class: center, middle, inverse, title-slide # Reverse sensitivity testing with compound Poisson processes ### Emma Kroell¹ <br /> Joint work with Silvana Pesenti¹ and Sebastian Jaimungal¹ ### ¹ Department of Statistical Sciences <br /> University of Toronto --- <style>.shareagain-bar { --shareagain-foreground: rgb(255, 255, 255); --shareagain-background: rgba(0, 0, 0, 0.5); --shareagain-twitter: none; --shareagain-facebook: none; --shareagain-linkedin: none; --shareagain-pinterest: none; --shareagain-pocket: none; --shareagain-reddit: none; }</style> `\(\newcommand{\Q}{{\mathbb{Q}}}\)` `\(\renewcommand{\P}{{\mathbb{P}}}\)` `\(\renewcommand{\R}{{\mathbb{R}}}\)` `\(\renewcommand{\E}{{\mathbb{E}}}\)` `\(\renewcommand{\F}{{\mathcal{F}}}\)` `\(\newcommand{\Qset}{{\mathcal{Q}}}\)` `\(\newcommand{\dQP}{\frac{d\Q}{d\P}}\)` `\(\newcommand{\boldeta}{{\boldsymbol{\eta}}}\)` `\(\DeclareMathOperator*{\essinf}{ess\,inf}\)` `\(\DeclareMathOperator*{\esssup}{ess\,sup}\)` `\(\renewcommand{\L}{{\mathcal{L}}}\)` ## Motivation <p style="margin-bottom:2cm;"> </p> - Suppose you have a portfolio `\(X_t\)` for `\(t \in [0,T]\)`. - Define an **adverse event** to `\(X_T\)`, for example an increase in a risk measure of `\(X_T\)`. - What type a stress at an earlier time would cause this adverse event to occur? - We restrict to the most-likely scenarios by finding the measure that achieves the adverse event while being "closest" to the reference measure using the **Kullback-Leibler (KL) divergence**. --- ## Literature review <p style="margin-bottom:2.5cm;"> </p> - Our work extends Pesenti et al. (2019): random variable case, VaR and ES constraints - Other distances: `\(f\)`-divergence (Cambou and Filipovic, 2017), Wasserstein (Pesenti, 2021), `\(\chi^2\)` divergence (Makam, Millossovich, and Tsanakas, 2021) - Another perspective: looking at worst-case outcome within a certain radius of the reference measure: Breuer and Csiszar (2013), Glasserman and Xu (2014), Blanchet and Murthy (2019) --- class: inverse, center, middle # Mathematical Preliminaries --- ### KL Divergence <p style="margin-bottom:2.5cm;"> </p> The KL divergence of `\(\Q\)` with respect to `\(\P\)` (also known as the relative entropy) is defined as `\begin{equation} D_{KL}(\mathbb{Q} \, \Vert \, \mathbb{P} ) = \begin{cases} \mathbb{E} \left[ \frac{d\mathbb{Q}}{d\mathbb{P}} \; \log \left( \frac{d\mathbb{Q}}{d\mathbb{P}} \right) \right] & \text{if } \Q \ll \P\\ \infty & \text{otherwise}\,, \end{cases} \end{equation}` where we use the convention `\(0 \log 0 = 0\)` and `\(\dQP\)` denotes the Radon-Nikodym (RN) derivative of `\(\Q\)` with respect to `\(\P\)`. --- ### Model <p style="margin-bottom:2.5cm;"> </p> - We take as given a filtered probability space `\((\Omega, \P, \F, \{\F_t\}_{t \in [0,T]})\)` - We consider jump processes of the form `\begin{equation} dX_t = \int_{\R} x\, \mu(dx,dt) \end{equation}` where `\(\mu\)` is a Poisson random measure - Assume `\(X\)` is compound Poisson; the mean measure is of the form `\begin{equation} \nu(dx,dt) = \kappa \, G(dx) dt. \end{equation}` --- class: inverse, center, middle # Main Problem --- ### Minimally perturbed process under stress We impose the stress on the process `\(X_t\)`, `\(t \in [0,T]\)` at the terminal time `\(T\)`. <p style="margin-bottom:0.8cm;"> </p> <div class="problem" number="1"}> <h1 style="font-size:0.5vw"> <span style="color:#dedee7">skip</span> </h1> Let \(f_i \colon \R \to \R\) and \(c_i \in \R\) for \(i \in [n]\), where \([n] := \{1, \ldots, n \}\), and consider \[ \inf_{\Q\in\Qset} D_{KL}(\mathbb{Q} \, \Vert \, \mathbb{P} ) \quad \text{s.t.} \quad \E^\Q\left[f_i(X_T)\right]=c_i, \quad i \in [n]\, , \] where \(\Qset\) is the class of equivalent probability measures given by \[ \mathcal{Q}:=\left\{\Q_h\; \Big| \; \frac{d\Q_h}{d\P}=\mathfrak{E}\left(\int_0^T \int_\R \left[ h_t(y) - 1 \right] \, \tilde \mu(dy,dt)\right) \right\} , \] where \(\mathfrak{E}(\cdot)\) denotes the stochastic exponential, \(\tilde \mu (dy, dt) := \mu (dy, dt) - \nu (dy, dt)\) the compensated measure, and \(h_t\) is a predictable, non-negative random fields satisfying \[ \E \left[ \exp \left( \int_0^T \int_\R (1 - h_t(y))^2 \, \mu(dy,dt) \right) \right] < \infty . \] </div> --- <p style="margin-bottom:1.5cm;"> </p> <div class="theorem"> <h1 style="font-size:0.5vw"> <span style="color:#dedee7">skip</span> </h1> If there exists \(\boldeta^* = (\eta_1^*, \ldots, \eta_n^*) \in \R^n\) such that \(\E[\exp \left( -\sum_{i=1}^n \eta_i^* \, f_i(X_T) \right)] < \infty\) and \[0 = \E \left[ \exp \left( -\sum_{j=1}^n\eta^*_j\,f_j(X_T) \right)\left( f_i(X_T)-c_i \right)\right], \quad i = 1, \ldots, n,\] then Optimization Problem 1 has a solution. It is the measure \(\Q^*\) characterized by the measure-change function \[h^*(t,x,y) = \frac{\E_{t,x+y}\left[\exp \left( -\sum_{i=1}^n\eta^*_i\,f_i(X_T) \right)\right]}{\E_{t,x}\left[\exp \left( -\sum_{i=1}^n\eta^*_i\,f_i(X_T) \right)\right]},\] where \(\E_{t,x}[\cdot]\) denotes the \(\P\)-expectation given that the process \(X\) has initial condition \(X_{t^-}=x\). One can write the corresponding Radon-Nikodym derivative as \[\frac{d\Q^*}{d\P} = \frac{\exp \left( -\sum_{i=1}^n\eta^*_i\,f_i(X_T) \right)}{\E[\exp \left( -\sum_{i=1}^n\eta^*_i\,f_i(X_T) \right)]}.\] The solution is unique. </div> --- ### Sketch of proof The time `\(t\)` version of the associated **value function**, with given **Lagrange multipliers** `\(\boldeta = (\eta_1, \ldots, \eta_n)\)`, is defined as `\begin{equation} J^\boldeta(t,x) := \inf_{\Q\in\Qset} \E^\Q_{t,x}\left[ \int_t^T\int_\R \big(1-(1-\log h(t,y))\;h(t,y) \big)\,\kappa \, G(dy) dt + \sum_{i=1}^n\eta_i \left(f_i(X_T)-c_i\right)\right] , \end{equation}` where `\(\E^\Q_{t,x}[\cdot]\)` denotes the `\(\Q\)`-expectation given that the process `\(X\)` has initial condition `\(X_{t^-}=x\)`. <p style="margin-bottom:1cm;"> </p> Using the **dynamic programming principle**, `\(J^\boldeta(t,x)\)` satisfies the Hamilton-Jacobi-Bellman equation. Applying the first order conditions to obtain the optimal `\(h\)` in feedback form, we obtain `\begin{equation} h^\boldeta(t,x,y) = \exp(J^\boldeta(t,x)-J^\boldeta(t,x+y)), \end{equation}` <p style="margin-bottom:1cm;"> </p> Letting `\(J^\boldeta(t,x)=-\log \omega^\boldeta(t,x)\)`, we find via the **Feynman-Kac representation** that `\begin{equation} \omega^\boldeta(t,x) = \E_{t,x}\left[ \exp\left( - \sum_{i=1}^n\eta_i \left(f_i(X_T)-c_i\right) \right) \right]. \end{equation}` --- ### What does X look like under the stressed measure? <p style="margin-bottom:1.5cm;"> </p> The **intensity** is now `$$\kappa^*(t,x) = \kappa \int_\mathbb{R} h^*(t,x,y) G(dy).$$` The **jump size** is now distributed as `$$G^*(t,x,dy) := \frac{\kappa \, h^*(t,x,dy) G(dy)}{\kappa^* (t,x)} = \frac{h^*(t,x,dy) G(dy)}{\int_\R h^*(t,x,dy') G(dy')}.$$` <p style="margin-bottom:1cm;"> </p> Note: `\(X\)` is no longer a true compound Poisson process due to the time and state dependence of both the intensity and the severity distribution. --- class: inverse, center, middle # Value at Risk --- <p style="margin-bottom:3.5cm;"> </p> The VaR of a random variable `\(Z\)` under measure `\(\Q\)` at level `\(\alpha \in (0,1)\)` is defined as `$$\text{VaR}^\Q_\alpha(Z) = \inf\{z \in \R | F_Z^\Q(z) \geq \alpha \}$$` where `\(F^\Q_Z\)` denotes the distribution function of `\(Z\)` under `\(\Q\)` and we use the convention that `\(\inf \emptyset = + \infty\)`. We implement this using the probability constraint: `\begin{equation} \Q(X_T < q) = \alpha. \end{equation}` i.e. let `\(f(x) = 1_{x < q}\)` and `\(c=\alpha\)`. --- <p style="margin-bottom:2.5cm;"> </p> <div class="proposition"> <h1 style="font-size:0.5vw"> <span style="color:#dedee7">skip</span> </h1> Let \(\alpha \in (0,1)\) and \(\essinf X_T < q < \esssup X_T\). The solution to Optimization Problem 1 with constraint given by \(f(x) = 1_{\{x < q\}}\) and \(c = \alpha\) is the measure \(\Q^*\) characterized by measure-change function \[ h^*(t,x,y) = \frac{1 + \left( e^{-\eta^*} - 1 \right) \P\left(X_T - X_{t-} < q - x - y \right)}{1 + \left( e^{-\eta^*} - 1 \right) \P\left(X_T - X_{t-} < q - x \right)}, \] where the Lagrange multiplier \(\eta^*\) is given by \[ \eta^* = \log \left(\frac{(1-\alpha)\P(X_T < q)}{\alpha \, \P(X_T \geq q)}\right) \, . \] The solution is unique. </div> --- ### VaR example: process intensity under the stressed measure .pull-left-mod[ - Assume `\(X_t\)` is compound Poisson with intensity `\(\kappa=5\)` and jumps distributed as `\(\Gamma(2,1)\)` - Impose a 10% upward stress on VaR. Since `\(\text{VaR}^\P_{0.9}(X_T) = 17.4\)`, this means we set `\(q=19\)` - On the right, we have the intensity `\(\kappa^*(t,x)\)` for a grid of time `\(t \in [0,1]\)` and state space `\(x \in [0,24]\)` ] .pull-right-mod[ <left> <div id="htmlwidget-e941b209781dd8f14bff" style="width:504px;height:504px;" class="plotly html-widget"></div> <script type="application/json" data-for="htmlwidget-e941b209781dd8f14bff">{"x":{"visdat":{"294c564721f8":["function () 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</left> ] --- ### VaR example: severity distribution under the stressed measure <p style="margin-bottom:1cm;"> </p> <center> <img src="data:image/png;base64,#slides_Waterloo_files/figure-html/unnamed-chunk-2-1.png" width="1008" /> </center> --- ### VaR example: sample paths <p style="margin-bottom:1cm;"> </p> .pull-left[ <center> Paths \(X_t, \, t \in {[0,1]} \text{ under } \mathbb{Q}^*\) <p style="margin-bottom:1cm;"> </p> <img src="data:image/png;base64,#slides_Waterloo_files/figure-html/unnamed-chunk-4-1.png" width="504" /> </center> ] .pull-right[ <center> Intensity process \(\kappa^*(t,x)\) <p style="margin-bottom:1cm;"> </p> <img src="data:image/png;base64,#slides_Waterloo_files/figure-html/unnamed-chunk-5-1.png" width="432" /> </center> ] --- class: inverse, center, middle # Aggregate Portfolios and "What-if" Scenarios --- <p style="margin-bottom:3cm;"> </p> - Suppose you have a **portfolio** of claim processes, `\(\boldsymbol{X} = (X^1,\ldots,X^d)\)` and one of them undergoes a stress. - Model is now a `\(d\)`-dimensional compound Poisson process, i.e., under `\(\P\)`, `\(\boldsymbol{X}\)` has mean measure `\begin{equation} \nu(\boldsymbol{x},dt) = \kappa \, G(d\boldsymbol{x}) dt , \end{equation}` where `\(G\)` is the `\(d\)`-dimensional severity distribution and `\(\kappa>0\)` is the scalar intensity. - For simplicity of notation, we stress the first component of `\(\boldsymbol{X}\)`, `\(X^1\)`. - In addition, suppose that instead of imposing constraints at the terminal time `\(T\)`, we impose constraints at an earlier time, `\(T^\dagger \in (0,T]\)`. --- <p style="margin-bottom:3cm;"> </p> <div class="problem" number="2"}> <h1 style="font-size:0.5vw"> <span style="color:#dedee7">skip</span> </h1> Let \(0 < T^\dagger \leq T < \infty\), \(f_i \colon \R \to \R\) and \(c_i \in \R\) for \(i \in [n]\), and consider \[ \inf_{\Q\in\Qset} D_{KL}(\mathbb{Q} \, \Vert \, \mathbb{P} ) \quad \text{s.t.} \quad \E^\Q\left[f_i(X^1_{T^\dagger})\right]=c_i, \quad i \in [n]\,, \] where \(\Qset\) is the class of equivalent probability measures induced by Girsanov's theorem \[ \mathcal{Q}:=\left\{\Q_h\; \Big| \; \frac{d\Q_h}{d\P}=\mathfrak{E}\left(\int_0^T \int_{\R^d} \left[ h_t(\boldsymbol{y}) - 1 \right] \, \tilde \mu(d\boldsymbol{y},dt)\right) \right\} ,\] and \(h: \R_+ \times \R^d \to \R\) is a predictable, non-negative process satisfying Novikov's condition on \([0,T]\). </div> --- <p style="margin-bottom:2cm;"> </p> <div class="theorem"> <h1 style="font-size:0.5vw"> <span style="color:#dedee7">skip</span> </h1> If there exists \(\boldeta^* = (\eta_1^*, \ldots, \eta_n^*) \in \R^n\) such that \(\E[\exp \left( -\sum_{i=1}^n \eta_i^* \, f_i(X^1_{T^\dagger}) \right)] < \infty\) and \[ 0 = \E \left[ \exp \left( -\sum_{j=1}^n\eta^*_j\,f_j(X_{T^\dagger}^1) \right)\left( f_i(X_{T^\dagger}^1)-c_i \right)\right] \quad \text{for } i \in [n],\] then Optimization Problem 2 has a solution. The solution is the measure \(\Q^*\) characterized by the measure-change function \[ h^*(t,\boldsymbol{x},\boldsymbol{y}) = \begin{cases} \frac{\E_{t,\boldsymbol{x}+\boldsymbol{y}}\left[\exp \left( -\sum_{i=1}^n\eta^*_i\,f_i(X^1_{T^\dagger}) \right)\right]}{\E_{t,\boldsymbol{x}}\left[\exp \left( -\sum_{i=1}^n\eta^*_i\,f_i(X^1_{T^\dagger}) \right)\right]} & \text{if} \quad t \leq T^\dagger \\ 1 & \text{if} \quad t > T^\dagger\,.\end{cases} \] The solution is unique. </div> --- ## Stress testing example <p style="margin-bottom:3cm;"> </p> - Suppose we have a bivariate process `\(\boldsymbol{X} = (X^1_t, X^2_t)_{t\in[0,T]}\)` - We consider the outcome of a **5% increase in the VaR of the aggregate portfolio** at the terminal time. We seek the conditions at the midpoint that would cause such an outcome. - In particular, we consider what level of stress `\(X^1\)` would need to undergo at the **midpoint** for there to be a 5% increase in the terminal aggregate portfolio. --- <p style="margin-bottom:3cm;"> </p> <center> Table: Required percentage increase in \(\text{VaR}_\alpha(X^1_{T/2})\) under \(\Q^*\) for various levels of \(\alpha\) <br/> to achieve a 5% increase in \(\text{VaR}_{0.9}(X^1_{T} + X^2_T)\) under \(\Q^*\) compared to under \(\P\). </center> <p style="margin-bottom:1cm;"> </p> <table> <thead> <tr> <th style="text-align:right;"> \(\alpha\) </th> <th style="text-align:right;"> Stress (% increase in \(\text{VaR}_\alpha(X_{T/2}^1\)) </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 0.3 </td> <td style="text-align:right;"> 50.22 </td> </tr> <tr> <td style="text-align:right;"> 0.4 </td> <td style="text-align:right;"> 36.04 </td> </tr> <tr> <td style="text-align:right;"> 0.5 </td> <td style="text-align:right;"> 25.57 </td> </tr> <tr> <td style="text-align:right;"> 0.6 </td> <td style="text-align:right;"> 19.73 </td> </tr> <tr> <td style="text-align:right;"> 0.7 </td> <td style="text-align:right;"> 17.10 </td> </tr> <tr> <td style="text-align:right;"> 0.8 </td> <td style="text-align:right;"> 15.44 </td> </tr> </tbody> </table> <p style="margin-bottom:1cm;"> </p> Parameters: `\(\xi_1 \sim \Gamma(2,1)\)`, `\(\xi_2 \sim \text{Exp}(2)\)`, `\(\kappa=5\)`, `\(t\)`-copula with corr=0.8 and 3 d.f. --- ## Conclusion - We introduce a framework for reverse sensitivity testing with compound Poisson processes, extending existing results with random variables - We explore two risk measure constraints: VaR and Expected Shortfall + VaR - Other possible constraints: mean, mean + variance - What-if scenarios: how big of a stress you would need to exceed a certain terminal risk threshold <p style="margin-bottom:5cm;"> </p> <center> <b> <h1 style="font-size:4vw"> <span style="color:#416aa3">Thank you!</span> </h1> </b> </center> --- ## References .small[ Blanchet, Jose and Karthyek Murthy (2019). “Quantifying Distributional Model Risk via Optimal Transport”. 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