Stochastic calculus is a way to describe quantities that evolve over time with an uncertain component. In finance, these quantities might be an asset price, an interest rate or the value of a trading strategy. The aim is not to predict the next random movement exactly. It is to describe possible movements consistently and understand their consequences for valuation and risk.

The first lessons use coin flips to introduce probability, expectation and the information available at a particular time. The path then moves from discrete random walks to Brownian motion, which provides a continuous-time model of uncertainty. You will see why ordinary calculus needs an additional term when it is applied to these paths, and how the Itô integral and Itô formula account for it.

The later lessons connect this mathematics to replication, risk-neutral pricing, Black–Scholes valuation, changes of probability measure and pricing equations. Mean reversion, stopping times and jumps show where a simple continuous stock-price model is no longer enough. Follow the path in order if these ideas are new to you. Each chart is an illustration of the stated model; one simulated path is not evidence that a market will follow that path.

Continue your learning: Risk sensitivities · Yield curves and bootstrapping

Look up a term as you work through the examples. Definitions explain the terminology; the source references and assumptions beside each lesson explain the scope of its calculation.

54 definitions

Adapted and integrable

A process is adapted when its value at each time is determined by the information available then. Integrability is a mathematical condition ensuring that the relevant expected absolute value is finite.

Asset

An asset is a resource or contractual right with economic value. For a bank, examples include cash, securities and loans that customers are expected to repay.

Barrier

A barrier is a specified level whose crossing changes a contract’s payoff or ends a stopping-time experiment. Crossing risk depends on the path, not only the final value.

Binomial distribution

A binomial distribution counts successes in a fixed number of independent trials with the same success probability. It is used here to make probabilities and replication in discrete outcomes easier to inspect.

Black–Scholes

The Black–Scholes model values suitable European options under assumptions including continuous trading and constant model volatility. Its formula is useful for teaching how time, price and volatility affect option value, but its assumptions are restrictive.

Brownian motion

Brownian motion is a continuous stochastic process with independent, normally distributed increments whose variance equals the elapsed time. Its paths are continuous but are not differentiable in the ordinary calculus sense.

Call and put

A call gives the holder the right to buy the underlying on the contract’s terms. A put gives the right to sell it. The holder’s exercise choice makes the payoff asymmetric.

Cash flow

A cash flow is a payment or receipt at a specified time. Its amount, date, currency and legal owner affect both valuation and liquidity.

Compounding

Compounding describes how interest accumulates on principal and previously earned interest. Under continuous compounding, accumulation is represented by an exponential function rather than a sequence of periodic additions.

Constrained and unconstrained expected shortfall

Unconstrained expected shortfall reflects the permitted joint-portfolio calculation. The constrained component restricts diversification across broad risk classes. The FRTB models-based capital calculation combines the specified measures rather than allowing arbitrary diversification assumptions.

Credit spread

A credit spread is the additional yield or premium associated with credit exposure relative to a reference rate. It can reflect default risk, liquidity and other market effects, so it is not a direct default probability.

Curvature

Curvature describes the part of a value change that a first-order sensitivity does not capture. Its regulatory construction differs by framework: FRTB uses specified repricing shocks, whereas SIMM derives a curvature exposure from volatility-weighted vega and expiry scaling.

Delta

Delta is the first-order sensitivity of value to a specified underlying input. A delta hedge offsets that local exposure at the current point but may need adjustment when the market moves.

Discount rate

A discount rate is a rate used to value future cash flows in present terms. The appropriate rate depends on the valuation framework and the characteristics of the cash flow.

Distribution

A probability distribution assigns probabilities to possible values or outcomes. It describes a range of possibilities rather than one realised observation.

Drift and diffusion

Drift describes the predictable local direction of a stochastic process under a chosen probability measure. The diffusion term determines the magnitude of its continuous random fluctuations.

Expectation

Expectation is an average weighted by the probabilities in a model. Conditional expectation updates that average using specified information and considers only the outcomes consistent with it.

Expiry

Expiry is the time at which an option’s exercise right ends. A European option can be exercised only at expiry, while other exercise styles allow different choices.

Feynman–Kac and PDE

The Feynman–Kac relationship connects certain conditional expectations of future payoffs with solutions of partial differential equations. In pricing, it links a probabilistic valuation to a differential-equation formulation under suitable conditions.

Geometric Brownian motion

Geometric Brownian motion models proportional changes with constant drift and volatility parameters. It keeps prices positive but omits jumps and changing volatility in its basic form.

Girsanov’s theorem

Girsanov’s theorem describes how a suitable change of probability measure alters the drift of a stochastic process. The probability-density weighting must satisfy the theorem’s conditions; it is not an arbitrary reassignment of outcomes.

Hazard rate

A default hazard rate describes the instantaneous rate of default conditional on survival up to that time. A spread divided by LGD is a simplified hazard approximation, not a full credit-curve calibration.

Hedge

A hedge is a position intended to offset part of another exposure. It can leave basis, timing, legal or non-linear risks and can introduce its own funding and counterparty demands.

Idiosyncratic and systematic risk

Idiosyncratic risk is specific to a name or position, while systematic risk is driven by factors shared across many positions. The distinction affects diversification assumptions and some regulatory aggregation treatments.

Independence

Variables are independent when learning the value of one does not change the probability distribution of the other. This is stronger than having zero linear correlation.

Interest rate

An interest rate expresses the cost of borrowing or the return on lending over a stated period. Its meaning depends on the currency, term, compounding convention and credit or collateral conditions.

Interpolation

Interpolation supplies values between known curve points. Linear interpolation in rates and linear interpolation in logarithms of discount factors need not produce the same intermediate prices or sensitivities.

Itô formula

Itô’s formula is the chain rule for suitable functions of stochastic processes. It includes an additional second-derivative term that accounts for the process’s quadratic variation.

Itô integral

An Itô integral accumulates a process against stochastic increments using information available before each increment. This non-anticipating construction differs from choosing an arbitrary ordinary-calculus integration rule.

Liquidity and solvency

Liquidity concerns the ability to meet obligations when due. Solvency concerns the ability of assets and loss-absorbing resources to cover liabilities and losses; a solvent institution can still face a timed cash shortage.

Long and short positions

A long position generally benefits from ownership or a rise in the relevant instrument’s value. A short position has the opposite exposure. For complex derivatives, the precise payment and sign convention should always be checked.

Martingale

A martingale is an integrable, adapted process whose conditional expected future value equals its current value. Individual realised paths can still rise or fall substantially.

Mean reversion

Mean reversion is the tendency of a modelled variable to move back towards a reference level. The Vasicek short-rate model uses a Gaussian mean-reverting process, which can produce negative rates.

Monte Carlo estimation

Monte Carlo estimation approximates an expectation by averaging simulated outcomes. Sampling error decreases as more independent information is obtained, but additional simulations do not remove an incorrect model assumption.

Option

An option gives its holder a contractual choice, such as buying or selling an asset on specified terms. Optionality also occurs inside other products, for example when a borrower can repay early.

Payoff

A payoff is the amount a contract produces in a specified outcome, usually at exercise or maturity. Its present value is not generally equal to the payoff because timing and uncertainty also matter.

Poisson process

A Poisson process counts randomly timed events with a specified intensity under its assumptions. Adding jumps allows a model to represent discontinuous changes that a purely continuous diffusion cannot generate.

Portfolio

A portfolio is a collection of positions considered together. Its risk depends on their sizes, relationships and permitted offsets, not just the sum of their standalone risks.

Quadratic variation

Quadratic variation is the limit, when it exists, of sums of squared increments over increasingly fine partitions. Brownian motion has non-zero quadratic variation, which explains the extra term in Itô calculus.

Random seed

A random seed fixes the sequence used by a pseudorandom simulation, making examples reproducible. Changing the seed generates a different sample without changing the model’s stated probability law.

Random variable

A random variable assigns a numerical value to each outcome in a probability model. Its expected value can be a weighted average that is not itself a possible individual outcome.

Random walk

A random walk is a process formed by adding successive random steps. Appropriate scaling of increasingly frequent independent steps can lead to a Brownian-motion limit.

Replication and no-arbitrage

Replication constructs a portfolio with the same payoffs as a claim in the modelled states. No-arbitrage requires matching attainable payoffs to have consistent prices under the relevant financing and trading assumptions.

Risk-neutral measure

A risk-neutral measure is a pricing probability measure under which suitably discounted traded asset prices are martingales under the model’s conditions. Its probabilities are pricing weights and need not represent a real-world forecast.

Sensitivity

A sensitivity measures how a value changes when a specified input changes, under a stated bump and unit convention. Greeks are familiar option sensitivities such as delta, gamma and vega.

Sigma-algebra and filtration

A sigma-algebra represents the events that can be distinguished with specified information. A filtration is a sequence of such information sets that grows through time.

Standard deviation

Variance is the probability-weighted squared deviation from a mean. Standard deviation is its square root and is expressed in the original units of the variable.

Stopping time

A stopping time is a random time whose occurrence can be determined from information available up to that time. Optional-stopping results need additional conditions; stopping a strategy does not automatically create a profitable martingale strategy.

Strike

The strike is the price specified in an option for buying or selling the underlying. Its position relative to the current underlying price helps determine the option’s value and sensitivities.

Underlying

The underlying is the asset, rate, index or other reference quantity on which a derivative’s payments depend. A derivative can depend on more than one underlying risk factor.

Value at Risk

Value at Risk is a loss threshold at a chosen confidence level over a specified horizon. It does not describe how severe losses beyond that threshold may be. A quantile is the corresponding threshold in a probability distribution.

Variable selection

Variable selection chooses the information used by a model based on relevance, availability, stability and validation. A predictive association does not by itself show that a variable causes default or will remain useful after conditions change.

Volatility

Volatility measures the variability of a price, return or other risk factor over a stated period. It describes dispersion, not the direction in which the next move will occur.

Yield curve

A yield curve or term structure describes how rates or discount factors vary with maturity. Projection curves and discount curves can serve different purposes within one valuation.