Antoine JacquierDepartment of Mathematics, Imperial College London, and the Alan Turing Institutea.jacquier@imperial.ac.ukandMugad OumgariUniversity College London and Lloyds BankingMugad.Oumgari@lloydsbanking.com
(Date: March 13, 2024)
Abstract.
The contributions of this paper are twofold:we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it.
Key words and phrases:
interest rates, fractional Brownian motion, convexity adjustment
2010 Mathematics Subject Classification:
60G15, 91-10
The authors would like to thank Damiano Brigo for helpful comments.AJ is supported by the EPSRC grants EP/W032643/1 and EP/T032146/1.‘For the purpose of open access, the author(s) has applied a Creative Commons Attribution (CC BY) licence (where permitted by UKRI, ‘Open Government Licence’ or ‘Creative Commons Attribution No-derivatives (CC BY-ND) licence’ may be stated instead) to any Author Accepted Manuscript version arising’.
1. Introduction and notations
1.1. Introduction
In fixed income markets, thedifferent schedules of payments and the diverse currencies, margins require specific adjustments in order to price all interest rate products consistently.This is usually referred to as convexity adjustment and has a deep impact on interest rate derivatives.Starting from[7, 9, 19],academics and practitioners alike have developed a series of formulae for this convexity adjustment in a variety of models,from simple stochastic rate models[16] to some incorporating stochastic volatility features[2].Recently, Garcia-Lorite and Merino[11] used Malliavin calculus techniques to compute approximations of this convexity adjustment for various interest rate products.Motivated by the new paradigm of rough volatility in Equity markets[4, 5, 8, 10, 12, 14, 15],we consider here stochastic dynamics for the short rate, driven by a general Gaussian Volterra process, providing more flexibility than standard Brownian motion.In the framework of the change of measure approach in[18],we introduce a clear definition of convexity adjustment for zero couponbonds, in Proposition2.11, namely as the non-martingale correction of ratios of zero-coupon prices under the forward measure, for which we are able to derive closed-form expressionsor asymptotic approximations.We introduce the model, derive its properties in Section2.In Section2.2, we define convexity adjustment and provide formulae for it, the main result of the paper, which we illustrate in some specific examples.Section3 provides some further expressions for liquid interest rate products, and we highlight some numerical aspects of the results in Section4.
1.2. Model and notations
On a given filtered probability space ,we are interested in short rate dynamics of the form
(1.1) |
with a deterministic function and a continuous Gaussian process adapted to the filtration .Here and below, given a function and a stochastic process, we write,and omit whenever .For some fixed time horizon ,define further, for ,
(1.2) |
as well as .We consider a given risk-neutral probability measure, equivalent to, so that the price of the zero-coupon bond at time is given by
(1.3) |
and we define the instantaneous forward rate process as
(1.4) |
Remark 1.1.
For modelling purposes, we shall consider kernels of convolution type, namely
(1.5) |
1.3. Empirical motivation
The modelling framework above (and in particular the introduction of a potentially singular kernel)is motivated by empirical observations.Assume that the kernel is given by a power-law form with , and that is a standard Brownian motion.To estimate the Hurst exponent, we follow the methodology devised in[12]for the instantaneous log volatility(although more refined and robust statistical estimation techniques are now available,we leave a detailed empirical analysis for future work) and compute it via the linear regression
for some constant.Of course such a linear regression hinges on some assumptions on the form of but a detailed analysis of short rate data is beyond the scope of the present paper,and we only provide here short insights into the potential roughness of short rates dynamics.We consider the sport interest rate data from Option Metrics111data available atWRDS/OptionMetrics.We consider the data from 4/1/2010 until 28/2/2023.For different dates within this period, Figures1 show the available data points (circles) as well as the interpolation by splines (the extrapolation is assumed flat).In Figure2,we compute the time series of the yield curves,for each (interpolated) maturities.The estimation of the Hurst exponent for each maturity is shown in Figure3.
![Interest rate convexity in a Gaussian framework (3) Interest rate convexity in a Gaussian framework (3)](https://i0.wp.com/arxiv.org/html/2307.14218v2/x3.png)
![Interest rate convexity in a Gaussian framework (4) Interest rate convexity in a Gaussian framework (4)](https://i0.wp.com/arxiv.org/html/2307.14218v2/x4.png)
A similar analysis on the US Daily Treasury Par Yield Curve Rates222data available athome.treasury.gov/resource-center/data-chart-center/interest-rates yields Figures4 and5.
![Interest rate convexity in a Gaussian framework (5) Interest rate convexity in a Gaussian framework (5)](https://i0.wp.com/arxiv.org/html/2307.14218v2/x5.png)
![Interest rate convexity in a Gaussian framework (6) Interest rate convexity in a Gaussian framework (6)](https://i0.wp.com/arxiv.org/html/2307.14218v2/x6.png)
2. Gaussian martingale driver
2.1. Dynamics of the zero-coupon bond price
We assume first that is a continuous Gaussian martingale with finite for all .In this case, the (predictable) quadratic variation process is clearly deterministic, but also continuous and increasing, and therefore its derivative exists almost everywhere.In order to ensure existence of the rate process in(1.1), we assume the following (we write for the Lebesgue measure on):
Assumption 2.1.
For each , , and is of convolution type(1.5).
Lemma 2.2.
Under Assumption2.1, is an Gaussian semimartingale.
Proof.
From(1.2), is in general not in convolution form(1.5).However, since is, we can write
where the functionis defined as.The stochastic integral then reads
which corresponds to a two-sided moving average process in the sense of[3, Section 5.2].Assumption2.1 then implies that for each , the function is absolutely continuous on and and the statement follows from[3, Theorem 5.5].∎
Remark 2.3.
- •
The property ensures that the stochastic integral is well defined.
- •
The assumption does not imply that the short rate itself, while Gaussian, is a semimartingale.
Proposition 2.4.
The price of the zero-coupon bond at time reads
and the discounted bond priceis a -martingale satisfying
Corollary 2.5.
The instantaneous forward rate satisfies and, for all ,
In differential form, for any fixed , for , this is equivalent to
Algorithm 2.6.
For simulation purposes, we consider a time grid and discretise the stochastic integral along this grid with left-point approximations as
The vector of stochastic integralscan then be simulated along the grid directly as
where the middle matrix is lower triangular (we omit the null terms everywhere for clarity).
Example 2.7.
With , for , and a Brownian motion,we recover exactly the Vasicek model[21], namely.
Example 2.8.
Consider the extension of the Vasicek model proposed by Hull and White[13], wherewhere and are sufficiently smooth deterministic functions of time.Direct computations yield the solution,with ,
Letting
makes it coincide exactly with our setup in(1.1).Now Assumption2.1 holds if and only if for all ,namely when the function is linear or constant.Note that, as mentioned in[6, Section3.3], the function is often assumed constant in practice.
Proof of Proposition2.4.
The price of the zero-coupon bond at time then reads
(2.1) |
Using Fubini, we can write
(2.2) | ||||
using(1.2).Plugging this into(2.1), the zero-coupon bond then reads
Conditional on, is centered Gaussian with,henceBy Fubini and Assumption2.1,
This is an -Dirichlet process[20, Definition 2],written as a decomposition of a local martingale and a term with zero quadratic variation.Therefore and
(2.3) |
Now, Itô’s formula with , using(2.3) yields, hence, for each ,,and therefore, since ,
The dynamics of the discounted zero-coupon bond price in the lemma follows immediately.∎
Proof of Corollary2.5.
It follows by direct computation starting from the instantaneous forward rate(1.4):
∎
Remark 2.9.
The two lemmas above correspond to the two sides of the Heath-Jarrow-Morton framework.From the expression of the instantaneous forward rate, letand ,so that,and consider the discounted bond price
Itôs’ formula then yields
(2.4) |
From the differential form of , we can write, for any ,
so that, using stochastic Fubini, we obtain
Now,
using Fubini, so that
and.Therefore,
and(2.4) gives
The discounted process is a local martingale if and only if its drift is null:for ,
which is equal to zero by definition of the functions.Therefore the drift (as a function of) is constant.Since it is trivially equal to zero at , it is null everywhereand isa -local martingale.
2.2. Convexity adjustments
We now enter the core of the paper, investigating the influence of the Gaussian driver on the convexity of bond prices.We first start with the following simple proposition:
Proposition 2.10.
For any ,
and there exists a probability measure such that is a -Gaussian martingale and
(2.5) |
under ,where .
Note that, from the definition of in(1.2), is non-negative whenever .In standard Fixed Income literature, the probability measure corresponds to the -forward measure.
Proof.
From the definition of the zero-coupon price(1.3) and Proposition2.4, is strictly positive almost surely and
and therefore Itô’s formula implies that, for any ,
Therefore
Define now the Doléans-Dade exponential
and the Radon-Nikodym derivative.Girsanov’s Theorem[17, Theorem 8.6.4] implies thatis a Gaussian martingale and satisfies(2.5) under.∎
The following proposition is key and provides a closed-form expression for the convexity adjustments:
Proposition 2.11.
For any let .We then have
whereis the convexity adjustment factor.
Remark 2.12.
- •
When or or is constant,there is no convexity adjustment,i.e. .
- •
More interestingly, if , then and
and the process is a -martingale on .
- •
Regarding the sign of the convexity adjustment, we have
Since is strictly positive, then.Furthermore, since
then ,and therefore, assuming strictly positive (as will be the case in all the examples considered here),
negative positive positive negative Considering without generality , the convexity adjustmentis therefore greater than for and less than above.
Proof of Proposition2.11.
Under, the process defined as satisfies , is clearly lognormaland hence Itô’s formula implies
so that
and therefore
With successively and , we can then write
so that
The first exponential is a Doléans-Dade exponential martingale under, thus has-expectation equal to one, and the proposition follows.∎
2.3. Examples
Let be a standard Brownian motion,so that and .
2.3.1. Exponential kernels
Assume that for some , then the short rate process is of Ornstein-Uhlenbeck type and
We can further compute, and
Therefore the diffusion coefficientand the Girsanov drift read
Finally, regarding the convexity adjustment,
Note that, as tends to zero,namely (in the limit), we obtain
2.3.2. Riemann-Liouville kernels
Let and .If , with , the short rate process(1.1) is driven by a Riemann-Liouville fractional Brownian motion with Hurst exponent.Furthermore, with ,
Therefore the diffusion coefficient and Girsanov drift read
Regarding the convexity adjustment, we instead have
Unfortunately, there does not seem to be a closed-form simplification here. We can however provide the following approximations:
Lemma 2.13.
The following asymptotic expansions are straightforward and provide some closed-form expressions that may help the reader grasp a flavour on the roles of the parameters:
- •
As tends to zero,
- •
For any , as tend to zero,
Proof.
From the explicit computation of above, we can write, as tends to zero,
As a function of, is continuously differentiable.Because we are integrating over the compact , we can integrate term by term, so that
where we can check by direct computations that the term is indeed non null.∎
2.4. Extension to smooth Gaussian Volterra semimartingale drivers
Let now in(1.1) be a Gaussian Volterra process with a smooth kernel of the form
for some standard Brownian motion.Assuming that is a convolution kernel absolutely continuous with square integrable derivative,it follows by[3] that is a Gaussian semimartingale (yet not necessarily a martingale) with the decomposition
where is a process of bounded variation satisfyingand hence the Itô differential of reads,and its quadratic variation is.The short rate process(1.1) therefore reads
where and .If satisfies Assumption2.1, then the analysis above still holds.
2.4.1. Comments on the Bond process
Let be the integrated short rate processand the bond price process on .
Lemma 2.14.
The process satisfies and, for ,
Proof.
For any , we can write
and therefore
(2.6) |
Itô’s formula[1, Theorem 4] then yields
so that, since ,the lemma follows from
∎
Remark 2.15.
We can also write in integral form as follows, using stochastic Fubini:
with and.As a consistency check, we have
which corresponds precisely to(2.6).
2.4.2. Specific example
Consider the kernel with ,so that and .In this case,,so that,which is an Ornstein-Uhlenbeck process,with covariance, for all ,
The short rate dynamics in(1.1) then reads
with,and the zero-coupon bond dynamics(Proposition2.4) reads
with
Applying stochastic Fubini, we then obtain
We note that the convexity adjustment in Proposition2.11 is only affected by a different weighting scheme in the integral given by the function.In our case, from the covariance computation above,,and therefore.
3. Pricing OIS products and options
3.1. Simple compounded rate
Using Proposition2.4, we can compute several OIS products and optionsConsider the simple compounded rate
(3.1) |
where is the day count fraction and the number of business days in the period.The following then holds directly:
where the superscript refers to reset dates;we use the superscript to refer to accrual dates below.
3.2. Compounded rate cashflows with payment delay
The present value at time zero of a compounded rate cashflow is given by
where denotes the compounded RFR rate.In the case where there is no reset delays, namely for all , then
where and ,using the convexity adjustment formula given in Proposition2.11.
3.3. Compounded rate cashflows with reset delay
Assuming now that , we can write,from(3.1),where
and is implied from the decomposition above.Therefore
Assume now that ,so that we can simplify the above as
4. Numerics
4.1. Zero-coupon dynamics
In Figure6, we analyse the impact of the parameter, in the Exponential kernel case (Section2.3.1)and in the Riemann-Liouville case (Section2.3.2),on the dynamics of the zero-coupon bond over a time span and considering a constant curve .In order to compare them properly, the underlying Brownian path is the same for all kernels.Unsurprisingly, we observe that the Riemann-Liouville case creates a lot more variance of the dynamics.
4.2. Impact of the roughness on convexity
We compare in Figure7 the impact of the (roughness of the) kernel on the convexity adjustment.We consider a constant curve and.As tends to zero (exponential kernel case) and as tends to (Riemann-Liouville case),the convexity adjustments converge to the same value (as expected),approximately equal to.In Figure8,we consider Example2.4.2,shifting away from a standard Brownian driver.
Disclosure of interest
We confirm that there are no relevant financial or non-financial competing interests to report.
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