Reverse sensitivity testing with compound Poisson processesEmma Kroell1 
 Joint work with Silvana Pesenti1 and Sebastian Jaimungal11 Department of Statistical Sciences 
 University of Toronto1 / 25

Motivation

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.

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Literature review

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), $χ^{2}$ 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)

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Mathematical Preliminaries4 / 25

KL Divergence

The KL divergence of $Q$ with respect to $P$ (also known as the relative entropy) is defined as $D_{K L} (Q ‖ P) = {\begin{cases} E [\frac{d Q}{d P} \log (\frac{d Q}{d P})] & if Q ≪ P \\ \infty & otherwise, \end{cases}$ where we use the convention $0 \log 0 = 0$ and $\frac{d Q}{d P}$ denotes the Radon-Nikodym (RN) derivative of $Q$ with respect to $P$ .

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Model

We take as given a filtered probability space $(Ω, P, F, {F_{t}}_{t \in [0, T]})$
We consider jump processes of the form $d X_{t} = \int_{R} x μ (d x, d t)$ where $μ$ is a Poisson random measure
Assume $X$ is compound Poisson; the mean measure is of the form $ν (d x, d t) = κ G (d x) d t .$

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Main Problem7 / 25

Minimally perturbed process under stress

We impose the stress on the process $X_{t}$ , $t \in [0, T]$ at the terminal time $T$ .

skip

Let

f_{i} : R \to R

and

c_{i} \in R

for

i \in [n]

, where

[n] := {1, \dots, n}

, and consider

inf_{Q \in Q} D_{K L} (Q ‖ P) s.t. E^{Q} [f_{i} (X_{T})] = c_{i}, i \in [n],

where

Q

is the class of equivalent probability measures given by

Q := {Q_{h} | \frac{d Q_{h}}{d P} = E (\int_{0}^{T} \int_{R} [h_{t} (y) - 1] \tilde{μ} (d y, d t))},

where

E (\cdot)

denotes the stochastic exponential,

\tilde{μ} (d y, d t) := μ (d y, d t) - ν (d y, d t)

the compensated measure, and

h_{t}

is a predictable, non-negative random fields satisfying

E [\exp (\int_{0}^{T} \int_{R} (1 - h_{t} (y))^{2} μ (d y, d t))] < \infty .

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skip

If there exists

η^{*} = (η_{1}^{*}, \dots, η_{n}^{*}) \in R^{n}

such that

E [\exp (- \sum_{i = 1}^{n} η_{i}^{*} f_{i} (X_{T}))] < \infty

and

0 = E [\exp (- \sum_{j = 1}^{n} η_{j}^{*} f_{j} (X_{T})) (f_{i} (X_{T}) - c_{i})], i = 1, \dots, 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} [\exp (- \sum_{i = 1}^{n} η_{i}^{*} f_{i} (X_{T}))]}{E_{t, x} [\exp (- \sum_{i = 1}^{n} η_{i}^{*} f_{i} (X_{T}))]},

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 (- \sum_{i = 1}^{n} η_{i}^{*} f_{i} (X_{T}))}{E [\exp (- \sum_{i = 1}^{n} η_{i}^{*} f_{i} (X_{T}))]} .

The solution is unique.

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Sketch of proof

The time $t$ version of the associated value function, with given Lagrange multipliers $η = (η_{1}, \dots, η_{n})$ , is defined as $J^{η} (t, x) := inf_{Q \in Q} E_{t, x}^{Q} [\int_{t}^{T} \int_{R} (1 - (1 - \log h (t, y)) h (t, y)) κ G (d y) d t + \sum_{i = 1}^{n} η_{i} (f_{i} (X_{T}) - c_{i})],$ where $E_{t, x}^{Q} [\cdot]$ denotes the $Q$ -expectation given that the process $X$ has initial condition $X_{t^{-}} = x$ .

Using the dynamic programming principle, $J^{η} (t, x)$ satisfies the Hamilton-Jacobi-Bellman equation.

Applying the first order conditions to obtain the optimal $h$ in feedback form, we obtain $h^{η} (t, x, y) = \exp (J^{η} (t, x) - J^{η} (t, x + y)),$

Letting $J^{η} (t, x) = - \log ω^{η} (t, x)$ , we find via the Feynman-Kac representation that $ω^{η} (t, x) = E_{t, x} [\exp (- \sum_{i = 1}^{n} η_{i} (f_{i} (X_{T}) - c_{i}))] .$

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What does X look like under the stressed measure?

The intensity is now $κ^{*} (t, x) = κ \int_{R} h^{*} (t, x, y) G (d y) .$

The jump size is now distributed as $G^{*} (t, x, d y) := \frac{κ h^{*} (t, x, d y) G (d y)}{κ^{*} (t, x)} = \frac{h^{*} (t, x, d y) G (d y)}{\int_{R} h^{*} (t, x, d y^{'}) G (d y^{'})} .$

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.

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Value at Risk12 / 25

The VaR of a random variable $Z$ under measure $Q$ at level $α \in (0, 1)$ is defined as ${VaR}_{α}^{Q} (Z) = inf {z \in R | F_{Z}^{Q} (z) \geq α}$ where $F_{Z}^{Q}$ denotes the distribution function of $Z$ under $Q$ and we use the convention that $inf \emptyset = + \infty$ .

We implement this using the probability constraint: $Q (X_{T} < q) = α .$ i.e. let $f (x) = 1_{x < q}$ and $c = α$ .

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skip

Let

α \in (0, 1)

and

e s s i n f X_{T} < q < e s s s u p X_{T}

. The solution to Optimization Problem 1 with constraint given by

f (x) = 1_{{x < q}}

and

c = α

is the measure

Q^{*}

characterized by measure-change function

h^{*} (t, x, y) = \frac{1 + (e^{- η^{*}} - 1) P (X_{T} - X_{t -} < q - x - y)}{1 + (e^{- η^{*}} - 1) P (X_{T} - X_{t -} < q - x)},

where the Lagrange multiplier

η^{*}

is given by

η^{*} = \log (\frac{(1 - α) P (X_{T} < q)}{α P (X_{T} \geq q)}) .

The solution is unique.

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VaR example: process intensity under the stressed measure

Assume $X_{t}$ is compound Poisson with intensity $κ = 5$ and jumps distributed as $Γ (2, 1)$
Impose a 10% upward stress on VaR. Since ${VaR}_{0.9}^{P} (X_{T}) = 17.4$ , this means we set $q = 19$
On the right, we have the intensity $κ^{*} (t, x)$ for a grid of time $t \in [0, 1]$ and state space $x \in [0, 24]$

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VaR example: severity distribution under the stressed measure

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VaR example: sample paths

Paths

X_{t}, t \in [0, 1] under Q^{*}

Intensity process

κ^{*} (t, x)

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Aggregate Portfolios and "What-if" Scenarios18 / 25

Suppose you have a portfolio of claim processes, $X = (X^{1}, \dots, X^{d})$ and one of them undergoes a stress.
Model is now a $d$ -dimensional compound Poisson process, i.e., under $P$ , $X$ has mean measure $ν (x, d t) = κ G (d x) d t,$ where $G$ is the $d$ -dimensional severity distribution and $κ > 0$ is the scalar intensity.
For simplicity of notation, we stress the first component of $X$ , $X^{1}$ .
In addition, suppose that instead of imposing constraints at the terminal time $T$ , we impose constraints at an earlier time, $T^{†} \in (0, T]$ .

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skip

Let

0 < T^{†} \leq T < \infty

f_{i} : R \to R

and

c_{i} \in R

for

i \in [n]

, and consider

inf_{Q \in Q} D_{K L} (Q ‖ P) s.t. E^{Q} [f_{i} (X_{T^{†}}^{1})] = c_{i}, i \in [n],

where

Q

is the class of equivalent probability measures induced by Girsanov's theorem

Q := {Q_{h} | \frac{d Q_{h}}{d P} = E (\int_{0}^{T} \int_{R^{d}} [h_{t} (y) - 1] \tilde{μ} (d y, d t))},

and

h : R_{+} \times R^{d} \to R

is a predictable, non-negative process satisfying Novikov's condition on

[0, T]

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skip

If there exists

η^{*} = (η_{1}^{*}, \dots, η_{n}^{*}) \in R^{n}

such that

E [\exp (- \sum_{i = 1}^{n} η_{i}^{*} f_{i} (X_{T^{†}}^{1}))] < \infty

and

0 = E [\exp (- \sum_{j = 1}^{n} η_{j}^{*} f_{j} (X_{T^{†}}^{1})) (f_{i} (X_{T^{†}}^{1}) - c_{i})] 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, x, y) = {\begin{cases} \frac{E_{t, x + y} [\exp (- \sum_{i = 1}^{n} η_{i}^{*} f_{i} (X_{T^{†}}^{1}))]}{E_{t, x} [\exp (- \sum_{i = 1}^{n} η_{i}^{*} f_{i} (X_{T^{†}}^{1}))]} & if t \leq T^{†} \\ 1 & if t > T^{†} . \end{cases}

The solution is unique.

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Stress testing example

Suppose we have a bivariate process $X = (X_{t}^{1}, X_{t}^{2})_{t \in [0, T]}$
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.

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Table: Required percentage increase in

{VaR}_{α} (X_{T / 2}^{1})

under

Q^{*}

for various levels of

α

to achieve a 5% increase in

{VaR}_{0.9} (X_{T}^{1} + X_{T}^{2})

under

Q^{*}

compared to under

P

$α$	Stress (% increase in ${VaR}_{α} (X_{T / 2}^{1}$ )
0.3	50.22
0.4	36.04
0.5	25.57
0.6	19.73
0.7	17.10
0.8	15.44

Parameters: $ξ_{1} \sim Γ (2, 1)$ , $ξ_{2} \sim Exp (2)$ , $κ = 5$ , $t$ -copula with corr=0.8 and 3 d.f.

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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

Thank you!

24 / 25

References

Blanchet, Jose and Karthyek Murthy (2019). “Quantifying Distributional Model Risk via Optimal Transport”. In: Mathematics of Operations Research 44.2, pp. 565–600

Breuer, Thomas and Imre Csiszar (2013). “Systematic stress tests with entropic plausibility constraints”. In: Journal of Banking & Finance 37.5, pp. 1552–1559.

Cambou, Mathieu and Damir Filipovic (2017). “Model Uncertainty and Scenario Aggregation”. In: Mathematical Finance 27.2, pp. 534-567.

Glasserman, Paul and Xingbo Xu (2014). “Robust risk measurement and model risk”. In: Quantitative Finance 14.1, pp. 29–58.

Hofert, Marius et al. (2018). Elements of copula modeling with R. Springer.

Jackson, Kenneth R, Sebastian Jaimungal, and Vladimir Surkov (2008). “Fourier space time-stepping for option pricing with Lévy models”. In: Journal of Computational Finance 12.2, p. 1.

Kullback, Solomon and Richard A Leibler (1951). “On information and sufficiency”. In: The Annals of Mathematical Statistics 22.1, pp. 79–86.

Makam, Vaishno Devi, Pietro Millossovich, and Andreas Tsanakas (2021). “Sensitivity analysis with χ2-divergences”. In: Insurance: Mathematics and Economics 100, pp. 372–383.

Pesenti, Silvana M, Pietro Millossovich, and Andreas Tsanakas (2019). “Reverse sensitivity testing: What does it take to break the model?” In: European Journal of Operational Research 274.2, pp. 654–670.

Pesenti, Silvana M (2021). “Reverse Sensitivity Analysis for Risk Modelling” Available at SSRN:http://dx.doi.org/10.2139/ssrn.3878879.

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Motivation

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.

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Reverse sensitivity testing with compound Poisson processes

Emma Kroell¹
Joint work with Silvana Pesenti¹ and Sebastian Jaimungal¹

¹ Department of Statistical Sciences
University of Toronto

Motivation

Literature review

Mathematical Preliminaries

KL Divergence

Model

Main Problem

Minimally perturbed process under stress

skip

skip

Sketch of proof

What does X look like under the stressed measure?

Value at Risk

skip

VaR example: process intensity under the stressed measure

VaR example: severity distribution under the stressed measure

VaR example: sample paths

Aggregate Portfolios and "What-if" Scenarios

skip

skip

Stress testing example

Conclusion

Thank you!

References

Motivation

Help

Reverse sensitivity testing with compound Poisson processes

Emma Kroell1 Joint work with Silvana Pesenti1 and Sebastian Jaimungal1

1 Department of Statistical Sciences University of Toronto

Motivation

Literature review

Mathematical Preliminaries

KL Divergence

Model

Main Problem

Minimally perturbed process under stress

skip

skip

Sketch of proof

What does X look like under the stressed measure?

Value at Risk

skip

VaR example: process intensity under the stressed measure

VaR example: severity distribution under the stressed measure

VaR example: sample paths

Aggregate Portfolios and "What-if" Scenarios

skip

skip

Stress testing example

Conclusion

Thank you!

References

Motivation

Help

Emma Kroell¹
Joint work with Silvana Pesenti¹ and Sebastian Jaimungal¹

¹ Department of Statistical Sciences
University of Toronto