to understand can be learned on European options.
The four basic option payoffs
The clearest way to see an option is by its value at expiry as a function of the underlying price S, with strike K.
| Position | Payoff at expiry | Maximum gain | Maximum loss |
|---|---|---|---|
| Long call | max(S − K, 0) | Unlimited | Premium paid |
| Long put | max(K − S, 0) | K, less premium | Premium paid |
| Short call | −max(S − K, 0) | Premium received | Unlimited |
| Short put | −max(K − S, 0) | Premium received | K, less premium |
- Long call: max(S − K, 0). Flat at zero below the strike, then rising at 45 degrees. Loss capped at the premium; gain unbounded.
- Long put: max(K − S, 0). Falling to the left of the strike, flat at zero to the right. Maximum gain is K (the underlying going to zero).
- Short call: the long call flipped. Premium collected, unbounded loss above the strike.
- Short put: premium collected, large loss if the underlying collapses.
The kink at the strike is the mathematical heart of the instrument. A forward’s payoff is a straight line and differentiable everywhere; an option’s payoff has a corner. Everything difficult about options — the changing hedge ratio, the sensitivity to volatility, the need for continuous rebalancing — traces to that corner.
A worked case
A three-month call on a stock trading at 100, strike 105, premium 3.
- Stock at 103 at expiry: option pays 0. You lose the premium of 3.
- Stock at 105: option pays 0. Same loss. The strike itself is a break-even for the payoff, not for you.
- Stock at 108: option pays 3. Net zero — this is your true break-even, strike plus premium.
- Stock at 130: option pays 25. Net gain 22.
Notice the seller’s position in that last line. They collected 3 and paid out 25. Selling options is a business of collecting small amounts frequently and occasionally paying out a great deal, which is exactly why the pricing has to be right.
Moneyness
Moneyness describes where the underlying sits relative to the strike, and it is standard vocabulary on any derivatives desk.
- In-the-money (ITM): exercising now would produce a positive payoff. Call with S > K, put with S < K.
- At-the-money (ATM): S ≈ K. In practice desks often mean at-the-money-forward, comparing the strike to the forward price rather than spot.
- Out-of-the-money (OTM): exercising now would be worthless. Cheapest options, and the ones retail buyers gravitate toward for the same reason people buy lottery tickets.
Why an option is worth more than its payoff today
An option’s value splits into two components, and the second is where the interesting behaviour lives.
Intrinsic value is what you would get exercising immediately: max(S − K, 0) for a call. Never negative, because you would simply not exercise.
Time value is everything else — the market price minus intrinsic value. Take that 105-strike call with the stock at 100. Intrinsic value is zero, yet it trades at 3. Someone is paying 3 for something worth nothing today, because before expiry the stock might move above 105.
Time value is the price of that possibility, and it depends on two things:
- Time remaining. More time, more chance of a favourable move. Time value decays toward zero at expiry, and the decay accelerates as expiry approaches rather than running linearly.
- Volatility. The critical one. A stock that swings widely is far more likely to travel past the strike than a placid one.
The asymmetry that makes volatility valuable
Here is the point that separates people who understand options from people who have merely memorised the payoff diagram.
Take that 105-strike call, stock at 100. Suppose the stock moves ±20 instead of ±5. On the upside at 120, the option pays 15 instead of 0. On the downside at 80, the option pays zero — exactly the same as it would have paid at 95. The extra downside movement costs the holder nothing, because the loss was already capped at the premium.
Higher volatility increases the upside without increasing the downside. So an option on a volatile asset is worth more than an identical option on a stable one, holding everything else equal. This is true for both calls and puts.
Which yields a conclusion that surprises most newcomers: volatility is the primary input to an option’s price. Not the expected return of the underlying. Not whether you think the stock is going up. Two analysts with opposite directional views but the same volatility estimate should agree on the option’s value.
Why that is so — and why an option’s price turns out not to depend on the underlying’s expected return at all — is the subject of the next post, and it is best understood by building the price up from a two-state model rather than starting from a formula.
Try it yourself
Chart a payoff diagram yourself, in this order:
- In a spreadsheet, build a column of possible expiry prices from 60 to 140.
- Add a call payoff column using =MAX(S−K,0).
- Subtract the premium to get your net position, then chart it.
- Build a second column for a put at the same strike.
- Add the two payoff columns together and chart the result.
The result is a V-shaped profile that profits from a large move in either direction and loses if nothing happens. That is a straddle, and constructing it from two payoff columns teaches more about options than reading a definition of it.
Frequently asked questions
Why do option payoffs have a kink at the strike?
Because the holder has a right rather than an obligation, and will simply not exercise when doing so would lose money. That choice creates the corner.
What is the difference between intrinsic value and time value?
Intrinsic value is what you would receive exercising immediately. Time value is everything else — the price of the chance that the position improves before expiry.
Why does volatility raise the value of both calls and puts?
Higher volatility widens the upside while the downside stays capped at the premium. Consequently the expected payoff rises for either type.
Does the expected return of the stock affect the option price?
No. Two analysts with opposite directional views but the same volatility estimate should agree on the value, as the binomial tree post demonstrates.
Next in this series: binomial tree option pricing — replication in five steps, using nothing beyond secondary-school arithmetic. Later, the Greeks turn this theory into a hedging job.