Foundations of Interest Rate modelling — the simplified way
Background
The idea of interest rate is something that we all have been introduced at a very early age of our life. Back in school we were taught ideas like simple interest, compound interest etc. Subsequently, at undergraduate / post graduate levels many of us had the opportunity to learn advanced concepts related to interest rates be it through- fixed income mathematics, study of trading strategies, learning derivatives etc. Some of us who moved into the core quant finance roles in the industry have had the opportunity to further learn and apply advanced interest rates concepts used in the market.
The level of interest rates and the volatility in rates is a matter of “great interest” in the industry. Actions of industry participants get influenced by the prevailing interest rate environment and also on the expectation of rates movement in the near future. For example, say a company wishes to raise debt funding for its business, the company will do an extensive analysis of the interest rate market because the market is going to determine the yield at which the company can issue bonds to the market participants. This will subsequently have an impact on the weighted average cost of capital i.e. WACC of the firm which in turn impacts company value. This is just one example, there are multiple such examples as to why industry participants spend so much time analyzing the interest rate environment.
Interest rate theory and related concepts are one of the key building block in study of quantitative finance. It’s a very widely researched topic in both — industry and academia. There are a lot of wonderful reference books on interest rate theory, however, at times these references may become immensely heavy from a math perspective; thus a reader who may not come from a math background may find a few concepts to be esoteric. I will keep this article light on mathematical treatment while still attempting to explain the key concepts pertaining to rates.
Clarity on concepts pertaining to equities is very handy!
Before one embarks on the journey to explore interest rate models, it’s important to have a background on quant concepts pertaining to equities. Further, equities as an asset class is comparatively easy to understand vis-à-vis interest rates. Fundamental ideas learnt in equity i.e. especially equity derivatives play a vital role in the process to learn interest rates. As we will see in the below sections, we can find parallels between our good old Black Scholes Merton (BSM) partial differential equation (PDE) and the Bond pricing equation (BPE). It’s this BPE which can be extended to understand the popular spot rate models for interest rates. We will attempt to combine these ideas in the sections to follow.
We will not be deriving results in this article, but we will use the results/ functional forms / partial differential equations directly and understand the logic behind them. Also, we will not be linking rates modelling with the Martingales approach.
Brushing up on Black Scholes Merton (BSM) PDE
The BSM model that was found by Black, Scholes and Merton a few decades back, continues to be relevant in today’s market too. BSM PDE is an important result which was arrived at by combining multiple concepts as mentioned below. We will try to summarize these ideas in bullets so that readers who may not be very familiar with BSM, would be able to familiarize themselves:
o Asset price process: Our underlying stock price has its own process which can be represented by a stochastic differential equation like:
dS = r.S.dt + σ.S.dX ; where dX represents the Brownian motion governing the randomness in the asset price process dS
o Our portfolio is: π = V — ΔS; we are long an option represented by V, and short Δ number of shares. We hold Δ constant over a very small time step
o Delta Hedging: We make our portfolio risk-less over a small time step. As it is risk-less over that time step, all that we can expect to earn is the risk-free rate of interest
o Ito’s calculus: The change in the value given by dV can be expressed using Ito’s. For readers who may not have read about Ito’s , it has its grounding in Taylor Series expansion
o Equating our portfolio to one earning risk free rate of return: In the process of deriving the BSM PDE, we equate our delta hedged portfolio to be equivalent to one that earns risk free rate of interest.
This leads us to the standard BSM PDE as below:
Extending the ideas to understand the interest rates process:
Interest Rates (“IR”) is a slightly complicated asset class as compared to Equity. Further, for IR, we encounter multiple models for rates (these models are generally named after the mathematicians / scientists who found them), whereas for equities we have one principal model i.e. the GBM. Models for IR can be broadly classified into equilibrium models and no-arbitrage models. Equilibrium models usually start with assumptions about economic variables and derive a process for the short rate, r. They then explore what the process for r implies about bond prices and option prices. Whereas, a No-arbitrage model is designed to be consistent with today’s term structure of interest rates. The chief difference between an equilibrium and a no-arbitrage model is: In an equilibrium model, today’s term structure of interest rates is an output. Whereas, in a no-arbitrage model, today’s term structure of interest rates is an input.
Interest Rates modelling theory is very vast and covering everything in one single article is impossible. In this article, we will focus on only two important concepts namely:
a. Bond Pricing Equation
b. One particular class of solutions called as the affine solutions
Bond Pricing Equation:
We take inspiration from the ideas used for the BSM PDE for deriving the Bond Pricing Equation (“BPE”). We follow similar ideas of constructing a portfolio that is riskless over a short time interval. However, there is one main difference between them: In deriving the BSM PDE, we had an underlying i.e., stock price that was tradable and could be used for delta hedging in order to create a risk-less portfolio. However, for interest rates, we focus on a quantity called as the short rate. The short rate is not a tradable quantity. Therefore, we will need to modify our earlier portfolio a little bit in order to reach our goal of creating a portfolio that is risk-less over a short time step. Our new portfolio is now given as: π = V1 — ΔV2
The main difference in this portfolio and the BSM portfolio is that we are long one bond V1 and short Δ units of bond V2. Similar to the treatment done for BSM, lets walkthrough the points specific to BPE below:
o Our portfolio is: π = V1 — ΔV2; We hold Δ constant over a very small time step
o Delta Hedging: We make our portfolio risk-less over a small time step. As it is risk-less we can expect to earn is the risk-free rate of interest
o Ito’s calculus: The change in the portfolio value given by dπ can be expressed using Ito’s.
o We then use arbitrage arguments to equate the portfolio return equal to risk free rate. This risk free rate is just the spot rate.
The resulting BPE in the PDE form looks like this:
The above is a linear parabolic partial differential equation. The derivation of the above PDE is a pretty standard process which one can find in literature
Market Price of Risk
In the interest modelling exercise, there is one additional term to study (that does not exist in BSM analysis) which we call as the market price of risk. In layman terms one can consider this as the compensation that traders will demand for taking on risk of trading bonds to construct our portfolio. The underlying stochastic quantity i.e., the spot rate is not a tradable asset, it’s for this reason that we need to introduce the idea of market price of risk.
For equities we have a real drift rate denoted by μ. However, we price as if the asset grows with a rate r, the risk-free rate.
For fixed-income the real growth of the spot interest rate may be u(r, t) but we price as if it were u(r, t)−λ(r, t)w(r, t). The latter is the risk-adjusted drift rate.
Therefore in our modelling exercise say via Monte Carlo simulation, our risk neutral spot rate process we would be as below:
After looking at the above, many of us may have rightly guessed there are opportunities to study this through the idea of equivalent martingale measures which would allow us to move conveniently between real and risk neutral worlds.
Affine Solutions
We have built up the pricing equation for an arbitrary model. That is, we have not specified the risk-neutral drift, u − λw, or the volatility, w.
We now attempt to choose these functions to give us a good model. We make choices for the risk-neutral drift and volatility that lead to tractable models, i.e. models for which the solution of the pricing equation for zero-coupon bonds can be found analytically.
We explore the affine solution approach to arrive at the solution. Under the affine solution approach, we look for a solution of the functional form:
V(r, t; T) = exp(A(t) — r*B(t))
Let’s look at the Vasicek model as an example to understand this:
The Vasicek model takes the form:
Where dr is the SDE for interest rates under the Vasicek model.
The bond pricing equation looks like:
This is a backward kolmogorov equation. We define the final condition for the bond price as:
Now we go back to our functional form of the affine solution mentioned above and calculate the partial derivatives i.e.
We can solve for value of A and B. For brevity, we take the resulting values of A and B directly for the Vasicek model.
The above can be plugged into the functional form of the affine solution to get to the theoretical bond price.
Similar treatment may be applied to say the Cox Ingersoll Ross (CIR) model as well, only that doing the calculation by hand may get a bit lengthy and tedious.
We have discussed a one factor model above. There are other models that one may study including — a two factor model (say HW2F) or the Heath, Jarrow and Merton models etc. as the next steps.
Conclusion
In this article, we discussed a couple of fundamental ideas pertaining to interest rates modelling. As one would have gathered, interest rates modelling is a very extensive topic which cannot be covered in one single article. However, the ideas discussed in the article will be of help to interested readers to get started in their journey towards exploring more advanced ideas in interest rates modelling.
IR modelling is a very exciting area to work in. It’s probably one of those ‘evergreen’ roles in the industry, the reason being — be it a bank, a financial institution, insurance firm etc. everyone would want to have some estimation on how they expect rates to evolve so that they can make trading / investment / business decisions accordingly.
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