Sig-SDEs model for quantitative finance

Introduction

The question of finding a parsimonious model that well represents empirical data has been of paramount importance in quantitative finance. The modelling choice is dictated by the desire to fit and explain the available data, but is also subject to computational considerations. Inevitably, all models can only provide an approximation to reality, and the risk of using inadequate ones is hard to detect. A classical approach consists in fixing a class of parametric models, with a number of parameters that is significantly smaller than the number of available data points. Next, in the process called calibration, the goal is to solve a data-dependent optimization problem yielding an optimal choice of model parameters. The main challenge, of course, is to decide what class of models one should choose from. The theory of statistical learning (Vapnik, 2013) tell us that to simple models cannot fit the data, and to complex one are not expected to generalise to unseen observations. In modern machine learning approaches, one usually starts by defining a highly oveparametrised model from some universality class, exhibiting a number of parameters often exceeding the number of data points, and let (stochastic) gradient algorithms find the best configuration of parameters yielding a calibrated model. In this work, we find a middle ground between the two approaches. We develop a new framework for systematic model selection that exhibits universal approximation properties, and we provide a explicit solution to the optimization used in its calibration, that completely removes the need to deploy expensive gradient descent algorithms. Importantly the class of models that we consider builds upon classical risk models that are well underpinned by research on quantitative finance.

The mathematical object at the core of this work is the expected signature of a path, whose properties are well-understood in the field of stochastic analysis. It allows to identify a linear structure underpinning the high non-linearity of the sequential data we work with. This linear structure leads to a massive speed-up of calibration, pricing, and generation of future scenarios. Our approach provides a new systematic model selection mechanism, that can also be deployed to calibrate classical non-Markovian models in a computationally efficient way. Signatures have been deployed to solve various tasks in mathematical finance, such as options pricing and hedging (Lyons et al., 2019, 2020), high frequency optimal execution (Kalsi et al., 2020; Cartea et al., 2020) and others (Lyons et al., 2014; Gyurkó et al., 2013). They have also been applied in several areas of machine learning (Kidger et al., 2019; Yang et al., 2015, 2016a, 2016c, 2016b, 2017; Xie et al., 2017; Li et al., 2017; Chevyrev and Oberhauser, 2018; Király and Oberhauser, 2016).

A natural question would be whether one can find a model for the volatility process $\Sigma_{t}$ that is large enough to include all the classical models such as the ones mentioned above and that would allow for systematic a data driven model selection. We will require such a model to satisfy the following requirements:

Universality. The model should be able to approximate arbitrarily well the dynamics of classical models.

Efficient calibration. Given market prices for a family of options, it should be possible to efficiently calibrate the model so that it correctly prices the family of options.

Fast pricing. Ideally, it should be possible to quickly price (potentially exotic) options under the model without using Monte Carlo techniques.

Efficient simulation. Sampling trajectories from the model should be computationally cheap and efficient.

An example of a model that satisfies point 1. above is a neural network model, where the volatility process $\Sigma_{t}$ is approximated by a neural network $\mathcal{NN}^{\theta}(t,(W_{s})_{s\in[0,t]})$ with parameters $\theta$. Such a model would be able to approximate a rich class of classical models. However, the calibration and pricing of such models would involve performing multiple Monte Carlo simulations on each epoch, which might be expensive if done naively. See however, (Cuchiero et al., 2020; Gierjatowicz P., 2020).

The aim of this paper is to propose a model for asset price dynamics that, we believe, satisfies all four points above. Our technique models the volatility process $\Sigma_{t}$ as

In other words, any continuous function on a compact set of paths can be uniformly well approximated by a linear combination of terms of the signature. This universal approximation property is similar to the one provided by Neural Networks (NN). However, as we will discuss below, NN models depend on a very large collection of parameters that need to be optimized via expensive back-propagation-based techniques, whilst the optimization needed in our Sig-SDE model consists of a simple linear regression on the terms of the signature. In this way, the signature can be thought of as a feature map for paths that provides a linear basis for the space of continuous functions on paths. In the setting of SDEs, sample paths are Brownian and solutions are images of these sample trajectories by a continuous functions that one wishes to approximate from a set of observations. Our Sig-SDE model will rely upon the universality of the signature to approximate such functions acting on Brownian trajectories. Importantly, the signature of a realisation of a semimartingale provides a unique representation of the sample trajectory (Hambly and Lyons, 2010; Boedihardjo et al., 2016). Similarly, the expected signature – i.e. the collection of the expectations of the iterated integrals – provides a unique representation of the law of the semimartingale (Chevyrev et al., 2016).

Notation and preliminaries

We begin by introducing some notation and preliminary results that are used in this paper.

Let $I=(i_{1},\ldots,i_{p})$ and $J=(j_{1},\ldots,j_{q})$ be any two multi-indices in $\mathcal{I}_{d}$. Their concatenation product $\otimes$ as the multi-index $I\otimes J=(i_{1},\ldots,i_{m},j_{1},\ldots,j_{n})\in\mathcal{I}_{d}$.

2. Linear functionals

For a given $d\geq 1$, a linear functional is a (possibly infinite) sequence of real numbers indexed by multi-indices in $\mathcal{I}_{d}$ of the following form

We note that a multi-index $K\in\mathcal{I}_{d}$ is always a linear functional. Both concatenation $\otimes$ and can be extended by linearity to operations on linear functionals. We will now define two basic operations on linear functionals that will be used throughout the paper.

3. Signatures

Rough paths theory can be briefly described as a non-linear extension of the classical theory of controlled differential equations which is robust enough to allow a deterministic treatment of stochastic differential equations controlled by much rougher signals than semi-martingales (Lyons et al., 2007).

where the integral is to be interpreted in the Stratonovich sense.

A more detailed overview of signatures is included in Appendix A.

Signature Model

In this section we define the Signature Model for asset price dynamics that we propose in this paper. The goal is to approximate the volatility process $\Sigma_{t}$ (that is a continuous function on the driving Brownian path) by a linear functional on the signature of the Brownian path.

The Signature Model possesses the universality property, in the sense that given a classical model, there exists a Signature Model that can approximate its dynamics to a given accuracy (Levin et al., 2013).

We show in the upcoming Sections 5-7 that (a) the Signature Model is efficient to simulate, (b) it is efficient to calibrate, and (c) exotic options can be priced fast under the Signature Model.

The Signature Model introduced in Definition 3.1 assumes that the source of noise (i.e. the Brownian motion $W$) is one-dimensional. This was done for simplicity, but the authors would like to emphasise that the model generalises in a straightforward way to multi-dimensional Brownian motion.

Numerical experiments

We now demonstrate the feasibility of our methodology as outlined in Sections 5-7. Throughout this section, we work with the Signature Model

In this section we will show experiments for the calibration of the model, pricing of options under the signature model and simulation. Sections 5-7 will then include the technical details of how calibration, pricing and simulation of signatures model are done.

We assume that the family of options available on the market are a mixture of vanilla and exotic options, given as follows:

Vanilla call options with strikes $K=0.5,0.6,\ldots,1,1.$ and maturities $t=0.4,0.45,0.5,\ldots,0.9,0.95,1$:

Variance options with strikes $K=0.01,0.015,\ldots,0.035,0.04$ and maturities $t=0.4,0.45,0.5,\ldots,0.9,0.95,1$:

where $\langle X\rangle$ is the quadratic variation of $X$.

Down-and-Out barrier call options with maturity 1, strikes $K=0.9,0.92,0.94,\ldots,1.01,1.03$ and barrier levels $L=0.6,0.62,0.64,\ldots,0.88,0.9$:

The option prices are generated from a Black-Scholes model with volatility $\sigma=0.2$:

Figure 1 shows the absolute error between the real option prices and the option prices of the calibrated model, for the different option types.

2. Simulation

Once the Signature Model has been calibrated to the available option prices, we can use Algorithm 1 to simulate realisations of the calibrated Signature Model. Figure 2 shows 1,000 realisations of the Signature Model.

3. Pricing

We will now use the calibrated Signature Model to price a new set of options that was not used in the calibration step. This set of option consists of Down-and-In barrier put options with barriers levels $L=0.7,0.71,\ldots,0.81,0.82$ and strikes $K=0.9,0.92,\ldots,1.01,1.03$:

Figure 3 shows the absolute error of the prices under the Signature Model, compared to the real prices.

As we see, the calibrated model is able to generate accurate prices for these new exotic options. The error is highest when the barrier is close to the strike price, as expected.

Simulation

Let $0\leq s\leq t$. Then, for each multi-index $K\in\mathcal{I}_{d}$ we have

These two results lead to Algorithm 1. We note there are a number of publicly available software packages to compute signatures, such as esig https://pypi.org/project/esig/, iisignature https://github.com/bottler/iisignature, (Reizenstein and Graham, 2020) and signatory https://github.com/patrick-kidger/signatory, (Kidger and Lyons, 2020).

Pricing

This section will show that exotic options can be priced fast under a Signature Model. This will be done via a two step procedure. First, it was shown in (Lyons et al., 2019, 2020) that prices of exotic options can be approximated with arbitrary precision by a special class of payoffs called signature payoffs, defined below. Hence, we will assume that the exotic option to be priced is a signature payoff, defined as follows.

Let $I=(i_{1},\ldots,i_{n})$ be any multi-index in $\mathcal{I}_{2}^{n}$ such that $n\leq N$. If $n=1$ then $I=(i_{1})$ and we necessarily one of the following two options must hold

Hence the statement holds for $n=1$. Let’s assume by induction that the statement holds for any $1\leq n\leq N$. We write $I=J\otimes(i_{n})$ with $i_{n}\in\{1,2\}$ and $J=(i_{1},\ldots,i_{n-1})\in\mathcal{I}_{2}^{n-1}$. Clearly $|(i_{n})|,|J|<n$, therefore by induction hypothesis

By definition of the signature (see 2.6) we know that

Calibration

Following Section 6, we assume that the options $\Phi_{i}$ are given by signature options. Therefore, we assume that we can write $\Phi_{i}$ by

The minimisation problem we aim to solve now is the following:

Conclusion

In this paper we have proposed a new model for asset price dynamics called the signature model. This model was develop with the objective of satisfying the following properties:

Efficiency of calibration to vanilla and exotic options.

Fast pricing of vanilla and exotic options.

Due to the rich properties of signatures, the signature model satisfies all four properties and is, therefore, capable of generating realistic paths without sacrificing the computational feasibility of calibration, pricing and simulation.

References

Appendix A Overview of signatures

In this section we state some of the main properties of signatures that are used in this paper.

For any two multi-indices $I,J\in\mathcal{I}_{d}$ and $1$-dimensional multi-indices $a,b\in\mathcal{I}_{d}^{1}=\{1,\ldots,d\}$ we define the shuffle product recursively as follows:

We have the following examples for $\mathcal{I}_{4}$:

$(1,2)\shuffle(3)=(1,2,3)+(1,3,2)+(3,1,2)$.

$(1,2)\shuffle(3,4)=2\cdot(1,2,2,4)+(1,2,4,2)+(2,1,2,4)+(2,1,4,2)+(2,4,1,2)$.

See main result in (Hambly and Lyons, 2010). ∎

Proposition 2.2 in (Lyons et al., 2007). ∎

For a given time interval $[0,T]$ we call a continuous, surjective, increasing function $\psi:[0,T]\to[0,T]$ a time-reparametrization.

We will now define the expected signature of a semimartingale.

The expected signature – i.e. the expectation of the iterated integrals (8) – behaves analogously to the moments of random variables, in the sense that under certain assumptions it characterises the law of the stochastic process:

Appendix B Time and Lead-lag transformation

The invariance of the signature of a semimartingale to time reparametrizations allows to handle irregularly sampled sample paths (prices etc.) by completely eliminating the need to retain information about the original time-parametrization. Nonetheless, for the pricing of many options, especially ones resulting from payoffs calculated pathwise (such as integrals for American options), the time represents an important information that we are required to retain. To do so it suffices to augment the state space of the input semimartingale $X$ by adding time $t$ as an extra dimension to get $X^{\text{add-time}}_{t}=(t,X_{t})$.

We report another basic transformation that can be applied to semimartingales and that will be useful in the sequel of the paper: the lead-lag transformation. This transformation allows us to write Itô integrals as linear functions on the signature of the lead-lag transformed path.

and linear interpolation in between. Figure 4 shows the lead-lag transformation of a Brownian motion. As we see, the lead component leads the lag component, hence the name. The lead component can be seen as the future of the path, and the lag component as the past.

The work in (Flint et al., 2016) showed the convergence of this limit and studied some of its properties.

Let $I=(i_{1},\ldots,i_{n})\in\mathcal{\mathcal{missing}}{I}^{n}_{3}$ be a multi-index. We denote by $\mathcal{P}(I)$ the set of all possible tuples of non-empty multi-indices from $\mathcal{I}^{n-1}_{3}$ such that their concatenation is equal to $I$ and their length doesn’t exceed $2$, i.e.

$\mathcal{P}((1,2,3))=\{(1,2,3),(1,(2,3)),((1,2),3)\}$.

$\begin{aligned} \mathcal{P}((1,3,2,2))=&\{(1,3,2,2),(1,(3,2),2),(1,3,(2,2)),\\ &((1,3),2,2),((1,3),(2,2))\}\end{aligned}$

where for any $k\geq 1,F^{\otimes k}=F\otimes\ldots\otimes F$ for $k$ times.

Define the function $\alpha:\mathcal{I}_{3}\to\mathcal{I}_{3}$ that maps a multi-index to another multi-index in the following way: $\forall I\in\mathcal{I}_{3}$

Given a final time $T$ define the linear functional $\mathbf{E}_{T}:=\exp(T+\frac{T}{2}(2,2))$. Then we have the explicit closed-form expression for the Expected Signature of the lead-lag Brownian motion: given any multi-index $I\in\mathcal{I}_{3}$

If $I=(3,2,3)$, $\mathcal{P}(I)=\{(3,2,3),((3,2),3),(3,(2,3))\}$. Hence,