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Short-rate model

Short-rate model is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Short-rate model rather than just read about it. In short: A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r t {\displaystyle r_{t}\,} . The short rate Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate.

Short-rate model — main illustration
Short-rate model — illustration

Key takeaways

  • Short-rate model belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Short-rate model to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Short-rate model from memory before moving on to harder problems.

Reference excerpt

A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r t {\displaystyle r_{t}\,} .

The short rate Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate. The short rate, r t {\displaystyle r_{t}\,} , then, is the (continuously compounded, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time t {\displaystyle t} . Specifying the current short rate does not specify the entire yield curve. However, no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of r t {\displaystyle r_{t}\,} as a stochastic process under a risk-neutral measure Q {\displaystyle Q} , then the price at time t {\displaystyle t} of a zero-coupon bond maturing at time T {\displaystyle T} with a payoff of 1 is given by

P ( t , T ) = E Q ⁡ [ exp ⁡ ( − ∫ t T r s d s ) | F t ] , {\displaystyle P(t,T)=\operatorname {E} ^{Q}\left[\left.\exp {\left(-\int _{t}^{T}r_{s}\,ds\right)}\right|{\mathcal {F}}_{t}\right],}

where F {\displaystyle {\mathcal {F}}} is the natural filtration for the process. The interest rates implied by the zero coupon bonds form a yield curve, or more precisely, a zero curve. Thus, specifying a model for the short rate specifies future bond prices. This means that instantaneous forward rates are also specified by the usual formula

f ( t , T ) = − ∂ ∂ T ln ⁡ ( P ( t , T ) ) . {\displaystyle f(t,T)=-{\frac {\partial }{\partial T}}\ln(P(t,T)).}

Short rate models are often classified as endogenous and exogenous. Endogenous short rate models are short rate models where the term structure of interest rates, or of zero-coupon bond prices T ↦ P ( 0 , T ) {\displaystyle T\mapsto P(0,T)} , is an output of the model, so it is "inside the model" (endogenous) and is determined by the model parameters. Exogenous short rate models are models where such term structure is an input, as the model involves some time dependent functions or shifts that allow for inputting a given market term structure, so that the term structure comes from outside (exogenous). Other authors use 'equilibrium' and 'no arbitrage' in place of 'endogenous' and 'exogenous'.

Particular short-rate models Throughout this section W t {\displaystyle W_{t}\,} represents a standard Brownian motion under a risk-neutral probability measure and d W t {\displaystyle dW_{t}\,} its differential. Where the model is lognormal, a variable X t {\displaystyle X_{t}} is assumed to follow an Ornstein–Uhlenbeck process and r t {\displaystyle r_{t}\,} is assumed to follow r t = exp ⁡ X t {\displaystyle r_{t}=\exp {X_{t}}\,} .

… excerpt ends here. Continue reading the full article.

Illustrations

Short-rate model: Tree returning the OAS (black vs red): the short rate is the top value; the development of the bond value shows pull to par clearly.
Tree returning the OAS (black vs red): the short rate is the top value; the development of the bond value shows pull to par clearly.

Worked examples

Example 1 — a first encounter with Short-rate model

Start with the simplest possible case. Write down what Short-rate model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Short-rate model before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Short-rate model ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Short-rate model

In research
Short-rate model appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Short-rate model in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Short-rate model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Interest rates, Mathematical finance, Short-rate models, so understanding it makes those chapters shorter.
In everyday life
Look for Short-rate model outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Short-rate model in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Short-rate model means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Short-rate model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Short-rate model in simple terms?

A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r t {\displaystyle r_{t}\,} . The short rate Under a short rate model, the stochastic…

Why does Short-rate model matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Short-rate model?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Short-rate model.

Tags

  • Interest rates
  • Mathematical finance
  • Short-rate models

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