Short Rate Models (Part 1: Introducing Merton's Model)
Table of Contents
- Introduction
- Recap of Bond Pricing from Short Rate Models
- Merson’s Model
- Wrapping Up
- Optional: Technical Definition of the Solution to SDE
Introduction
Welcome to my refresher series on short-rate models, dear reader! These mathematical marvels are not just academic curiosities but crucial tools in finance. They are among the main workhorses of interest rate modelling, used in the ivory towers of academic research and by the fast-paced world of Wall Street desk quants. This refresher series aims to help you understand, formulate, and solve these models by deriving and implementing well-known and newer models of varying complexity and generality. Code snippets included in this post (and more) can be found in the Github repository. The code library will evolve with the series as we build out more types of models and can exploit the benefits of more abstraction.
Despite their seeming simplicity, short-rate models are incredibly versatile for modelling interest rates. The short rate is also known as the instantaneous spot interest rate, at which one can borrow money for an infinitesimally short period in the market. A short-rate model specifies the evolution of the short rate according to a stochastic differential equation (SDE). Using the no-arbitrage principle, we can then use its solution to derive the price of a zero-coupon bond and other interest-rate derivatives. The wide array of stochastic processes one can articulate makes the short-rate models a highly versatile tool and a key component of many financial models and strategies.
In this first post, we start with some preliminaries. These are the foundational concepts and tools we will need to understand and implement the short-rate models. We’ll then go step-by-step to derive the solution of the short rates and the price of a zero-coupon bond for one of the simplest short-rate models: the Merton model. By the end of this post, we will have a good grasp of the basic concepts and tools needed to understand and implement the short-rate models.
Recap of Bond Pricing from Short Rate Models
Zero-coupon bonds (ZCB) form the foundation of term structure modelling. The price of a \(T\)-year ZCB at time \(t\) (\(P(t, T)\)) is the risk-adjusted present value of 1-dollar in \(T\)-years. The (continuously compounded) yield-to-maturity (YTM) \(y(t, T)\) at time \(t\) is given by
- YTM to Price:
- Price to YTM:
As we explicitly model the short rate dynamic, the YTM is the expected average (under the risk-neutral measure \(Q\)) of the short rate from \(t\) to \(T\).
- Short Rate to YTM:
- Short Rate to Price:
These fundamental relationships will be used to calculate the price and yield of ZCB using our specific short-rate process. Readers familiar with asset pricing will notice that the price of a ZCB is calculated with respect to the risk-neutral measure \(Q\). We will not be concerned with risk-neutral and physical measures just yet. For the first four posts, we assume the risk-neutral measure in bond pricing and the physical measure in the context of short-rate simulation and parameter estimation. A fuller treatment will be postponed to Post 5, where we establish the relationship between the risk-neutral and physical measures.
Merton’s Model (1973)
Merton’s model is one of the first short-rate models and is arguably the simplest one; it laid the foundation for many subsequent short-rate models. It was introduced in 1973 by the renowned economist Robert C. Merton, who extended the concepts of equity option pricing to the bond market by modelling short-rate dynamics as a simple Gaussian process.
Model Specification
The following stochastic differential equation defines Merton’s model:
\[dr_t = \mu dt + \sigma dW^Q_t\]where:
- \(r_t\): short-term interest rate at time \(t\)
- \(\mu\): constant drift term
- \(\sigma\): constant volatility
- \(W^Q_t\): Wiener process (standard Brownian motion) under the risk-neutral measure \(Q\).
Key Concepts
Continuous-Time Modeling
Merton’s model was one of the first to apply continuous-time stochastic processes to interest rate modelling, paving the way for more sophisticated models. It is a one-factor model, as the only source of randomness is the short rate. This will later be generalized to the multi-factor model, where multiple factors drive the dynamic of the short rate.
Gaussian Process
The model assumes that the short rate follows a Gaussian process, allowing for positive and negative levels. It also assumes that rate volatility is constant and independent of the level of the short rate. This modelling choice needs to be consistent with empirical results. While rates can go negative (though mostly inconceivable in 1973), the short-rate volatility generally depends on the short rate’s level. Short rates also exhibit mean reversion as opposed to having a constant drift. Later generations of short-rate models are invented to address these issues.
Equilibrium Short-Rate Model
Merton’s model belongs to a subclass of the short-rate model called the equilibrium model. This class of model cannot fit to the initial term structure exactly. This is because the yield curve generated by some short-rate models can assume only certain forms. There is another class of short-rate models called the arbitrage-free model. which allows both the drift \(\mu = \mu_t\) and \(\sigma = \sigma_t\) to depend on time. For the Merton model, if we let \(mu\) to a deterministic function of time, we obtain the Ho-Lee Model that can fit the initial term structure exactly by varying \(\mu_t\) deterministically over time.
Derivation of the solution to the short-rate model
The short rate is assumed to follow the SDE:
\[dr_t = \mu dt + \sigma dW^Q_t\]The solution to the SDE used by the Merton model is given by
\[r_t = r_s + \mu (t-s) + \int_s^t \sigma dW^Q_u\]We can verify this solution satisfies the SDE by applying Itô’s lemma. As a refresher, Itô’s lemma is a fundamental result in stochastic calculus. It determines how a function \(f(t, r_t)\) evolves. It allows us to derive the SDE for \(f\) given the SDE of \(r_t\).
Mathematically, if \(r_t = \mu(t, r_t)dt + \sigma(t, r_t) dW_t\), Itô’s lemma says that the dynamic of \(f\) is
\[df(t, r_t) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial r_t} dr_t + \frac{1}{2}\sigma(t, r_t)^2\frac{\partial^2 f}{\partial r_t^2} dt\]To verify our solution, we apply Itô’s lemma to the function \(f(t, r_t) = r_t\) and expand to get
\[dr_t = \left[\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial r} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial r^2}\right]dt + \sigma \frac{\partial f}{\partial r} dW_t\]and arrive trivially at
\[dr_t = \mu dt + \sigma dW_t\]which is the same as the SDE of the short rate.
We will not be overly concerned with the technical definition of the solution to a stochastic differential equation. There are plenty of resources offering rigorous treatment of this topic. However, we will include more technical details in the optional section at the end of each post.
Derivation of the Bond Pricing in Merton’s Model
The price of a \(T\)-year ZCB is given by:
\[P(t,T) = \exp\left(-(T-t)r_t - \frac{\mu \left(T-t\right)^2}{2} + \frac{\sigma^2\left(T-t\right)^3}{6}\right)\]We will derive this formula in two different ways. The first is to solve stochastic integral directly, and the second is to “guess” the form of the solution and solve it by matching the bond price formula to the guessed solution. Even though I call the second way the “guessed” solution, it is based on the observation that the Merton model falls in the category of the affine term-structure model, whose prices are all of the form \(A(t, T)exp(-B(t, T) r_t)\). This will be covered in later posts.
Direct Solution
Start with the general bond pricing equation: \(P(t,T) = \mathbb{E}_t^Q\left[\exp\left(-\int_t^T r_s ds\right)\right]\) where \(\mathbb{E}_t^Q\) denotes the expectation under the risk-neutral measure.
Since \(r_t\) follows a Gaussian process, the \(\int_t^T r_s ds\) is also normally distributed.
For a normally distributed variable \(X\) with mean \(m\) and variance \(v\), we have:
If you forget this identity, you can derive it from the definition of expectation.
\[\mathbb{E}\left[\exp\left(-X\right)\right] = \frac{1}{\sqrt{2\pi v}}\int_{-\infty}^{\infty}\exp\left(-x\right)\exp\left(-\frac{\left(x - m\right)^2}{2v}\right)dx\]or you happen to remember the moment-generating function of a normal distribution:
\[M_X(t) = \mathbb{E}\left[\exp\left(tX\right)\right] = \exp\left(\mu t +\frac{\sigma^2t^2}{2}\right)\]- Calculate the mean and variance of \(\int_t^T r_s ds\), note that
The integrals in the last term can be interchanged (why? what condition must be satisfied?), and after integrating all deterministic terms, we have
\[\int_t^T r_s ds = r_t\left(T-t\right) + \mu\frac{\left(T-t\right)^2}{2} + \sigma\int_t^T \left(s-t\right) dW_s\]which has mean and variance of
\[m = \mathbb{E}_t^Q\left[\int_t^T r_s ds\right] = \left(T-t\right)r_t + \frac{1}{2}\left(T-t\right)^2\mu\]and
\[v = \text{Var}_t^Q\left[\int_t^T r_s ds\right] = \text{E}_t^Q\left[\int_t^T r_s^2 ds\right] =\frac{1}{3}\left(T-t\right)^3\sigma^2\]- Apply the formula from step 3: \(P(t,T) = \exp\left(-(T-t)r_t - \frac{1}{2}(T-t)^2\mu + \frac{1}{6}(T-t)^3\sigma^2\right)\)
The Guessed Solution
- Assume the bond price has the form
- Apply Itô’s lemma to \(P(t,T)\) (Try!)
- Under the risk-neutral measure, the expected return of the bond \(dP/P\) is the risk-free rate $r$; substitute the guessed solution into the equation:
- Equate coefficients of \(r\) and constant terms, we get two equations:
- Solve for \(B\left(t,T\right)\)
From \(-\frac{\partial B}{\partial t} = 1\), we get \(B\left(t,T\right) = T - t\). We need the boundary conditions for \(B\left(T,T\right)\) to arrive at this result. Can you deduce what it is from the assumed bond price formula from point 1 above?
- Solve for \(A\left(t,T\right)\)
Substitute \(B\left(t,T\right)\) into the second equation:
\[\frac{\partial A}{\partial t}\frac{1}{A} - \mu B + \frac{1}{2}B^2\sigma^2 = 0\]Solve for the integral, we get
\[\ln\left(A\left(t,T\right)\right) = -\mu\left(T-t\right)^2/2 + \sigma^2\left(T-t\right)^3/6 + C\]Where \(C\) is a constant of integration.
- Apply the boundary conditionstion \(P\left(T,T\right) = 1\)
Therefore, \(\ln\left(A\left(T,T\right)\right) = 0 = C\)
- Write the final expression for \(A(t,T)\)
- Apply the formula to the bond price
Dynamic of the zero-coupon bond yield
For all one-factor affine short-rate models, the risk-neutral dynamic of the zero-coupon bond yield \(y(t, T)\) can be derived by applying the Itô’s lemma to the zero-coupon bond yield formula. This gives
\[dy = \frac{\partial y}{\partial t} dt + \frac{\partial y}{\partial r} dr + \frac{1}{2}\frac{\partial^2 y}{\partial r^2} \left(dr\right)^2\]where
\[\frac{\partial y}{\partial t} = \frac{1}{T-t}\left(y_t - \frac{dA/dt}{A\left(t, T\right)} + \frac{dB\left(t,T\right)}{dt} r_t\right)\] \[\frac{\partial y}{\partial r} = \frac{B\left(t, T\right)}{T-t} = 1\] \[\frac{\partial^2 y}{\partial r^2} = 0\]The resulting dynamic for \(y\) is a stochastic process that depends on both \(y\) and \(r\) in its diffusion term. Let \(\tau = T-t\) then
\[dy = \left[\frac{y_t - r_t}{\tau} - \frac{1}{2}\tau\sigma^2\right] dt + \sigma dW_t\]The drift term can be interpreted as the sum of the annualized carry/rolldown \(\left(y_t - r_t\right)/\tau\) and the convexity adjustment term \(\frac{1}{2}\tau\sigma^2\). The diffusion term is the volatility of the short rate \(\sigma\), which is the same regardless of \(\tau\). This is inconsistent with empirical observation: the short rates are much more volatile than long rates in general. For short-rate models that include mean-reversion, such as the Vasicek model, tend to have the opposite problem where long-rate volatility is too low. As we will see when we introduce the Vasicek model.
Wrapping Up
While simplistic in its assumptions, Merton’s model serves as an excellent starting point for understanding short-rate models in terms of technicality and historical development. It already exhibits some of the key features and limitations of short-rate models. In the next post, we’ll review how to simulate short rates from \(\mu\) and \(\sigma\), as well as how to calibrate Merton’s model to market observed short rates.
Optional: Technical Definition of the Solution to SDE
This section is adapted from the lecture notes
Let \(\mu(t, r_t)\) and \(\sigma(t, r_t)\) be the drift and diffusion terms of the SDE:
\[dr_t = \mu(t, r_t) dt + \sigma(t, r_t) dW_t, \quad t \in (0, T],\]The (strong) solution to the SDE is the process \(r(t)\) that satisfies the integral equation:
- \[\mu(t, r_t) \in \mathrm{L}_{ad}\left(\Omega, L^1\left([0, T\right) \right)\]
- \[\sigma(t, r_t) \in \mathrm{L}_{ad}\left(\Omega, L^2\left([0, T\right) \right)\]
- and \(r_t\) satisfies the following integral equation
A sufficient condition for the existence and uniqueness of the solution is that the coefficients \(\mu(t, r_t)\) and \(\sigma(t, r_t)\) are Lipschitz continuous in \(r_t\) and grow at most linearly in \(r_t\), i.e.
- Lipschitz condition: there exists a constant \(K\) such that for all \(t \in [0, T]\) and \(x, y \in \mathbb{R}\),
- Linear growth condition: there exists a constant \(K\) such that for all \(t \in [0, T]\) and \(x \in \mathbb{R}\),
We can view these two conditions as ensuring the boundedness of the coefficients. Intuitively, the Lipschitz condition ensures that the coefficients do not change too rapidly as the state variable \(x\) changes. Given an initial condition \(r_0\), there is a well-defined path for the stochastic process r_t.The linear growth condition ensures that the coefficients do not explode to infinity in fitnite time. It also ensures that small changes in the initial condition will not cause large, unstable changes in the behavior of \(r_t\). We will see these conditions again in Post 4 when we discuss the strong convergence of the Euler-Maruyama discretization.