Option pricing: where fair option prices come from

Advanced13 min read · updated 3 September 2026 · 18 sources

Option pricing answers a question that looks impossible at first: what is the right to buy a stock a year from now at a fixed price worth, when nobody knows the future price? The answer comes not from a forecast but from the absence of arbitrage: an option can be replicated with stock and borrowed money, so it must cost as much as that replica. This article takes you from the basic idea through Black-Scholes and binomial trees to the Greeks, the volatility smile, exotic options and strategies, with recalculated examples.

The idea: price through replication

A call gives you the right to buy the underlying at the exercise price (strike) K; at expiration it is worth max(S_T − K, 0), and a put is worth max(K − S_T, 0). The key point: this payoff can be replicated with the right number of shares and a loan. If such a replica exists, the option must cost exactly the same. Otherwise, you could make money without risk. Black and Scholes (1973) put this principle at the very start of their paper:

A calculation with risk-neutral probabilities gives the same result: if you choose the probability of a rise so that the stock earns exactly the risk-free rate on average, i.e., (1.05 − 0.8) / (1.2 − 0.8) = 0.625, the call is worth the discounted expected value 0.625 × 20 / 1.05 = 11.90. This does not mean investors are risk-neutral; these probabilities are just a computational tool. Cox et al. (1979) built the binomial model with many small steps on this idea.

Black-Scholes: formula, inputs and assumptions

As the steps become smaller and smaller, the price follows a geometric Brownian motion, and the replica turns into a continuous hedge. The result is the formula of Black and Scholes (1973), which Merton (1973) extended, among other things, to dividends:

The formula has only five inputs, and four of them are observable. The stock's expected return is missing because it is already built into the replica. The only unknown is volatility, which is why option traders are, at heart, trading volatility. The formula rests on these assumptions:

• lognormal prices with constant volatility and no jumps,
• continuous, costless trading with infinitely divisible positions and short selling allowed,
• a constant risk-free rate, with dividends as a continuous yield,
• European exercise, i.e., only on the expiration date.

For other underlyings, a single parameter, the cost of carry, is enough: for futures it is zero (Black 1976), and for currencies it equals the domestic minus the foreign interest rate (Garman and Kohlhagen 1983). Violated assumptions do not make the formula useless; they push the errors into the implied volatility (more on that below).

Put-call parity

A long call and a short put with the same strike and the same expiration together always pay S_T − K at expiration, just like a forward purchase. So for European options, the following holds without any model at all:

If parity is clearly violated in the market, an arbitrage trade pays off; after bid-ask spreads, borrowing fees and taxes, there is rarely anything left. Conversely, you can back out the forward from calls and puts, and with it the dividend yield the market expects. For American options, parity holds only as an inequality, because the put can be exercised early.

Binomial trees, convergence and early exercise

The tree from the first example becomes realistic when you split the time to expiration into hundreds of steps. In the model of Cox et al. (1979), the price rises in each step by the factor u = e^(σ√Δt) or falls by d = 1/u. The option value is computed backward from expiration, step by step, as the discounted risk-neutral expected value. As the number of steps grows, the price approaches the Black-Scholes value of 10.4506, but with Cox-Ross-Rubinstein it zigzags on the way:

The error flips sign between even and odd numbers of steps, because the strike sits sometimes closer to an upper and sometimes closer to a lower terminal node. Leisen and Reimer (1996) place the nodes so that the tree matches the normal distribution values of the Black-Scholes formula (with an odd number of steps; for 100, the tree uses 101). Their trees converge smoothly with order two and, in their comparison, reached the same accuracy on average 1,400 times faster.

Where trees really earn their keep is American options, which can be exercised at any time. At each node, the tree checks whether exercising immediately is worth more than holding. For a call on a stock that pays no dividends, it never is: the call is worth more alive than exercised, as Merton (1973) showed. For a put, by contrast, it can pay to collect the strike early and earn interest on it.

The critical stock price below which exercising the put pays off rises toward the strike as the remaining time to expiration shrinks. Bermudan options, with a few fixed exercise dates, are worth somewhere between a European and an American option.

The Greeks and hedging

The Greeks measure how the option price reacts to changes in the inputs. For the call from the example:

Delta is also the hedge ratio: if you have sold 1,000 of these calls, you buy about 637 shares and are then neutral to small price moves. Because delta changes with the price (gamma), the hedge has to be adjusted continually. Gamma and theta pull in opposite directions: if you buy options, you gain from large moves and lose time value every day; if you sell them, you earn the time value and lose on jumps. In the Black-Scholes model, the two offset each other exactly when the actual volatility matches the volatility priced in.

Practice bears out the theory only in part. According to Bakshi and Kapadia (2003), delta-hedged long options on the S&P 500 lost money on average, and the more so the higher the volatility. This points to a negative volatility risk premium. Coval and Shumway (2001) found average losses of about 3% per week for at-the-money straddles on the S&P 500 that are neutral to market moves. Buying options is therefore more like paying an insurance premium than making a fair bet.

Implied volatility, smile and skew

If you plug the market price into the formula and solve for σ, you get the implied volatility. If the call from the example costs 12 instead of 10.45, the market is pricing in 24.11% volatility. It is not a pure forecast, because it contains the premium for protection (see VIX).

If the Black-Scholes model were right, all strikes of one expiration would have the same implied volatility. In fact, they form a smile, and for stock indexes mainly a skew: puts far out of the money are considerably more expensive. The market expects fatter tails on the downside than the lognormal distribution allows (see skewness and kurtosis). Rubinstein (1994) showed how to back out a risk-neutral distribution and a matching tree from the prices of many strikes. However, according to Dumas et al. (1998), models in which volatility is a fixed function of price and time did no better at forecasting or hedging than simply smoothing implied volatilities across strikes and expirations.

In practice, the smile is therefore described by a smooth curve, often with SVI: a formula with five parameters for the total variance σ²T as a function of log-moneyness ln(K/F). Gatheral and Jacquier (2014) show how to fit SVI without static arbitrage, that is, so that neither a butterfly spread nor a calendar spread ends up with a negative price. Gatheral (2006) gives an overview of the entire volatility surface.

Exotic options at a glance

Exotic options change the payoff or make it depend on the price path. For the standard example (price and strike 100, one year, 5% interest rate, 20% volatility, standard call 10.45), the results are:

Before rounding, the out and in barriers add up to exactly the standard call. Barriers make options cheaper because they expire worthless on some paths. The formulas usually assume continuous monitoring; if the barrier is only checked daily, the correction by Broadie et al. (1997) helps: you shift the barrier away from the price by the factor exp(0.5826 σ √Δt), with Δt as the interval between observations. Asian options are cheaper because an average fluctuates less. For the geometric average there is an exact formula, which Kemna and Vorst (1990) used as a control variate for simulating the arithmetic average. Lookbacks are expensive because, with hindsight, they deliver the best price (Goldman et al. 1979; Conze and Viswanathan 1991).

Typical strategies and their payoffs

Combining several options can create almost any payoff profile. The logic is easiest to read on the expiration date: each option adds a kink at its strike. The figures assume a price of 100, 91 days, a 3% interest rate and 20% volatility; the cost is today's model value.

Long spreads limit both cost and gain. Short structures such as the iron condor earn the premium as long as the price stays within a range, and lose a multiple of it on large moves. Straddles and strangles are bets on volatility: they win if the actual move is larger than the one priced in. More on the covered call and protective put in the glossary.

Practical pitfalls

• Bid-ask spread: for individual stocks and strikes far out of the money, the spread can eat up a large part of the premium. Use the price at which you can actually trade, not the midpoint.
• Dividends and early exercise: American calls are sometimes exercised shortly before a dividend. If you have sold calls, you may then have to deliver the stock and miss out on the dividend.
• Theta and gamma shortly before expiration: time value does not decay evenly but faster and faster toward the end. At the money, gamma then becomes very large: small price moves change delta sharply.
• Implied versus realized: options are expensive when implied volatility is above the volatility later realized. On average that is the case, but in crashes it is the other way around. Sellers earn steadily for a long time and then lose a lot at once.
• Model risk: constant volatility and lognormal prices are approximations. For exotics, especially barriers, the volatility near the barrier is decisive; a single smile value at the strike can be well off the mark.
• Margin: short options require margin. If margin requirements rise in a crash, you may have to close positions at the worst possible moment.
• Contract size: exchange-traded stock options usually cover 100 shares; quoted prices are per share.

How Portfolio Lab calculates

The option pricer values European options with the generalized Black-Scholes-Merton model for stocks, futures and currencies, and American and Bermudan options with binomial trees: Leisen-Reimer provides the main price, with Cox-Ross-Rubinstein and Jarrow-Rudd shown alongside for comparison. The page shows the convergence of the trees and the early exercise boundary. It prices barriers in closed form, including the correction for discrete monitoring, arithmetic Asian options with Monte Carlo and a control variate, and lookbacks and digitals in closed form. On top of that come all the Greeks, implied volatility, an SVI smile (parametric or fitted to option chains) and strategies with up to eight legs. Formulas, defaults and limitations are described in the option pricer methodology. Haug (2007) serves as the formula reference and Hull (2021) as the textbook.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.