Measuring Financial Resilience Using Backward Stochastic Differential Equations

TL;DR

The paper introduces the resilience rate, a dynamic-risk metric defined as $mdot ho_ au(X)= obreak\lim_{ obreak\varepsilon o0^+}rac{1}{ obreak obreak \varepsilon} obreak\mathbb{E}igl[ ho_{( au+ obreak \varepsilon)igr extrm{ ext{}}} obreak- ho_ au(X)ig| au<Tigr]$, capturing the instantaneous expected recovery of a dynamic risk measure after a breach, and showing that it can be represented as the conditional expectation of the BSDE generator $g$ at $ au$. The authors develop a stochastic-calculus framework for BSDEs with jumps, establish representation theorems for the time-derivative of the first BSDE component, and connect resilience to properties of the risk measure and its driver. They introduce resilience-acceptance sets, prove several structural properties (time-consistency, cash-insensitivity, monotonicity, continuity, convexity relationships), and provide canonical financial examples (put options, Vasicek bond prices, entropic risk) illustrating computation and interpretation. A key contribution is the notion of resilience neutrality and the resilience risk adjustment (RRA), which model policy adjustments to enforce desired short-term resilience and quantify the cost of such adjustments. The framework offers a principled way to evaluate and regulate financial resilience by linking local risk preferences to the dynamics of risk evaluation via BSDEs with jumps.

Abstract

We introduce the resilience rate as a measure of financial resilience. It captures the expected rate at which a dynamic risk measure recovers, i.e., bounces back, when the risk-acceptance set is breached. We develop the corresponding stochastic calculus by establishing representation theorems for expected time-derivatives of solutions to backward stochastic differential equations (BSDEs) with jumps, evaluated at stopping times. These results reveal that the resilience rate can be represented as a suitable expectation of the generator of a BSDE. We analyze the main properties of the resilience rate and the formal connection of these properties to the BSDE generator. We also introduce resilience-acceptance sets and study their properties in relation to both the resilience rate and the dynamic risk measure. We illustrate our results in several canonical financial examples and highlight their implications via the notion of resilience neutrality.

Figures (3)

• Figure 1: For a time interval of one year ($1.0\, y$), we simulate $10^6$ random trajectories of the asset price $S$, which follows the Black and Scholes model with initial value $S_0 = 1.0 k\, \text{\euro}$, deterministic drift ${\mu=0.10\, y^{-1}}$, deterministic volatility $\sigma=0.10\, y^{-1/2}$, and zero risk-free interest rate. The time step is set to ${\,\mathrm{d} t=y/252}$. Based on these simulations, we use the Black and Scholes formula to calculate the value of the replicating portfolio $V(X)$ for a put option with payoff $X=(K-S_T)^+$, maturity $T=1.0 \, y$ and strike price $K=1.0 k\, \text{\euro}$. The colored solid lines in the graph are a selection of put option price trajectories. The black dotted line depicts the expected time evolution of the put option price, namely $[0,T]\ni t\mapsto \mathbb{E}[V_t(X)]$, numerically computed by averaging over the $10^6$ trajectories of $V(X)$. The red dashed line is the time evolution of the resilience rate, i.e., $(0,T)\ni t\mapsto \bm\dot V_t(X)$, see equation \ref{['EQ:resilience_put_t']}. Here, the expectation was computed via numerical integration with respect to the probability distribution of $S_t$. Moreover, using equation \ref{['EQ:resilience_put']}, we computed the resilience rate $\bm\dot V_\tau(X)\approx-78\, \text{\euro}y^{-1}$ at the stopping time $\tau$ defined in equation \ref{['EQ:stopping_time']}, with $c=80\, \text{\euro}$, where the expectation was numerically estimated by averaging over the $10^6$ trajectories. In the graph, the put option price trajectories that breached the $80\, \text{\euro}$ barrier were truncated. At each truncation point, the resilience rate $\bm\dot V_\tau(X)$ is represented as the slope of an incident line.
• Figure 2: For a time interval of one year ($1.0\, y$), we simulate $10^6$ random trajectories for a risk-free interest rate $r$ that follows a standard Vasicek model with speed of mean reversion $a=1.0 \,y^{-1}$, long-term mean level $b=2.0\% \,y^{-1}$, initial value $r_0 = 2.0\% \,y^{-1}$ and instantaneous volatility ${\sigma} = 1.0\% \, y^{-3/2}$. The time step is set to ${\,\mathrm{d} t=y/252}$. Based on these simulations, we compute the price $P$ of a zero-coupon bond available in the market. The colored solid lines in the graph are a selection of bond price trajectories. The black dotted line depicts the expected time evolution of the bond price, namely $[0,T]\ni t\mapsto \mathbb{E}[P_t]$, numerically computed by averaging over the $10^6$ trajectories of $P$. The red dashed line is the time evolution of the resilience rate, i.e., $(0,T)\ni t\mapsto \bm\dot P_t$, see equation \ref{['EQ:vasicek_dot_Pt']}. Using equation \ref{['EQ:vasicek_resilience_rate']}, we computed the resilience rate $\bm\dot P_\tau\approx0.050\, \text{\euro}y^{-1}$ at the stopping time $\tau:=T\wedge \inf\{t\in[0,T] \ :\ r_t\geq 5.0\%\,y^{-1}\}$, where the expectation was numerically estimated by averaging over the $10^6$ trajectories. In the graph, the bond price trajectories for which the underlying risk-free interest rate breached the $5.0\%\,y^{-1}$ barrier, were truncated at the breaching time. At each truncation point, the resilience rate $\bm\dot P_\tau$ is represented as the slope of an incident line.
• Figure 3: Dependence of the resilience rate on the speed of mean reversion in Example \ref{['EX:vasicek']}.

Theorems & Definitions (55)

• Definition 1: Dynamic risk measure
• Definition 2
• Remark 3
• Remark 4
• Definition 5: Resilience rate
• Example 6: VaR and ES
• Remark 7
• Remark 8
• Proposition 9
• proof