Economic Modelling · Stochastic models for security prices

Binomial, Jump Diffusion and Mean-Reverting Security Price Models

Q: What is the main difference between lognormal and jump diffusion models?

The lognormal model has continuous paths and constant volatility. Jump diffusion adds random sudden jumps, usually from a Poisson process. This produces fatter tails and skewed returns.

Q: Why is the Ornstein-Uhlenbeck process called mean-reverting?

Its drift α(μ − X) points towards the long-run mean μ. If X is above μ the drift is negative, and if below it is positive. Random shocks still move it around.

Q: What are the limitations of Black-Scholes that stochastic volatility models address?

Black-Scholes assumes constant volatility and lognormal prices. Real markets show changing volatility and an implied volatility smile. Stochastic volatility models allow volatility to vary randomly, so they fit option prices better.

Q: Is the binomial model only a simplification of Black-Scholes?

It is also a model in its own right. It is flexible for early exercise and path-dependent features. With many small steps and suitable u and d, it approaches the lognormal model.

Updated 11 October 2026 · Fact-checked

These models relax the lognormal (geometric Brownian motion) assumption. Binomial models move prices up or down in discrete steps. Jump diffusion adds sudden jumps. Mean-reverting models, such as Ornstein-Uhlenbeck, pull values back to a long-run level. Stochastic volatility lets volatility itself vary. You choose the model by matching it to the feature of the data.

Understand Other Models: Discrete Time, Jump and Mean-Reverting

The standard model for a share price is geometric Brownian motion: dS = μS dt + σS dB. It gives lognormal prices, constant volatility and continuous paths. It is easy to use, but real prices do not fully behave this way.

Real data show fat tails, sudden large moves (crashes), volatility that changes over time, and, for some series, a pull back towards a typical level. Each alternative model fixes one of these problems.

A discrete-time model moves price only at set dates. In the binomial model, over each step the price either multiplies by u (up) or by d (down). It is simple, needs no calculus, and handles American-style features well. With small steps and suitable u and d, it approaches the lognormal model.

A jump diffusion model adds random jumps to the continuous diffusion. Jumps usually arrive as a Poisson process with rate λ, and each jump has a random size. This creates fatter tails and skewed returns, which matches crashes better than lognormal does.

Mean-reverting models suit series that should not drift away forever, such as interest rates, spreads or volatility. The Ornstein-Uhlenbeck (OU) process is dX = α(μ − X) dt + σ dB. When X is above μ the drift is negative, and when below it is positive. It is not usually used for share prices, because values can go negative and prices have no reason to revert. Stochastic volatility models let σ follow its own random process, often mean-reverting. They explain volatility clustering and the implied volatility smile that Black-Scholes cannot.

Key rules to remember

Binomial step: S(t+Δt) = S(t) × u with probability p, or S(t) × d with probability 1 − p; Take u > d > 0. Prices stay positive.
Binomial moments for the one-step return: E[S(t+Δt) | S(t)] = S(t) × [pu + (1 − p)d]; Variance of the next price = S(t)² × p(1 − p)(u − d)².
Risk-neutral up probability: q = (e^(rΔt) − d) ÷ (u − d); Valid for no arbitrage when d < e^(rΔt) < u. Use for pricing, not real-world forecasting.
Lognormal model (GBM): dS = μS dt + σS dB, so ln S(t) ~ N(ln S(0) + (μ − σ²/2)t, σ²t); Continuous paths, constant σ, no jumps.
Jump diffusion: dS/S = μ dt + σ dB + (J − 1) dN, where N is Poisson with rate λ; J is the random jump multiplier. The expected number of jumps in time t is λt.
Ornstein-Uhlenbeck process: dX = α(μ − X) dt + σ dB; α > 0 is the speed of reversion, μ the long-run mean.
OU mean and variance: E[X(t)] = μ + (X(0) − μ)e^(−αt); Var[X(t)] = σ²(1 − e^(−2αt)) ÷ (2α); Long-run variance is σ² ÷ (2α). X(t) is normal.

How to solve Other Models: Discrete Time, Jump and Mean-Reverting questions

Use this method for any question comparing or applying these models.

1Identify the model named or implied: binomial, jump diffusion, OU or stochastic volatility.
2Write the model in standard notation and define every symbol, including u, d, p, λ, α, μ and σ.
3State the assumptions explicitly, such as step length, independence of jumps, or constant long-run mean.
4Do the calculation: build the tree, find the moments, or evaluate the mean and variance formulas.
5Check that the answer is sensible: probabilities between 0 and 1, variances positive, reversion towards μ.
6For discussion parts, link each model to the feature of data it captures and say what it still cannot do.
7Conclude with the practical implication, such as pricing, risk measurement or simulation.

Quickest way: Match the feature to the model

When to use it: Use for MCQs and short discussion questions where you must pick or compare models.

Sudden large moves or fat tails: jump diffusion.
Discrete dates, tree pricing or early exercise: binomial.
Series pulled back to a level, such as rates or spreads: OU or other mean-reverting model.
Volatility changing or smile in implied volatility: stochastic volatility.
Then name the one drawback: more parameters, harder calibration, or possible negative values for OU.

Common mistakes in Other Models: Discrete Time, Jump and Mean-Reverting

Using the real-world probability p to price an option on a binomial tree.
Both are probabilities of an up move, so they look interchangeable.
Fix: Price with the risk-neutral probability q = (e^(rΔt) − d) ÷ (u − d). Use p only for real-world projections.
Saying a mean-reverting model makes share prices predictable and so always profitable.
Students overstate the idea of reversion.
Fix: Say the process reverts in expectation only. Random shocks still occur, and reversion speed must be estimated.
Forgetting that OU values can go negative.
Students compare it with lognormal prices, which are always positive.
Fix: State that OU is normal, so it suits rates or spreads. It is a poor choice for a share price level.
Treating jump diffusion as just a larger σ.
Both raise the chance of big moves.
Fix: Jumps add discontinuous paths, skewness and excess kurtosis. Higher σ in GBM keeps paths continuous and returns normal.
Mixing up α and σ in the OU variance.
The formula has both in different places.
Fix: Long-run variance is σ² ÷ (2α). Higher α means faster reversion and lower variance.
Claiming stochastic volatility models are always better than Black-Scholes.
They fit data better, so students assume they are superior.
Fix: Say they fit better but add parameters, are harder to calibrate and often lack closed-form prices.

Worked examples

Example 1

A share is priced at ₹100. Over one year it rises to ₹120 or falls to ₹90. The continuously compounded risk-free rate is 5% per year. Find the risk-neutral probability of an up move, giving q to 3 decimal places.

Show the solution

u = 120 ÷ 100 = 1.2 and d = 90 ÷ 100 = 0.9.
e^(0.05) = 1.051271.
Check no arbitrage: 0.9 < 1.0513 < 1.2, so it holds.
q = (1.051271 − 0.9) ÷ (1.2 − 0.9) = 0.151271 ÷ 0.3 = 0.50424.

Answer: q ≈ 0.504

Example 2

An interest rate follows dX = 0.4(6 − X) dt + 0.02 dB with X(0) = 4 (units in %, σ in the same units per √year). Find E[X(2)] and the long-run variance. Then explain why this model is more suitable than GBM for a short-term rate series that stays near a typical level.

Show the solution

E[X(t)] = μ + (X(0) − μ)e^(−αt), with μ = 6, α = 0.4, t = 2.
e^(−0.8) = 0.449329.
E[X(2)] = 6 + (4 − 6) × 0.449329 = 6 − 0.898658 = 5.101342.
Long-run variance = σ² ÷ (2α) = 0.0004 ÷ 0.8 = 0.0005.
Explanation: the drift pulls X towards 6, so the series stays near a typical level. GBM has no pull back and its variance grows without limit.

Answer: E[X(2)] ≈ 5.10 (%). Long-run variance = 0.0005. OU suits a series that stays near a level; GBM does not revert.

Exam tips

Write the model equation and define all symbols before calculating. Marks are given for notation and assumptions.
In discussion questions, give one feature, one model and one limitation for each model. This structure earns marks quickly.
Always state whether a probability is real-world or risk-neutral.
Check the no-arbitrage condition d < e^(rΔt) < u before using q.
For OU questions, comment on the sign of the drift relative to μ to show you understand reversion.

Practice questions from Stochastic models for security prices

Other Models: Discrete Time, Jump and Mean-Reverting: frequently asked questions

What is the main difference between lognormal and jump diffusion models?

The lognormal model has continuous paths and constant volatility. Jump diffusion adds random sudden jumps, usually from a Poisson process. This produces fatter tails and skewed returns.

Why is the Ornstein-Uhlenbeck process called mean-reverting?

Its drift α(μ − X) points towards the long-run mean μ. If X is above μ the drift is negative, and if below it is positive. Random shocks still move it around.

What are the limitations of Black-Scholes that stochastic volatility models address?

Black-Scholes assumes constant volatility and lognormal prices. Real markets show changing volatility and an implied volatility smile. Stochastic volatility models allow volatility to vary randomly, so they fit option prices better.

Is the binomial model only a simplification of Black-Scholes?

It is also a model in its own right. It is flexible for early exercise and path-dependent features. With many small steps and suitable u and d, it approaches the lognormal model.