Risk-Neutral Pricing & Model Foundations

Suppose a stock trades at $100. Over the next year you believe two outcomes are equally likely: a 50% chance it is $120, and a 50% chance it is $80. It is tempting to value an option with those probabilities directly. That is not how standard no-arbitrage pricing works.

Derivative pricing uses a risk-neutral measure. The name does not mean investors are actually risk-neutral. It is a mathematical framework. The probability distribution is adjusted so that, after carrying costs, the expected growth of a tradable asset is consistent with the risk-free rate.

Under that measure, the discounted expected payoff of a derivative equals its theoretical no-arbitrage price. In words: derivative price equals the present value of the expected future payoff, and the expectation uses the risk-neutral probability measure.

The split is the point of the framework. Real-world probability asks what we believe the stock will do. Risk-neutral probability asks which distribution is consistent with today’s market prices and with no arbitrage. That distinction is the start of quantitative derivatives pricing.

A stock can have a risk-neutral expected return tied to the risk-free rate. That does not mean the stock will earn the risk-free rate. It means the probability framework used to price derivatives was built so that discounted tradable prices satisfy the martingale, or no-arbitrage, condition.

Risk-neutral probabilities are a pricing tool, not a forecast of what will happen. The distinction matters as soon as you run a Monte Carlo. A path drawn under the pricing measure is there to value a contract. It is not a prediction of next year’s stock price.

Pricing many derivatives needs a mathematical model of how the underlying can move through time. A stochastic process is that model: something that changes over time and includes randomness.

From today the stock can take a random step, then another, then another. You do not know the path. You model a distribution of possible paths, and the derivative’s value comes from that distribution under the pricing measure.

Geometric Brownian motion is the classic continuous model for a stock price. A simplified version under the risk-neutral measure is a small change in the stock equal to a drift at the risk-free rate plus a random shock scaled by volatility.

Read it as future stock movement = systematic component + random component. The notation can wait. Volatility is what controls how widely the simulated paths spread. A low volatility stays near the start. A high volatility wanders. The Monte Carlo in Options lab is a sketch of this idea: you supply a return and a volatility, and it draws paths.

The model is useful, and real markets do not behave exactly like it. A smooth path might walk $99, $102, $101, $104. A real session can go from $100 to $82 overnight.

• Sudden price jumps
• Large unexpected moves
• Volatility that changes
• Volatility clustering
• Fat tails
• Market crashes
• Overnight gaps

A jump-diffusion model keeps two kinds of movement. Diffusion is the small continuous steps. A jump is an occasional large discontinuous move. Together: normal movement plus occasional jumps. The resulting distribution can reflect extreme moves better than basic geometric Brownian motion.

Merton’s jump-diffusion model is one named version. More elaborate jump models exist. The lesson here is the gap in the path, not the extra algebra.

Realized volatility describes how much the underlying actually moved over a historical period. Apple’s realized volatility over the last 30 days might be 28%. It is calculated from past prices. It looks backward.

Implied volatility is the volatility that, plugged into an option-pricing model, produces a price consistent with the observed market price of the option. The market gives a price. The model solves for the volatility that makes the model match that price. That solved number is implied volatility.

Realized volatility can be 25% while options trade at prices that correspond to 32% implied volatility. That does not automatically mean the option is expensive. It means the market’s option prices imply about 32% under the chosen model assumptions.

• Expected future volatility
• Volatility risk premium
• Supply and demand
• Event risk
• Hedging demand
• Model assumptions
• Market liquidity

Implied volatility is not simply a forecast of future realized volatility.

It is tempting to say a stock has one volatility, as in “Apple volatility is 30%.” The options market does not work that way. Different options imply different volatilities depending on strike, expiration, and moneyness.

Plot implied volatility across strikes and expirations and you get a volatility surface: a landscape whose height is implied volatility, with strike in one direction and expiration in the other. The surface is how the market prices volatility across the listed options.

A volatility smile means options with strikes far from the current stock price have higher implied volatility than options near the money. The wings sit above the middle.

A volatility skew means the shape is not symmetric. In many equity markets, downside puts trade with higher implied volatility than comparable upside calls. That is market pricing and demand for protection and for exposure. It is one reason a single volatility number for every option is inadequate.

Volatility also varies by expiration. A teaching term structure:

Term structure is how implied volatility changes with time to expiration. A major event can bend the near end. Earnings tomorrow can leave 30-day implied volatility at 45% while 6-month implied volatility sits at 28%.

An options trader cannot simply ask “what is the stock’s implied volatility?” The better question is the implied volatility for this strike and this expiration.

Combine the two dimensions, strike and expiration, and each point on the surface is the implied volatility of one strike at one expiration. That is much closer to how an institutional desk thinks about volatility than a single number for the ticker.

Suppose the model prices an Apple $200 call at $7.25 and the market trades that option at $8.10. The model is not on the market. Calibration finds the parameters that let the model reproduce observed prices.

1. 1Start from market option prices
2. 2Select a model
3. 3Estimate the parameters
4. 4Run the pricing model
5. 5Compare the model with the market
6. 6Adjust the parameters and repeat

In Black–Scholes, implied volatility is the volatility that makes the model price equal the market price. One option, one implied volatility.

Institutional models can have many parameters. Heston is one stochastic-volatility framework: starting variance, a long-run level, a pull-back speed, vol-of-vol, and a price/variance correlation. The local and stochastic volatility lesson is where those knobs are named. The calibration lesson is where a fit is validated, not only minimized.

If a model prices thousands of options and does not reproduce the market’s volatility structure, the valuations and the risk measures can be inconsistent with market prices. Calibration is the fit. Validation is whether that fit deserved trust. Continue on the model calibration page.

Every model is an approximation. Geometric Brownian motion assumes one kind of price behavior. Black–Scholes assumes a particular volatility structure. A richer model assumes something else. The model is not the market.

• Incorrect assumptions
• Poor calibration
• Bad market data
• Numerical approximations
• The wrong model for the contract
• A market that has changed structure
• Extrapolating past observed strikes and expirations

That list is part of institutional risk management, not a footnote.

Stock prices and option prices are the market data. Option prices become implied volatility, then a smile or a skew, and a term structure, then one surface. A pricing model — geometric Brownian motion, a jump-diffusion, or something richer — sits on that surface. Calibration adjusts the model until it matches. The output is a derivative price and the risk numbers, including the Greeks, around it.

The Monte Carlo in Options lab answers: what could happen if the underlying follows the assumptions you type? The next question is which assumptions to type, and how those assumptions relate to option prices you can see.

Open the Monte Carlo sketch