Hedging with small uncertainty aversion

Abstract

We study the pricing and hedging of derivative securities with uncertainty about the volatility of the underlying asset. Rather than taking all models from a prespecified class equally seriously, we penalise less plausible ones based on their “distance” to a reference local volatility model. In the limit for small uncertainty aversion, this leads to explicit formulas for prices and hedging strategies in terms of the security’s cash gamma.

Appendices

Appendix A: Calculus

The following result is an extension of the implicit function theorem that allows the defining function to depend on a parameter. In particular, it provides parameter-independent bounds for the first and second derivatives of the implicitly defined functions.

Lemma A.1

Let $\varLambda \neq \emptyset$ be a set, and $U$, $V$ open subsets of ℝ. For each $\lambda \in \varLambda$, let $F_{\lambda }: U \times V \to \mathbb{R}$ and $y_{\lambda }: U \to V$ be twice continuously differentiable functions with

$$\begin{aligned} F_{\lambda }\big(x,y_{\lambda }(x)\big) &= 0, \quad x \in U. \end{aligned}$$

(A.1)

If there are constants $M_{0} > 1$ and $M_{1} \geq 0$ such that for each $\lambda \in \varLambda$ and all $(x,y) \in U \times V$,

$$\begin{aligned} \begin{aligned} \left\vert \frac{\partial F_{\lambda }}{\partial x}(x,y)\right\vert , \left\vert \frac{\partial^{2} F_{\lambda }}{\partial x^{2}}(x,y) \right\vert & \leq M_{0} + M_{1} \vert y\vert , \\ \left\vert \frac{\partial^{2} F_{\lambda }}{\partial x \partial y}(x,y) \right\vert , \left\vert \frac{\partial^{2} F_{\lambda }}{\partial y ^{2}} \right\vert &\leq M_{0}, \quad \left\vert \frac{\partial F_{ \lambda }}{\partial y}(x,y)\right\vert \geq \frac{1}{M_{0}}, \end{aligned} \end{aligned}$$

(A.2)

then there is a constant $\widetilde{M} > 0$ such that for all $\lambda \in \varLambda$ and $x\in U$,

$$\begin{aligned} \vert y_{\lambda }'(x) \vert &\leq \widetilde{M}\big(1 + M_{1} \vert y_{\lambda }(x) \vert \big) \ \textit{and} \ \vert y_{\lambda }''(x) \vert \leq \widetilde{M} \big(1+M_{1}\vert y _{\lambda }(x)\vert + M_{1}\vert y_{\lambda }(x)\vert^{2}\big). \end{aligned}$$

Moreover, if $\frac{\partial^{2} F_{\lambda }}{\partial y^{2}} \equiv 0$, then for all $\lambda \in \varLambda$ and $x\in U$,

$$\begin{aligned} \vert y_{\lambda }''(x) \vert &\leq \widetilde{M} \big(1+M_{1}\vert y _{\lambda }(x)\vert \big). \end{aligned}$$

Proof

Taking the derivative of (A.1) with respect to $x$ yields

$$\begin{aligned} \frac{\partial F_{\lambda }}{\partial x}\big(x,y_{\lambda }(x)\big) + \frac{\partial F_{\lambda }}{\partial y}\big(x,y_{\lambda }(x)\big)y _{\lambda }'(x) &= 0, \quad \lambda \in \varLambda , x \in U. \end{aligned}$$

(A.3)

Solving this for $y_{\lambda }'(x)$ and using the bounds (A.2) then gives

$$\begin{aligned} \vert y_{\lambda }'(x)\vert &\leq M_{0}\big(M_{0} + M_{1} \vert y_{ \lambda }(x) \vert \big), \quad \lambda \in \varLambda , x \in U. \end{aligned}$$

(A.4)

Taking the derivative in (A.3), we obtain for all $x\in U$ that

$$\begin{aligned} &\frac{\partial^{2} F_{\lambda }}{\partial x^{2}}\big(x,y_{\lambda }(x) \big) + 2\frac{\partial^{2} F_{\lambda }}{\partial x \partial y} \big(x,y_{\lambda }(x)\big) y_{\lambda }'(x) \\ &\quad{}+\frac{\partial^{2} F_{\lambda }}{\partial y^{2}}\big(x,y_{\lambda }(x)\big) \big(y_{\lambda }'(x)\big)^{2} + \frac{\partial F_{\lambda }}{\partial y}\big(x,y_{\lambda }(x)\big)y_{\lambda }''(x) = 0. \end{aligned}$$

Again, solving for $y_{\lambda }''(x)$ and using the bounds (A.2) and (A.4) gives

$$\begin{aligned} \vert y_{\lambda }''(x) \vert &\leq \overline{M} \big(1+M_{1}\vert y _{\lambda }(x)\vert + M_{1}\vert y_{\lambda }(x)\vert^{2}\big), \quad \lambda \in \varLambda , x \in U, \end{aligned}$$

(A.5)

for some sufficiently large constant $\overline{M}$. Finally, if $\frac{\partial^{2} F_{\lambda }}{\partial y^{2}} \equiv 0$, then it is easily seen that the quadratic term in (A.5) vanishes. □

Appendix B: Stochastic differential equations

Fix an abstract filtered probability space $(\varOmega , \mathcal{F}, \mathbb{F}= (\mathcal{F}_{t})_{t\geq 0}, P)$ carrying an $r$-dimensional Brownian motion $W = (W^{1}_{t},\ldots , W^{r} _{t})_{t\geq 0}$. The goal of this section is to prove the existence of solutions for a class of stochastic differential equations (SDEs) whose coefficients change at a stopping time. More precisely, we consider SDEs of the form

$$\begin{aligned} X_{t} &= \xi + \int_{0}^{t} \sigma (s,X)\,\mathrm{d}W_{s} + \int_{0} ^{t} b(s,X)\,\mathrm{d}s, \quad t \geq 0, \end{aligned}$$

(B.1)

where $\xi$ is an $\mathbb{R}^{d}$-valued $\mathcal{F}_{0}$-measurable random vector, $X = (X^{1}_{t},\ldots ,X^{d}_{t})_{t\geq 0}$ is a continuous semimartingale in $\mathbb{R}^{d}$, and

$$\begin{aligned} \sigma (s,X) &= \sigma^{(1)}(s,X)\mathbf{1}_{\lbrace s< \tau (X)\rbrace } + \sigma^{(2)}(s,X_{s})\mathbf{1}_{\lbrace s\geq \tau (X)\rbrace }, \\ b(s,X) &= b^{(1)}(s,X)\mathbf{1}_{\lbrace s< \tau (X)\rbrace } + b ^{(2)}(s,X_{s})\mathbf{1}_{\lbrace s\geq \tau (X)\rbrace }. \end{aligned}$$

(B.2)

Here, $\tau$ is a stopping time for the filtration induced by the canonical process on $C(\mathbb{R}_{+};\mathbb{R}^{d})$, $\sigma^{(1)}$, $b ^{(1)}$ are functions on $\mathbb{R}_{+}\times C(\mathbb{R}_{+}; \mathbb{R}^{d})$ that are progressive for the same filtration, and $\sigma^{(2)},b^{(2)}$ are measurable functions on $\mathbb{R}_{+} \times \mathbb{R}^{d}$; all codomains are understood to be of suitable dimension.

First of all, note that we cannot apply general existence results directly to the coefficients $\sigma$ and $b$ since the stopping time $\tau$ is typically not a continuous function on $C(\mathbb{R}_{+}; \mathbb{R}^{d})$. However, existence for this type of SDE is of course expected, provided that solutions exist for both sets of coefficients separately. The obvious idea is to solve the SDE for the first set of coefficients, stop the solution at $\tau$, and solve from there the SDE with the second set of coefficients. This can be made precise as follows.

Theorem B.1

Suppose that the process $X^{(1)}$ on $(\varOmega ,\mathcal{F}, \mathbb{F},P)$ satisfies

$$\begin{aligned} X^{(1)}_{t} &= \xi + \int_{0}^{t} \sigma^{(1)}(s,X^{(1)}) \, \mathrm{d}W_{s} + \int_{0}^{t} b^{(1)}(s,X^{(1)}) \,\mathrm{d}s, \quad t \geq 0. \end{aligned}$$

(B.3)

Moreover, assume that there is a constant $K > 0$ such that for all $t, t' \geq 0$ and $x, x' \in \mathbb{R}^{d}$,

$$\begin{aligned} \vert \sigma^{(2)}(t,x) - \sigma^{(2)}(t',x')\vert + \vert b^{(2)}(t,x)-b ^{(2)}(t',x')\vert &\leq K \vert (t,x) - (t',x') \vert , \end{aligned}$$

(B.4)

$$\begin{aligned} \vert \sigma^{(2)}(t,x)\vert + \vert b^{(2)}(t,x) \vert &\leq K (1+ \vert (t,x) \vert ), \end{aligned}$$

(B.5)

where $\vert \cdot \vert$ denotes the Euclidean norm in the suitable dimension. Then there is a continuous, $\mathbb{F}$-adapted, $\mathbb{R}^{d}$-valued process $X$ satisfying (B.1).

Proof

The solution prior to time $\hat{\tau }:= \tau (X^{(1)})$ is already given. To construct the part of the solution after time $\hat{\tau }$, consider the time-shifted filtration $\widehat{\mathbb{F}}= ( \widehat{\mathcal{F}}_{t})_{t \geq 0}$ defined by $\widehat{\mathcal{F}}_{t} = \mathcal{F}_{\hat{\tau }+ t}$ and the time-shifted $(\widehat{\mathbb{F}},P)$-Brownian motion $\widehat{W} = (\widehat{W}_{t})_{t \geq 0}$ defined by $\widehat{W}_{t} := W_{ \hat{\tau }+ t} - W_{\hat{\tau }}$. By our assumptions (B.4) and (B.5), the coefficients of the SDE

$$\begin{aligned} \widehat{Y}_{t} &= \begin{pmatrix} \widehat{X} \\ \widehat{A} \end{pmatrix} _{t} = \zeta + \int_{0}^{t} \begin{pmatrix} \sigma^{(2)}(\widehat{A}_{s},\widehat{X}_{s}) \\ 0 \end{pmatrix} \,\mathrm{d}\widehat{W}_{s} + \int_{0}^{t} \begin{pmatrix} b^{(2)}(\widehat{A}_{s},\widehat{X}_{s}) \\ 1 \end{pmatrix} \,\mathrm{d}s \end{aligned}$$

fulfil the standard Lipschitz and linear growth assumptions that guarantee the existence of a $P$-a.s. unique strong solution for any $\widehat{\mathcal{F}}_{0}$-measurable random vector $\zeta$ in $\mathbb{R}^{d+1}$. In particular, there exists an $\widehat{\mathbb{F}}$-progressive process $\widehat{Y} = (\widehat{X}, \widehat{A})$ for the initial condition $\zeta := (X^{(1)}_{ \hat{\tau }},\hat{\tau })$. Clearly, $\widehat{A}_{t} = \hat{\tau }+ t$ plays the role of the shifted time variable. A simple time change now yields that $X^{(2)}_{t} := \widehat{X}_{t-\hat{\tau }} \mathbf{1} _{\lbrace t \geq \hat{\tau }\rbrace }$ is $\mathbb{F}$-progressive and satisfies

$$\begin{aligned} X^{(2)}_{t} &= X^{(1)}_{\hat{\tau }} + \int_{\hat{\tau }}^{t} \sigma ^{(2)}(s,X^{(2)}_{s}) \,\mathrm{d}W_{s} + \int_{\hat{\tau }}^{t} b ^{(2)}(s,X^{(2)}_{s}) \,\mathrm{d}s \quad \text{on } \lbrace t \geq \hat{\tau }\rbrace . \end{aligned}$$

(B.6)

Finally, we verify that the process $X_{t} := X^{(1)}_{t} \mathbf{1} _{\lbrace t < \hat{\tau }\rbrace } + X^{(2)}_{t}\mathbf{1}_{\lbrace t \geq \hat{\tau }\rbrace }$ is a solution to the original SDE (B.1). As $X^{(2)}_{\hat{\tau }} = X^{(1)}_{\hat{\tau }}$,

$$\begin{aligned} X_{t} &= X^{(1)}_{t \wedge \hat{\tau }} + X^{(2)}_{t \vee \hat{\tau }} - X^{(2)}_{\hat{\tau }}. \end{aligned}$$

Plugging in (B.3) and (B.6) gives

$$\begin{aligned} X_{t} &= \xi + \int_{0}^{t \wedge \hat{\tau }} \sigma^{(1)}(s,X^{(1)}) \,\mathrm{d}W_{s} + \int_{0}^{t\wedge \hat{\tau }} b^{(1)}(s,X^{(1)}) \,\mathrm{d}s \\ & \quad{} +\int_{\hat{\tau }}^{t \vee \hat{\tau }} \sigma^{(2)}(s,X^{(2)}_{s}) \,\mathrm{d}W_{s} + \int_{\hat{\tau }}^{t\vee \hat{\tau }} b^{(2)}(s,X ^{(2)}_{s}) \,\mathrm{d}s \\ &= \xi + \int_{0}^{t} \sigma^{(1)}(s,X^{(1)})\mathbf{1}_{\lbrace s < \hat{\tau }\rbrace } \,\mathrm{d}W_{s} + \int_{0}^{t} b^{(1)}(s,X^{(1)}) \mathbf{1}_{\lbrace s < \hat{\tau }\rbrace } \,\mathrm{d}s \\ & \quad{} +\int_{0}^{t} \sigma^{(2)}(s,X^{(2)}_{s})\mathbf{1}_{\lbrace s\geq \hat{\tau }\rbrace } \,\mathrm{d}W_{s} + \int_{0}^{t} b^{(2)}(s,X^{(2)} _{s})\mathbf{1}_{\lbrace s\geq \hat{\tau }\rbrace } \,\mathrm{d}s. \end{aligned}$$

(B.7)

As $X^{(1)}_{s} = X_{s}$ on $\lbrace s \leq \hat{\tau }\rbrace$, Galmarino’s test implies that $\hat{\tau }= \tau (X^{(1)}) = \tau (X)$. Moreover, since $\sigma^{(1)}$ is progressive, $\sigma^{(1)}(s,X^{(1)})$ only depends on the path of $X^{(1)}$ up to time $s$. Using also the definition of $\sigma$ in (B.2), we obtain

$$\begin{aligned} \sigma^{(1)}(s,X^{(1)})\mathbf{1}_{\lbrace s < \hat{\tau }\rbrace } + \sigma^{(2)}(s,X^{(2)}_{s})\mathbf{1}_{\lbrace s \geq \hat{\tau } \rbrace } &= \sigma (s,X). \end{aligned}$$

Using this and the analogous statement for the drift coefficients, we see from (B.7) that $X$ is a solution to (B.1). □