Derivatives pricing using signature payoffs

Introduction

This offers a wide range of applications in various fields, including finance. Suppose we have some (possibly high-dimensional) price paths, and we want to study the effects of an unknown real-valued continuous function on price paths. If all we know about the function is its image on a finite set of such price paths, we could infer more knowledge about it by taking a look to its linear effects on the signature of the price paths. We can then use this linear representation of the unkown function in terms of the signature to make predictions about the value of the function on unseen price paths.

The objective of this paper is to leverage this linear representation of functions on paths to quickly price arbitrary financial derivatives. We do so by introducing in Section 4 a family of derivatives called signature payoffs. Similarly to polynomials, these signature payoffs form an algebra: the sum of two signature payoff is a signature payoff; the product of two signature payoffs is a signature payoff. Moreover, signature payoffs are fast to price and can approximate continuous payoffs – i.e. continuous real-valued functions defined on price paths – arbitrarily well.

In Section 6 we applied the methodology to approximate prices of European options, American options, Asian options, lookback options and variance swaps using signature payoffs, and the signature approach proved to be very accurate at pricing them.

Notation

The signature of a path

We shall begin by defining the space of $d$-dimensional paths with bounded $p$-variation.

where the suppremum is taken over all partitions of $[0,T]$. The space of continuous paths of bounded $p$-variation is then defined as

Let $X$ be a path for which the signature is well defined. The first term of the signature is given by

In other words, the first term of the signature is equal to increment of the path over the period $[0,T]$. The tensor $X^{2}$, on the other hand, is just the Levy area of the path: the signed area between the path and the chord that joins the initial point $X_{0}$ and the ending point $X_{T}$. Higher order terms of the signature capture other aspects of the path, but their geometric intuition becomes less clear.

Signatures are unique representations of paths up to tree-like equivalences () that are parametrisation invariant ([9, Lemma 2.12]). The reduction of the infinite dimensional group of reparametrisations makes signatures a concise way of representating paths, so that it makes sense to try to describe functions on paths by taking a look to their effects on path signatures. This is the idea we followed in Section 4, where we define signature payoffs as payoffs on price paths that act through their signatures in a linear way. As we will see, this class of payoffs is rich enough to approximate continuous payoffs to arbitrary accuracy.

Similarly to the case of the polynomial map discussed in the introduction, the expected signature of a stochastic process turns out to contain enough information about the law of the process to completely determine it ().

Before defining signature payoffs, we will introduce an augmention of a one-dimensional path.

Therefore, the augmention of a path is a $3$-dimensional path. A payoff function will then be defined as a real-valued mapping on augmented paths.

The payoff of a lookback call option with floating strike is defined as $\mathscr{P}(\widehat{X}):=\max_{0\leq t\leq T}X_{t}-X_{T}$ for all $\widehat{X}\in\mathcal{A}^{p}([0,T])$.

Notice that in these two examples, the payoff function is continuous. This is true for many other derivatives such as American options, Asian options, variance swaps, etc.

Signature payoffs

The assumption that $S(\widehat{X})$ is a $p$-geometric rough path is used to ensure uniqueness of the signature, as per . If $X$ has bounded variation (i.e. $p=1$) then the signature, defined as Riemann–Stieltjes integrals, will always be a geometric rough path. If $X$ is a semimartingale, we need to define $S(\widehat{X})$ using Stratonovich integrals to make sure that the signature of $\widehat{X}$ is a geometric rough path.

It is important to have examples of payoffs that satisfy the conditions of Theorem 4.2. Most financial derivatives, both vanilla and exotic options, are included in the framework of Theorem 4.2. Some examples include European options, American options, Asian options, lookback options and variance swaps. Notice that barrier options, for example, are not included in this setting in principle because the discontinuity on the barrier implies that they do not satisfy the continuity hypothesis of Theorem 4.2. However, one could fix this by smoothening barrier payoffs – for example, with barrier bending.

A major application of Theorem 4.2 is that as we have found a family of derivatives that approximates continuous payoffs, we could price a given derivative or a basket of derivatives by approximation using signature payoffs. As we will see in Section 5, signature payoffs can be quickly priced. Therefore, pricing a derivative by approximation with signatures becomes especially interesting when pricing the derivative is computationally expensive. In Section 6 we will see that this approach to pricing arbitrary continuous payoffs is extremely accurate.

Pricing signature payoffs

However, since the signature payoffs were defined to be linear functions on signatures, we have:

Suppose now that one has a basket of derivatives $\{\mathscr{P}_{1},\ldots,\mathscr{P}_{N}\}$. These payoffs could be expensive to price, so that pricing the entire basket would take a considerable amount of time. Using Theorem 4.2, one could use the following procedure to price the basket:

with $r$ and $\sigma$ the interest rate and volatility respectively, which are assumed to be constant, and $W$ a standard Brownian motion. The augmented path $\widehat{X}_{t}=\left(t,X_{t},\frac{t}{T}X_{0}\right)$ will then be a diffusion process satisfying the SDE

and the initial condition is $\widehat{X}_{0}=(0,0,X_{0})$.

It turns out that the function $\Phi$ satisfies a parabolic partial differential equation, as shown in [7, Theorem 4.7].

Let $X$ be a $d$-dimensional Itô diffusion process

with $W$ a $k$-dimensional Brownian motion and

Moreover, $\Phi_{n}$ satisfies the initial condition $\Phi_{n}(0,\cdot)=0$ for $n\geq 1$, and $\Phi_{0}\equiv 1$.

Using the theorem above, we may now state and prove a theorem that gives an explicit recurrent relation for the expected signature of the Black–Scholes model.

Define $E_{n}(t):=\exp\left(\left(nr+n(n-1)\frac{\sigma^{2}}{2}\right)t\right)$ for $0\leq t\leq T$, and given a word $I$ with alphabet $\{1,2,3\}$ let $\alpha(I)$ denote the number of appearences of the letters 2 or 3 in $I$.

Moreover, the first order term of $F$ is given by $F_{1}(t)=te_{1}+X_{0}(\exp(rt)-1)e_{2}+X_{0}te_{3}/T$, and for $n\geq 2$ the nth term of $F$ is characterised by the following:

$\pi^{(1,i_{2},\ldots,i_{n})}(F(t))=\int_{0}^{t}E_{\alpha((1,i_{2},\ldots,i_{n}))}(s)\pi^{(i_{2},\ldots,i_{n})}(F(t-s))ds$.

$\pi^{(2,1,i_{3},\ldots,i_{n})}(F(t))=(r+\sigma^{2}\alpha((1,i_{3},\ldots,i_{n})))\int_{0}^{t}E_{\alpha((2,1,i_{3},\ldots,i_{n}))}(s)\pi^{(1,i_{3},\ldots,i_{n})}(F(t-s))ds$.

$\pi^{(2,2,i_{3},\ldots,i_{n})}(F(t))=(r+\sigma^{2}\alpha((2,i_{3},\ldots,i_{n})))\int_{0}^{t}E_{\alpha((2,2,i_{3},\ldots,i_{n}))}\pi^{(2,i_{3},\ldots,i_{n})}(F(t-s))ds$ $+\frac{1}{2}\sigma^{2}\int_{0}^{t}E_{\alpha((2,2,i_{3},\ldots,i_{n}))}(s)\pi^{(i_{3},\ldots,i_{n})}(F(t-s))ds$.

$\pi^{(3,i_{2},\ldots,i_{n})}(F(t))=\frac{1}{T}\pi^{(1,i_{2},\ldots,i_{n})}(F(t))$.

Assume $X$ follows the dynamics given in (3). Then, $X$ is of the form

where $A$ is the infinitesimal generator of $\widehat{X}$. Moreover, $\Phi_{0}\equiv 1$ and $\Phi_{n}(0,\cdot)=0$.

Set $f_{n}(t,x)=(e_{1}+rxe_{2}+X_{0}e_{3}/T)\otimes\Phi_{n-1}(t,x)+\sigma^{2}x^{2}e_{2}\otimes\partial_{x}\Phi_{n-1}(t,x)+\dfrac{1}{2}\sigma^{2}x^{2}e_{2}\otimes e_{2}\otimes\Phi_{n-2}(t,x)$ for $n\geq 2$, and $f_{1}(t,x)=e_{1}+rxe_{2}+X_{0}e_{3}/T$. By Feynman-Kac formula, $\Phi_{n}$ will then be given by

Therefore, for $I=(1)$, $I=(2)$ or $I=(3)$, we have $\pi^{I}(\Phi(t,X_{0}))=X_{0}^{\alpha(I)}\pi^{I}(F(t))$ with $\pi^{(1)}(F(t)):=t$, $\pi^{(2)}(F(t)):=\exp(rt)-1$ and $\pi^{(3)}(F(t)):=t/T$.

Assume now that $\pi^{I}(\Phi(t,X_{0}))=X_{0}^{\alpha(I)}\pi^{I}(F(t))$ holds for all words $I$ of length strictly less than $n$. We will show that it also holds for words of length $n$, with $n\geq 2$. We will distinguish four steps.

Case 1: $I=(1,i_{2},\ldots,i_{n})$. In this case $\pi^{(1,i_{2},\ldots,i_{n})}(f_{n}(t,x))=\pi^{(i_{2},\ldots,i_{n})}(\Phi_{n-1}(t,x))=\\ x^{\alpha((i_{2},\ldots,i_{n}))}\pi^{(i_{2},\ldots,i_{n})}(F(t))$, so that $\pi^{(1,i_{2},\ldots,i_{n})}(\Phi_{n}(t,X_{0}))=X_{0}^{\alpha((1,i_{2},\ldots,i_{n}))}\pi^{(1,i_{2},\ldots,i_{n})}(F(t))$ with $\pi^{(1,i_{2},\ldots,i_{n})}(F(t)):=\int_{0}^{t}E_{\alpha((1,i_{2},\ldots,i_{n}))}(s)\pi^{(i_{2},\ldots,i_{n})}(F(t-s))ds$.

Case 2: $I=(2,1,i_{3},\ldots,i_{n})$. We have

Hence, $\pi^{(2,1,i_{3},\ldots,i_{n})}(\Phi_{n}(t,X_{0}))=X_{0}^{\alpha((2,1,i_{3},\ldots,i_{n}))}\pi^{(2,1,i_{3},\ldots,i_{n})}(F(t))$ with

Case 3: $I=(2,2,i_{3},\ldots,i_{n})$. We have

Thus, $\pi^{(2,2,i_{3},\ldots,i_{n})}(\Phi_{n}(t,X_{0}))=X_{0}^{\alpha((2,2,i_{3},\ldots,i_{n}))}\pi^{(2,2,i_{3},\ldots,i_{n})}(F(t))$ with+

Case 4: $I=(3,i_{2},\ldots,i_{n})$. From the definition of $\pi^{(3,i_{2},\ldots,i_{n})}(S(\widehat{X}))$, we have:

Theorem 5.2 provides a way to explicitly find the expected signature. Therefore, we have now found a way of explicitly pricing under the Black–Scholes model any signature payoff. Since the terms in $F$ can be iteratively computed up to a given order in a fast way, we can price signature payoffs inexpensively:

which corresponds to the well-known fair value of forward contracts.

In this section we studied what the fair value of a signature payoff is, under the Black–Scholes model, and we found an explict closed formula. A natural question would be whether a similar procedure can be followed for more complex models. Assuming that the price path follows some diffusion process, such as a local volatility model, one should be able to apply Theorem 5.1 again in order to get a similar explicit expresion for the fair value.

Numerical experiments

We implemented the proposed approach of pricing using signature payoffs to compute the fair price of different derivatives. We considered European call options with moneyness $99\%$, American put options with moneyness $99\%$, Asian options with moneyness $102\%$, lookback options and variance swaps with strike $20\%$, all with 1 year maturity. All of these derivatives satisfy the conditions of Theorem 4.2.

For obvious computational reasons, we had to truncate signatures so that instead of considering linear functionals on the full signature, we considered linear functionals on the truncated signature of order $n\geq 1$. For this experiments, we fixed $n=4$, which for a 3-dimensional augmented path produces a linear functional of dimension $1+3+3^{2}+3^{3}+3^{4}=121$.

The linear functional was estimated applying linear regression against the truncated signature using a dataset of simulated market conditions following the Black–Scholes model (3). A dataset of $100$ different market conditions was used for this task – a tiny dataset for machine learning standards. We then priced the derivatives using the approximated signature payoff in an out-of-sample set of 100 market conditions, and we then compared the price we obtained with the corresponding real prices. As we see in Figure 1, the accuracy is remarkable: we obtained an $R^{2}$ higher than 0.99999 in all the derivatives we considered.

Conclusion

In this paper we introduce signature payoffs, a family of derivatives that pay to the holder an amount dependent on the signature of the price process. This dependence on the signature is given by a linear functional, which makes pricing them relatively easy – the task of computing the fair value of signature payoffs is reduced to the task of computing an expected signature. As we have seen, this expected signature is easy to compute if the price process is assumed to follow a specific model. In Section 5.1 we studied the particular case of a Black–Scholes model, but the computations shown in that section can easily be extended to other models. Moreover, one could easily extend the framework to price signature payoffs for dividend-paying underlyings by suitably modifying the definition of the augmention of a path (Definition 3) to incorporate information about the dividends. A similar approach can be followed to price multi-asset signature payoffs.

The power of signature payoffs comes from the capability of signatures to approximate continuous functions on paths. In Theorem 4.2 we show that signature payoffs can approximate arbitrary continuous functions to arbitrary accuracy. This makes pricing entire baskets of derivatives quick: using an accurate representation as a signature payoff of each derivative in the basket, which can be precomputed, we can use Section 5 to approximate the fair value of each derivative in the basket by the corresponding fair values of the signature payoffs. As we saw in Section 6, this approximation turns out to be remarkably accurate: we achieved an $R^{2}$ higher than $0.99999$ in all the payoffs we considered.

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