An Optimal Transport approach to arbitrage correction: Application to volatility Stress-Tests

As addressed above, in this work we propose a new method based on OT to remove arbitrage from option data. In Section 2, we show how we can associate a discrete signed measure to option prices affected by arbitrage. In Section 3, we recall the concepts and results from OT that we use to define a Wasserstein metric between signed measures, following the notion already introduced in Ambrosio et al. [2]. Then, in Section 4, we define subsets of martingales in the space of discrete signed measures. Using the previously defined metric, we give the statement of our problem: the correction of arbitrage as a projection of the signed measure defined in Section 2 onto the martingale subsets. Next, we show that this problem is equivalent to an OT problem (i.e. a linear program on couplings). Section 5 is devoted to the entropic regularization of the minimization problem. We provide several results: we prove strong duality, dual attainment, and convergence when the regularization parameter tends to zero. In Section 6, we propose a multi-constrained Sinkhorn algorithm to approximate the solution of the regularized problem, for which we establish convergence as the number of iterates becomes large. The steps of the algorithm are either explicit, or boil down to finding the root of an explicit scalar function. Finally, Section 7 gathers several numerical applications to, first, showcase the theoretical results of convergence and, second, to illustrate the correction of stressed implied volatility smiles obtained with our method.

The feasibility to build a consistent martingale model can be checked with model-independent tests, relying only on the available data $\left(k_{j}^{i},c_{j}^{i}\right)_{1\leq i\leq m,\,1\leq j\leq n_{i}}$ (see [14, 15] for the construction of a discrete model or [19] for a continuous model). This is equivalent to no-arbitrage by the First Fundamental Theorem of Asset Pricing. Cohen et al. showed (see [13, Section 2.4, Proposition 2]) that these tests boil down to verify that $c$ solves a linear system of inequalities: $\mathbf{A}c\leq\mathbf{b}$ with $\mathbf{A}\in\mathbb{R}^{R\times\sum_{i=1}^{m}n_{i}},\,\mathbf{b}\in\mathbb{R}^{R}$¹¹1See their github repository: https://github.com/vicaws/arbitragerepair for an implementation of $\mathbf{A}$ and $\mathbf{b}$ given arbitrary data.. If any inequality of the system is violated, then there is at least one (static) arbitrage in the data.

Cousot’s construction [14, 15] is based on the notion of convex order between marginals.

Being NDCO is in fact a necessary and sufficient condition for the existence of a martingale $\mu$, on the product space $\left(\mathbb{R}^{m},\mathcal{B}(\mathbb{R}^{m})\right)$, with marginals $\left(\mu_{i}\right)_{1\leq i\leq m}$ (see Strassen’s Theorem [32]).

Let us briefly emphasize Cousot’s procedure to build discrete marginals, NDCO and consistent with the data (assuming the tests mentioned above are passed). First, we need to artificially extend the (normalized) data with two more prices at each maturity: for all $i\in\llbracket 1,m\rrbracket$, $c_{0}^{i}:=1$ at strike $k_{0}^{i}:=0$ and $c_{n_{i}+1}^{i}=0$ at a strike $k_{n_{i}+1}^{i}>k_{n_{i}}^{i}$ carefully chosen (see [14, Section 4.2]). Note that it is possible to choose $k_{n_{i}+1}^{i}=k_{\mathrm{max}}$ for all $i\in\llbracket 1,m\rrbracket$ (see Lemma 4.1.2 for more details). Then, the (normalized) call pricing function $\pi_{i}:k\mapsto\pi_{i}(k)$, for the maturity $T_{i}$, is defined as the decreasing and piecewise-affine function drawn by the lower boundary of the convex hull of the points $\left\{(k_{j}^{i^{\prime}},c_{j}^{i^{\prime}})\,|\,i\leq i^{\prime},\,0\leq j\leq n_{i^{\prime}}+1\right\}$ (and zero after $k_{\mathrm{max}}$). Under the no-arbitrage assumption, $\pi_{i}$ verifies $\pi_{i}(k_{j}^{i})=c_{j}^{i}$ for all $i,j\in\llbracket 1,m\rrbracket\times\llbracket 1,n_{i}\rrbracket$. Furthermore, it defines a discrete probability measure $\mu_{i}$, the second derivative of $\pi_{i}$ in the sense of distributions, supported on a subset of $\bigcup\limits_{i=1}^{m}\left\{k_{1}^{i},\cdots,k_{n_{i}}^{i}\right\}\cup\left\{0,k_{\mathrm{max}}\right\}$, satisfying $\pi_{i}(k)=\int(x-k)^{+}d\mu_{i}$, for all $k\in\mathbb{R}^{+}$ (see [14, Lemma 3.1]). Thus defined, the probability measures $\left(\mu_{i}\right)_{1\leq i\leq m}$ are NDCO and their support is included in

the collection of all strikes.

Assume now that there are some arbitrages in the data, meaning that c does not solve Cohen’s system of inequalities. For example, they may be due to a local stress-test applied to the associated IVS (initially arbitrage-free). In some applications, we will need to be able to control which part of the IVS must be preserved during the process of removing arbitrage opportunities. We denote by $\bar{c}$ any vector of arbitrage-free (normalized) prices, on an arbitrary sub-grid of $\left(T_{i},k_{j}^{i}\right)_{1\leq i\leq m,\,1\leq j\leq n_{i}}$. For instance, $\bar{c}$ could correspond to the prices that were not impacted by the hypothetical stress-test mentioned above. On the contrary, it could also consist of the ones that were actually impacted, as long as, ignoring the rest of the data, they are arbitrage-free. In order to identify the maturities and the (normalized) strikes of the sub-grid, we introduce $\mathcal{I}\subset\llbracket 1,m\rrbracket$ and for any $i\in\mathcal{I}$, $\mathcal{J}_{i}=\left\{j_{1}<\cdots<j_{\bar{n}_{i}}\right\}\subset\left\{1,\cdots,n_{i}\right\}$, so that $\bar{c}=\left(\bar{c}_{j}^{i}\right)_{i\in\mathcal{I},\,j\in\mathcal{J}_{i}}$. Finally, we call $\bar{n}=\sum\limits_{i\in\mathcal{I}}\bar{n}_{i}$ the length of $\bar{c}$.

Even if there are arbitrages, we will see that it is still possible to associate discrete measures that are consistent with the data, through a simple adaptation of Cousot’s construction. However, the resulting measures might be signed or fail to be NDCO. Similarly, we can enlarge the data with additional prices for each $T_{i}$: $c_{0}^{i}:=1$ at $k_{0}^{i}:=0$ and $c_{n_{i}+1}^{i}:=0$ at $k_{n_{i}+1}^{i}=k_{\mathrm{max}}>k_{n_{i}}^{i}$ (see Lemma 4.1.2 to see how $k_{\mathrm{max}}$ should be chosen). As previously, we call $\Theta$ the collection of all strikes, namely $\Theta=\bigcup\limits_{i=1}^{m}\left\{k_{1}^{i},\cdots,k_{n_{i}}^{i}\right\}\cup\left\{0,k_{\mathrm{max}}\right\}$ and, to ease notations, we rewrite it as $\{k_{1},\cdots,k_{l}\}$, with $k_{1}<\cdots<k_{l}$, so that $l=\#\Theta$, $k_{1}=0$ and $k_{l}=k_{\mathrm{max}}$.

By construction, $\pi_{i}$ is piecewise-affine, continuous and $\pi_{i}(k_{j}^{i})=c_{j}^{i}$.

Note that each $\nu_{i}$ has a finite support included in $\{k_{1}^{i},\cdots,k_{n_{i}}^{i}\}\cup\{0,k_{\mathrm{max}}\}$, but can be seen as a measure distributed on $\Theta$, with zero weights for the elements of $\Theta$ that are not in its support. The $\nu_{i}$’s being specified, we can define what we call a “joint signed measure” $\nu$, on the finite product space $\Theta^{m}$, with marginals $\left(\nu_{i}\right)_{1\leq i\leq m}$. A simple choice would be to set $\nu=\nu_{1}\otimes\cdots\otimes\nu_{m}$, but this could be somehow unsatisfactory, for reasons that will become clearer in Section 4.2. From now, we just assume that $\nu$ is built. Note that it satisfies $\nu(\Theta^{m})=1$. In the following section, we will define a distance between signed measures with total mass equal to one, to formulate the correction of arbitrages as the projection of $\nu$ onto the subset of martingales. Just recall that, from Remark 2.1.2, it is sufficient to look for martingales on $\Theta^{m}$ to get rid of arbitrages in the data (up to a choice of $k_{\mathrm{max}}$).

Wasserstein distances are usually defined for elements in $\mathcal{M}_{+}^{\varrho}(1)$ i.e. for probability measures with finite moment of order $\varrho$ (see the monographs [33, 34] for more details), but it extends naturally for positive measures with some fixed finite mass $\alpha\in\mathbb{R}_{+}^{*}$. Their definition is based on the theory of optimal transport.

The following definition, motivated by Ambrosio et al. [2], extends the $1$-Wasserstein distance to signed measures.

All these constraints are linear in the variable $\mu$. Hence, seeing now measures as vectors in $\mathbb{R}^{N}$, there exists a matrix $A\in\mathbb{R}^{L\times N}$ and a vector $b\in\mathbb{R}^{L}$ (resp. a matrix $\bar{A}\in\mathbb{R}^{\bar{L}\times N}$ and a vector $\bar{b}\in\mathbb{R}^{\bar{L}}$) such that $\mathscr{M}=\left\{\mu\in\mathbb{R}^{N}\,|\,\mu\geq 0,\,A\mu=b\right\}$ (resp. $\mathscr{M}_{\bar{c}}=\left\{\mu\in\mathbb{R}^{N}\,|\,\mu\geq 0,\,\bar{A}\mu=\bar{b}\right\}$).

For the rest of the paper, we assume that $k_{\mathrm{max}}$ is properly chosen, so that Lemma 4.1.2 always applies.

As discussed in 2.2, the signed marginals $(\nu_{i})_{1\leq i\leq m}$ are uniquely defined in our setting. However, we have a degree of freedom in the choice of the joint signed measure $\nu$ with these marginals. Even though there are arbitrages, the choice $\nu=\nu_{1}\otimes\cdots\otimes\nu_{m}$ would correspond, in some way, to an independent framework which is unsatisfactory. Indeed, recall that our aim is to project $\nu$ onto the subset of martingales. Martingales never have independent marginals (except for constant martingales). Consequently, choosing $\nu$ as the product measure would be somehow “very far” from $\mathscr{M}$ (or $\mathscr{M}_{\bar{c}}$). Thus, we can try to build a signed measure that already looks like a martingale. We propose below a possible construction for this joint signed measure, but leave for future work the question of whether this choice can be improved.

Let us assume, for the moment, that $(\nu_{i})_{1\leq i\leq m}$ are true probability measures NDCO. Then, from Strassen’s Theorem, there exists a discrete probability measure $\nu$ on $\Theta^{m}$ satisfying (1), (2), (3), (4) and

for all $i\in\llbracket 1,m\rrbracket$, for all $p_{i}\in\llbracket 1,l\rrbracket$ ($ml$ equality constraints), where $(\nu_{i})_{p_{i}}$ is the $p_{i}$-th coefficient of the vector $\nu_{i}$. The constraint (6) is again linear in the variable $\nu=(\nu_{p_{1},\cdots,p_{m}})$. Therefore, we can write $\nu\in\left\{x\in\mathbb{R}^{N}\,|\,x\geq 0,A_{\nu}x=b_{\nu}\right\}$, with $A_{\nu}=\,\begin{bmatrix}A\\ B\end{bmatrix}$ for some $B\in\mathbb{R}^{ml\times N}$ and $b_{\nu}=\left(b,\nu_{1},\cdots,\nu_{m}\right)\in\mathbb{R}^{L+ml}$. However, there is some redundancy in the constraints satisfied by $\nu$. For instance, if (4) and (6) are verified, then (2) and (3) are automatically true by construction. Said differently, $A_{\nu}$ doesn’t have full rank. Removing the redundant rows (resp. coefficients) in $A_{\nu}$ (resp. $b_{\nu}$) and keeping the same notations for simplicity, we may assume that $\nu$ belongs to $\left\{x\in\mathbb{R}^{N}\,|\,x\geq 0,A_{\nu}x=b_{\nu}\right\}$, with $A_{\nu}\in\mathbb{R}^{L_{\nu}\times N}$, $b_{\nu}\in\mathbb{R}^{L_{\nu}}$, and $\mathrm{rank}(A_{\nu})=\min(L_{\nu},N)<L+ml$.

From this remark and coming back to our setting, we could try to find $\nu$ in the set $\mathscr{M}_{signed}=\left\{x\in\mathbb{R}^{N}\,|\,A_{\nu}x=b_{\nu}\right\}$ i.e. removing the positivity constraint (otherwise this set is empty because of arbitrages). It corresponds to the set of “signed martingales”, with fixed marginals $(\nu_{i})_{1\leq i\leq m}$. Besides, recall from the positive case that the product measure maximizes the entropy, or, said differently, the randomness among joint distributions with prescribed marginals. Because the entropy is not well defined for signed measures, an alternative could be to choose a signed martingale close to $\nu_{1}\otimes\cdots\otimes\nu_{m}$. To some extent, it corresponds to the least biased choice. Therefore, we choose $\nu$ to be the unique solution of the following quadratic constrained optimization problem

$\mathscr{M}_{signed}$ is closed and non-empty because $A_{\nu}$ has full rank. Moreover, $||.||^{2}$ is continuous and coercive, hence existence of a minimizer follows. Uniqueness comes from the strict convexity of $||.||^{2}$.

$\nu$ will remain fixed for the rest of the paper. We define $\nu^{+},\nu^{-}\in\mathbb{R}^{N}$ by $\nu^{+}=\left(\max(\nu_{p},0)\right)_{1\leq p\leq N}$ and $\nu^{-}=\left(\max(-\nu_{p},0)\right)_{1\leq p\leq N}$. Note that we have $\nu^{-}\neq 0_{\mathbb{R}^{N}}$, because of the presence of arbitrages. Since $\mu\mapsto\mathbb{W}_{1}(\mu,\nu)$ does not depend on the decomposition of $\nu$ (see Proposition 3.2.2), we may add the same positive vector to $\nu^{+}$ and $\nu^{-}$, to obtain a positive decomposition of $\nu$. Keeping the same notations for simplicity, we assume from now that we work with such decomposition. Finally, we call $\alpha\in\mathbb{R}^{*}_{+}$ the total mass of $\nu^{+}$ (which is nothing else than the $\mathrm{L}^{1}$-norm of $\nu^{+}\in\mathbb{R}^{N}$).

Now that $\nu$ is defined, we have all the ingredients to formulate the correction of arbitrages as a projection problem. More specifically, we will be interested in:

and

These infima are finite since $\mathscr{M}$ and $\mathscr{M}_{\bar{c}}$ are non-empty and $0\leq\mathbb{W}_{1}(.,\nu)<+\infty$.

In the discrete setting, Definition 3.2.1 reduces to

where $\Gamma(\mu^{+}+\nu^{-},\mu^{-}+\nu^{+})=\left\{M\in\mathbb{R}^{N\times N}_{+}\,|\,M\mathds{1}_{N}=\mu^{+}+\nu^{-}\text{ and }M^{\top}\mathds{1}_{N}=\mu^{-}+\nu^{+}\right\}$, $\mathds{1}_{N}=(1,\cdots,1)\in\mathbb{R}^{N}$, $\mu^{+}=\left(\max(\mu_{p},0)\right)_{1\leq p\leq N}$, $\mu^{-}=\left(\max(-\mu_{p},0)\right)_{1\leq p\leq N}$ and $\langle.,.\rangle_{F}:(A,B)\mapsto\mathrm{Tr}\left(A^{\top}B\right)$ is the Frobenius dot product between matrices. We also define two sets of couplings that will play a major role in what follows