Hedging under generalized good-deal bounds and model uncertainty

Abstract

We study a notion of good-deal hedging, that corresponds to good-deal valuation and is described by a uniform supermartingale property for the tracking errors of hedging strategies. For generalized good-deal constraints, defined in terms of correspondences for the Girsanov kernels of pricing measures, constructive results on good-deal hedges and valuations are derived from backward stochastic differential equations, including new examples with explicit formulas. Under model uncertainty about the market prices of risk of hedging assets, a robust approach leads to a reduction or even elimination of a speculative component in good-deal hedging, which is shown to be equivalent to a global risk minimization in the sense of Föllmer and Sondermann (1986) if uncertainty is sufficiently large.

Appendix

This section includes lemmas and derivations omitted from the paper’s main body.

Lemma 6

For $d< n$, let $\sigma \in {\mathbb {R}}^{d\times n}$ be of full-rank, $A\in {\mathbb {R}}^{n\times n}$ be symmetric and positive definite, and $h>0, Z\in {\mathbb {R}}^n, \xi \in \mathrm {Im}\ \sigma ^{\text {tr}}$. Let $\alpha '>0$ be a constant of ellipticity of $A^{-1}$ and assume that $\left| {\xi } \right| <h\sqrt{\alpha '}$ and $A^{-1}(\mathrm {Ker}\ \sigma ) = \mathrm {Ker}\ \sigma$. Then ${\bar{\phi }}:= {\varPi }(Z)+\left( {\varPi }^\bot (Z)^{\text {tr}}A^{-1}{\varPi }^\bot (Z)\right) ^{1/2}\left( h^2-\xi ^{\text {tr}}A\xi \right) ^{-1/2}A\xi$ is the unique minimizer of the function $\phi \mapsto F(\phi ):=-\xi ^*\phi +h\left( \left( Z-\phi \right) ^{\text {tr}}A^{-1}\left( Z-\phi \right) \right) ^{1/2}$ on $\mathrm {Im}\ \sigma ^{\text {tr}}$.

Proof

The proof is an application of the classical Kuhn–Tucker Theorem (cf. Rockafellar 1970, Sect. 28). For details see Kentia (2015, Lemma 3.35) and proof. $\square$

Lemma 7

Let $d< n, h>0$ be constant, $Z\in {\mathbb {R}}^n, A\in {\mathbb {R}}^{n\times n}$ a symmetric positive definite matrix, $\sigma \in {\mathbb {R}}^{d\times n}$ a full (d)-rank matrix, and $\xi ^0\in {\varPhi } := \text {Im}\ \sigma ^{\text {tr}}$. Let ${\varTheta }\subset {\mathbb {R}}^n$ be a convex-compact set, and $F: {\mathbb {R}}^n\times {\mathbb {R}}^n\ni (\phi ,\theta )\mapsto \theta ^\text {tr}(Z-\phi )-{\xi ^0}^\text {tr}\phi + h\left( (Z-\phi )^\text {tr}A^{-1}(Z-\phi )\right) ^{1/2}$. Then the minmax identity $\inf _{\phi \in {\varPhi }}\ \sup _{\theta \in {\varTheta }}F(\phi ,\theta ) = \sup _{\theta \in {\varTheta }}\ \inf _{\phi \in {\varPhi }}F(\phi ,\theta ).$ holds.

Proof

For all $\phi \in {\mathbb {R}}^n$, the function $\theta \mapsto F(\phi ,\theta )$ is concave, continuous. For all $\theta \in {\mathbb {R}}^n$ the function $\phi \mapsto F(\phi ,\theta )$ is convex and continuous. As ${\varTheta }\subset {\mathbb {R}}^n$ is convex and compact, and ${\varPhi } = \text {Im}\ \sigma ^{\text {tr}}$ is convex and closed, a minimax theorem (Ekeland and Temam 1999, Ch. VI, Prop. 2.3) applies and the minmax identity holds. $\square$

Proof (of Lemma 1)

Part (a) is classical (see Delbaen 2006 and cf. previously given other references). As for Part (b), m-stability and convexity of ${\mathcal {M}}^e$ follow from Delbaen (2006, Prop. 5). Convexity of ${\mathcal {Q}}^{\mathrm {ngd}}$ follows from that of ${\mathcal {M}}^e$ and the values of C. To show m-stability of ${\mathcal {Q}}^{\mathrm {ngd}}$, let $Z^i={\mathcal {E}}(\lambda ^i\cdot W)\in {\mathcal {Q}}^{\mathrm {ngd}},\ i=1,2, \tau \le T$ be a stopping time and $Z=I_{[0,\tau ]}Z^1_\cdot + I_{]\tau ,T]}Z^1_\tau Z^2_\cdot /Z^2_\tau$. Since ${\mathcal {M}}^e$ is m-stable, then $Z\in {\mathcal {M}}^e$ and one has $Z={\mathcal {E}}(\lambda \cdot W)$ for some predictable process $\lambda$. It remains to show that $\lambda$ is bounded and that $\lambda \in C$. From the expression of Z, writing the densities $Z,Z^1,Z^2$ as ordinary exponentials by distinguishing $t\le \tau$ and $t\ge \tau$, and taking the logarithm yields $(\lambda -I_{[0,\tau ]}\lambda ^1-I_{]\tau ,T]}\lambda ^2)\cdot W= \frac{1}{2}\int _0^\cdot \left( \left| {\lambda _s} \right| ^2-I_{[0,\tau ]}(s)\left| {\lambda ^1_s} \right| ^2-I_{]\tau ,T]}(s)\left| {\lambda ^2_s} \right| ^2\right) ds.$ Since ${\mathbb {F}}$ is the augmented Brownian filtration, then $[0,\tau ]$ and $]\tau ,T]$ are predictable and so is $\lambda -I_{[0,\tau ]}\lambda ^1-I_{]\tau ,T]}\lambda ^2$. Hence $\left( \lambda -I_{[0,\tau ]}\lambda ^1-I_{]\tau ,T]}\lambda ^2\right) \cdot W$ is a continuous local martingale of finite variation and is thus equal to zero. As a consequence $\lambda =I_{[0,\tau ]}\lambda ^1+I_{]\tau ,T]}\lambda ^2$ is bounded since $\lambda ^1,\lambda ^2$ are, and satisfies $\lambda \in C$ since C is convex-valued. $\square$

Proof (of Theorem 1)

As shown in Kentia (2015, Thm. 3.7), $\pi ^u_\cdot (X)$ admits under ${\widehat{Q}}$ the Doob–Meyer decomposition $\pi ^{u}_\cdot (X) = \pi ^{u}_0(X)+Z\cdot {\widehat{W}} - A=\pi ^{u}_0(X)+Z\cdot W + \int _0^\cdot \xi _t^{\text {tr}}Z_tdt- A,$ where $Z\in {\mathcal {H}}^2({\widehat{Q}})$ and A is a non-decreasing predictable process with $A_0=0$. Alternatively one rewrites $\displaystyle -d\pi ^u_t(X) =g_t(Z_t)dt - Z_t^{\text {tr}}dW_t+dK_t,$ with $K:= A -\int _0^\cdot \xi _t^{\text {tr}}Z_tdt - \int _0^\cdot \mathop {\text{ ess } \text{ sup }}_{\lambda \in {\varLambda }}\lambda _t^{\text {tr}}Z_t\ dt$ being finite-valued and predictable. For $(\pi ^u_\cdot (X),Z,K)$ to be a supersolution to the BSDE with parameters (g, X) it suffices to show that K is non-decreasing. For any $\lambda =-\xi +\eta \in {\varLambda }$, one can construct the sequence of $\lambda ^n=-\xi +\eta ^n\in {\varLambda }$ Girsanov kernels of measures $Q^n\in {\mathcal {Q}}^{\mathrm {ngd}}$ with $\eta ^n=\eta I_{\{|\eta |\le n\}}$ such that $\lambda ^n\rightarrow \lambda \ P\otimes dt\text {-a.e.}$ as $n\rightarrow \infty$. For each $Q^n$ it holds $\pi ^{u}_\cdot (X) = \pi ^{u}_0(X)+Z\cdot W^{Q^n} + \int _0^\cdot {Z_t}^{\text {tr}}\eta ^n_tdt- A.$ Since $\pi ^u_\cdot (X)$ is a bounded $Q^n$-supermartingale, then $dA_t - \xi _t^{\text {tr}}Z_tdt \ge Z_t^{\text {tr}}\lambda ^n_tdt,\ \text {for all } n\in {\mathbb {N}}.$ Taking the limit as $n\rightarrow \infty$ and using dominated convergence one obtains $dA_t - \xi _t^{\text {tr}}Z_tdt \ge Z_t^{\text {tr}}\lambda _tdt.$ Now taking the essential supremum over all $\lambda \in {\varLambda }$ yields $dK_t\ge 0$.

To show that the supersolution $(\pi ^u_\cdot (X),Z,K)$ is minimal, it suffices by the dynamic principle (cf. Kentia 2015, Lem. 3.6) to show that the Y-component of any other supersolution is a càdlàg Q-supermartingale for any $Q\in {\mathcal {Q}}^{\mathrm {ngd}}$. Let $({\bar{Y}},{\bar{Z}},{\bar{K}})$ be a supersolution of the BSDE for parameters (g, X), with ${\bar{Y}}\in {\mathcal {S}}^\infty$. By change of measure, under a $Q\in {\mathcal {Q}}^{\mathrm {ngd}}$ with Girsanov kernel $\lambda ^Q\in {\varLambda }$ we have

$$\begin{aligned}&-d{\bar{Y}}_t =\left( \mathop {\text{ ess } \text{ sup }}_{\lambda \in {\varLambda }}\lambda _t^{\text {tr}}{\bar{Z}}_t - {\bar{Z}}_t^{\text {tr}}\lambda ^Q_t\right) dt - {\bar{Z}}_t^{\text {tr}}dW^Q_t+d{\bar{K}}_t,\quad t\in [0,T], \end{aligned}$$

(68)

$$\begin{aligned}&\text {and get} \quad d{\bar{K}}_t+ \left( \mathop {\text{ ess } \text{ sup }}_{\lambda \in {\varLambda }}\lambda _t^{\text {tr}}{\bar{Z}}_t - {\bar{Z}}_t^{\text {tr}}\lambda ^Q_t\right) dt\ge 0, \quad t\in [0,T], \end{aligned}$$

(69)

by using that ${\bar{K}}$ is non-decreasing. By (68–69) and ${\bar{Y}}\in {\mathcal {S}}^\infty$, the local martingale ${\bar{Z}}\cdot W^Q$ is bounded from below, and thus is a supermartingale. As ${\bar{Y}}\in {\mathcal {S}}^\infty$, the integral of (69) on [0, T] is in $L^1(Q)$ and so ${\bar{Y}}$ is a Q-supermartingale. $\square$

Proof (Derivation of (28), (30))

The stochastic exponential ${\mathcal {E}}\left( (\varepsilon /\sqrt{\nu })\cdot W^{\nu }\right)$ is a uniformly integrable martingale which defines a measure ${\bar{Q}}\in \overline{{\mathcal {Q}}^{\text {ngd}}}\supseteq {\mathcal {Q}}^{\text {ngd}}$ (see (4) for definition of $\overline{{\mathcal {Q}}^{\text {ngd}}}\subset {\mathcal {M}}^e$) with Girsanov kernel ${\bar{\lambda }}:=\varepsilon /\sqrt{\nu }$, i.e. $d{\bar{Q}}/dP = {\mathcal {E}}((\varepsilon /\sqrt{\nu })\cdot W^{\nu })$. Indeed, applying Cheridito et al. (2005, Thm. 2.4 and Sect. 6) one gets that ${\mathcal {E}}\left( (\varepsilon /\sqrt{\nu })\cdot W^{\nu }\right)$ and $S=S_0{\mathcal {E}}\left( \sqrt{\nu }\cdot W^S\right)$ are uniformly integrable P- respectively Q-martingales. The variance process $\nu$ under ${\bar{Q}}$ is again a CIR process with parameters $({\bar{a}},b,\beta ,\rho )$ where ${\bar{a}}:=a+\beta \varepsilon \sqrt{1-\rho ^2}>a$ and the Feller condition $\beta ^2\le 2{\bar{a}}$ still holds. For a put option $X=(K-S_T)^+ \in L^\infty , {\bar{Y}}_t:=E^{{\bar{Q}}}_t[X]$ are given by the Heston formula (cf. Heston 1993), applied under ${\bar{Q}}$ (instead of P). Since the Heston price is non-decreasing in the mean reversion level of the variance process (Ould-Aly 2011, Prop. 5.3.1) one expects that $\pi ^u_t\left( X\right) ={\bar{Y}}_t= E^{{\bar{Q}}}_t[X]$. Let us make this precise. For $Q\in \overline{{\mathcal {Q}}^{\mathrm {ngd}}}$ with Girsanov kernel $\lambda$ satisfying $\left| {\lambda } \right| \le \varepsilon /\sqrt{\nu }$, one has $Y^Q_T={\bar{Y}}_T=X$ with $Y^Q_t=E^Q_t[X]$. Using Feynman–Kac, ${\bar{Y}}_t=u(t,S_t,\nu _t)$ for a function $u\in {\mathcal {C}}^{1,2,2}([0,T]\times {\mathbb {R}}^+\times {\mathbb {R}}^+)$ with $\frac{\partial u}{\partial \nu }\ge 0$ (see Ould-Aly 2011, Thm. 5.3.1, Cor. 5.3.1). By Itô’s formula and change of measure follows

$$\begin{aligned} d{\bar{Y}}_t&= \beta \sqrt{1-\rho ^2}\sqrt{\nu _t}\left( \lambda _t-\frac{\varepsilon }{\sqrt{\nu _t}}\right) \frac{\partial u}{\partial \nu }(t,S_t,\nu _t)dt\nonumber \\ {}&\quad +\beta \sqrt{1-\rho ^2}\sqrt{\nu _t}\frac{\partial u}{\partial \nu }(t,S_t,\nu _t)dW^{Q,\nu }_t\nonumber \\&\quad +\left( S_t\sqrt{\nu _t}\frac{\partial u}{\partial S}(t,S_t,\nu _t)+\beta \rho \sqrt{\nu _t}\frac{\partial u}{\partial \nu }(t,S_t,\nu _t)\right) dW^{S}_t,\quad t\in [0,T]. \end{aligned}$$

(70)

Since X is bounded, then ${\bar{Y}}$ is in ${\mathcal {S}}^\infty (Q)$ and a Q-supermartingale by (70) . Hence $Y^Q_t \le {\bar{Y}}_t$ for all $Q\in \overline{{\mathcal {Q}}^{\text {ngd}}}$, which by Part 1 of Kentia (2015, Thm. 3.7) implies the claim and thus we obtain the Heston type formula (28).

Since ${\bar{Q}}\in \overline{{\mathcal {Q}}^{\text {ngd}}}$ and $\pi ^u_0(X)=E^{{\bar{Q}}}[X]$ with $X\in L^\infty$, Corollary 1 implies that the good-deal bound is the Y-component of the minimal solution $({\bar{Y}},{\bar{Z}})\in {\mathcal {S}}^\infty \times {\mathcal {H}}^2$ (note $P={\widehat{Q}}$) of the BSDE (29) with generator $g_t(z)={\bar{\lambda }}_tz^2=\varepsilon z^2/\sqrt{\nu _t},$ for $z=(z^1,z^2)$, and terminal condition X. Now consider the strategy

$$\begin{aligned} {\bar{\phi }}_t={\bar{Z}}^1_t=S_t\sqrt{\nu _t}\frac{\partial u}{\partial S}(t,S_t,\nu _t)+\beta \rho \sqrt{\nu _t}\frac{\partial u}{\partial \nu }(t,S_t,\nu _t)=S_t\sqrt{\nu _t}\ {\varDelta }_t+\frac{\beta \rho }{2}{\mathcal {V}}_t. \end{aligned}$$

Clearly ${\bar{\phi }}$ is in the set ${\varPhi } = {\mathcal {H}}^2({\mathbb {R}})$ of permitted trading strategies since ${\bar{Z}}\in {\mathcal {H}}^2 ({\mathbb {R}}^2)$. Recall that ${\mathcal {P}}^{\text {ngd}}$ consists of $dQ/dP={\mathcal {E}}\left( (\lambda ^S,\lambda ^\nu )\cdot W\right)$ such that $\big |(\lambda ^S,\lambda ^\nu )\big |\le \varepsilon /\sqrt{\nu }$ with $(\lambda ^S,\lambda ^\nu )$ being bounded. For $Q\in {\mathcal {P}}^{\text {ngd}}$, any wealth process $\phi \cdot W^S, \phi \in {\varPhi }$, is thus in ${\mathcal {S}}^1(Q)$. As ${\mathcal {Q}}^{\text {ngd}}\subseteq {\mathcal {P}}^{\text {ngd}}$ holds, clearly $\pi ^u_t(X) \le \rho _t(X-\int _t^T\phi _sdW^S_s)$ for any strategy $\phi \in {\varPhi }$. To prove that ${\bar{\phi }}$ is a good-deal hedging strategy, we show the reverse inequality $\pi ^u_t(X) \ge E^Q_t\left[ X-\int _t^T{\bar{\phi }}_sdW^S_s\right]$ for all $Q\in {\mathcal {P}}^{\text {ngd}}$. Let $Q\in {\mathcal {P}}^{\text {ngd}}$ with Girsanov kernel $(\lambda ^S,\lambda ^\nu )$. Like in (70), we obtain for any stopping time $\tau$ that

$$\begin{aligned} {\bar{Y}}_{\tau \wedge T}- \int _{\tau \wedge t}^{\tau \wedge T}{\bar{\phi }}_sdW^S_s&= {\bar{Y}}_{\tau \wedge t} + \int _{\tau \wedge t}^{\tau \wedge T}\beta \sqrt{1-\rho ^2}\sqrt{\nu _s}\left( \lambda ^\nu _s-\frac{\varepsilon }{\sqrt{\nu _s}}\right) \frac{\partial u}{\partial \nu }(s,S_s,\nu _s)ds\nonumber \\&\quad + L_{\tau \wedge T}-L_{\tau \wedge t}, \end{aligned}$$

(71)

for the local Q-martingale $L := \int _0^\cdot \beta \sqrt{1-\rho ^2}\sqrt{\nu _s}\frac{\partial u}{\partial \nu }(s,S_s,\nu _s)dW^{Q,\nu }_s$. By $\frac{\partial u}{\partial \nu }\ge 0$ and $\big |\lambda ^\nu \big |\le \varepsilon /\sqrt{\nu }$ follows that ${\bar{Y}}_{\tau \wedge T}- \int _{\tau \wedge t}^{\tau \wedge T}{\bar{\phi }}_sdW^S_s$ is less than ${\bar{Y}}_{\tau \wedge t} + L_{\tau \wedge T}-L_{\tau \wedge t}$. Localizing L along a sequence of stopping times $\tau _n\uparrow \infty$ and taking conditional Q-expectations yields $E^Q_t[{\bar{Y}}_{\tau _n\wedge T}- \int _{\tau _n\wedge t}^{\tau _n\wedge T}{\bar{\phi }}_sdW^S_s] \le {\bar{Y}}_{\tau _n\wedge t}$. Using $X\in L^\infty$ and ${\bar{\phi }}\cdot W^S\in S^1(Q)$, the claim then follows by dominated convergence. Hence (30) holds for ${\mathcal {V}}_t:=\frac{\partial u}{\partial \sigma }(t,S_t,\nu _t) = 2\sigma _t\frac{\partial u}{\partial \nu }(t,S_t,\nu _t)$ and volatility $\sigma _t=\sqrt{\nu _t}$. $\square$