Robust Valuation, Arbitrage Ambiguity and Profit & Loss Analysis

Abstract

Model uncertainty is a type of inevitable financial risk. Mistakes on the choice of pricing model may cause great financial losses. In this paper we investigate financial markets with mean-volatility uncertainty. Models for stock market and option market with uncertain prior distributions are established by Peng’s G-stochastic calculus. On the hedging market, the upper price of an (exotic) option is derived following the Black–Scholes–Barenblatt equation. It is interesting that the corresponding Barenblatt equation does not depend on mean uncertainty of the underlying stocks. Appropriate definitions of arbitrage for super- and sub-hedging strategies are presented such that the super- and sub-hedging prices are reasonable. In particular, the condition of arbitrage for sub-hedging strategy fills the gap of the theory of arbitrage under model uncertainty. Finally we show that the term K of finite variance arising in the superhedging strategy is interpreted as the max Profit & Loss ($\mathrm{P} \& \mathrm{L}$) of shorting a delta-hedged option. The ask-bid spread is in fact an accumulation of the superhedging $\mathrm{P} \& \mathrm{L}$ and the sub-hedging $\mathrm{P} \& \mathrm{L}$.

Appendices

Peng’s G-stochastic Calculus

In this section we recall some necessary notions and lemmas of Peng’s G-stochastic calculus needed in this paper. Readers could refer to [37] for more systematic information.

For two stochastic processes $(X_{t})$ and $(Y_{t})$, let $\left\langle X,Y\right\rangle _{t}$ denote their mutual variance. We denote by S(n) the collection of $n\times n$ symmetric matrices, ${{S}}_{+}(d)$ the positive-semidefinite elements of S(d). We observe that S(n) is a Euclidean space with the scalar product $\left\langle A,B\right\rangle =tr[AB]$. Let $\Omega$ be a complete metrizable and separable space. Typically we can take $\Omega =C_{0}([0,+\infty ),{\mathbb {R}}^{d})$ with the topology of uniform convergence on compact subspaces. ${\mathcal {B}}(\Omega )$ denotes the Borel $\sigma$-algebra of $\Omega$. Let ${{\mathcal {H}}}$ be a linear space of real functions defined on $\Omega$ such that if $X_{1},\cdots ,X_{n}\in {{\mathcal {H}}}$ then $\varphi (X_{1},\cdots X_{n})\in {{\mathcal {H}}}$ for each $\varphi \in C_{l.\mathrm{Lip}({\mathbb {R}} ^{n})}$ where $C_{l.\mathrm{Lip}({\mathbb {R}}^{n})}$ denotes the linear space of (local Lipschitz) functions $\varphi$ satisfying

$$\begin{aligned} |\varphi \left( x\right) -\varphi \left( y\right) |\leqslant C(1+\left| x\right| ^{m}+\left| y\right| ^{m})|x-y|,\ \quad \forall x,y\in {\mathbb {R}}^{n}, \end{aligned}$$

for some $C>0$, $m\in N$ depending on $\varphi$. ${{\mathcal {H}}}$ is considered as a space of “random variables.” In this case $X=(X_{1},\cdots ,X_{n})$ is called an n-dimensional random vector, denoted by $X\in {\mathcal {H}}^{n}$. We also denote by $C_{b}^{k}({\mathbb {R}}^{n})$ the space of bounded and k-time continuously differentiable functions with bounded derivatives of all orders less than or equal to k; $C_{\mathrm{Lip}({\mathbb {R}}^{n})}$ the space of Lipschitz continuous functions.

Definition A.1

A sublinear expectation E on ${{\mathcal {H}}}$ is a functional ${{E}}:{\mathcal {H\mapsto }}{\mathbb {R}}$ satisfying the following properties: For all $X,Y\in {{\mathcal {H}}}$, we have

• (a) Monotonicity: If $X\geqslant Y$, then $E[X]\geqslant E[Y]$.

• (b) Constant preserving: $E[c]=c,\quad \forall c\in {\mathbb {R}}$.

• (c) Sub-additivity: $E[X+Y]\leqslant {E[}X] +{E[}Y]$.

• (d) Positive homogeneity: ${E[\lambda }X{]=\lambda E[} X{]},\quad \forall \lambda \geqslant 0$.

Definition A.2

Let $X_{1}$ and $X_{2}$ be two n-dimensional random vectors defined on nonlinear expectation spaces $(\Omega _{1},{\mathcal {H}}_{1},{{E}}_{1})$ and $(\Omega _{2},{\mathcal {H}}_{2},{{E}}_{2})$, respectively. They are called identically distributed, denoted by $X_{1}\overset{d}{=}X_{2}$, if

$$\begin{aligned} {{E}}_{1}[\varphi (X_{1})]={{E}}_{2}[\varphi (X_{2})],\ \quad \forall \varphi \in C_{l.\mathrm{Lip}}({\mathbb {R}}^{n}). \end{aligned}$$

Definition A.3

In a sublinear expectation space $(\Omega ,{\mathcal {H}},{{E}})$ a random vector $Y\in {{\mathcal {H}}}^{n}$ is said to be independent of another random vector $X\in {{\mathcal {H}}}^{m}$ under E if for each test function $\varphi \in C_{l.\mathrm{Lip}}({\mathbb {R}}^{m+n})$ we have

$$\begin{aligned} {{E}}[\varphi (X,Y)]={{E}}\left[ {{E}}[\varphi (x,Y)]_{x=X} \right] . \end{aligned}$$

Definition A.4

(G–normal distribution). A d-dimensional random vector $X=(X_{1},\cdots ,X_{d})$ in a sublinear expectation space $(\Omega ,{\mathcal {H}} ,{{E}})$ is called G-normal distributed if for each $a,b>0$ we have

$$\begin{aligned} aX+b{\overline{X}}\,\overset{d}{=}\sqrt{a^{2}+b^{2}}X, \end{aligned}$$

where ${\overline{X}}$ is an independent copy of X.

Remark A.1

It is easy to check that ${{E}}[X]=$ ${{E}}[-X]=0$. The so-called G is related to $G:{{S}}(d)\mapsto {\mathbb {R}}$ defined by

$$\begin{aligned} G\left( A\right) =\frac{1}{2}{E[}\left\langle AX,X\right\rangle {]}. \end{aligned}$$

Hu and Peng [38] proved that for a sublinear expectation E on $(\Omega ,{\mathcal {H}})$, there exists a family of linear expectation {$E_{P};P\in {\mathcal {P}}$} on $(\Omega ,{\mathcal {H}})$ such that ${{E}} \left[ \cdot \right] =\underset{P\in {\mathcal {P}}}{\sup }{\normalsize E} _{P}\left[ \cdot \right]$.

Definition A.5

For a given set of probability measures ${\mathcal {P}}$, we introduce the natural Choquet capacity

$$\begin{aligned} C(A):=\underset{P\in {\mathcal {P}}}{\sup }P(A),\ A\in {\mathcal {B}}(\Omega ). \end{aligned}$$

A property holds quasi-surely(q.s.) if it holds outside a polar set A, i.e., $C(A)=0$. A mapping X on $\Omega$ with values in a topological space is said to be quasi-continuous (q.c.) if $\forall \varepsilon >0$, there exists an open set O with $C(O)<\varepsilon$ such that $X|_{O^{c}}$ is continuous.

Definition A.6

(G–Brownian motion). A d-dimensional process $(B_{t})_{t\geqslant 0}$ on a sublinear expectation space $(\Omega ,{\mathcal {H}},{{E}})$ is called a G–Brownian motion if the following properties are satisfied:

• (i) $B_{0}(\omega )=0$;

• (ii) For each $t,s\geqslant 0$, the increment $B_{t+s}-B_{t}$ is independent from $(B_{t_{1}},B_{t_{2}},\cdots ,B_{t_{n}})$, for each $n\in N$ and $0\leqslant t_{1}\leqslant \cdots \leqslant t_{n}\leqslant t$;

• (iii) $B_{t+s}-B_{t}\overset{d}{=}\sqrt{s}X$, where X is G-normal distributed.

In the sequence, let $\Omega =C_{0}([0,+\infty ),{\mathbb {R}}^{d})$ denote the space of all ${\mathbb {R}}^{d}\hbox {-}$valued continuous paths $(\omega _{t})_{t\in \mathbb {R}^{+}}$ with $\omega _{0}=0$, by $C_{b}(\Omega )$ all bounded and continuous functions on $\Omega$. For each fixed $T\geqslant 0$, we consider the following space of random variables:

$$\begin{aligned} L_{ip}(\Omega _{T}):=\{X(\omega )=\varphi (\omega _{t_{1}\wedge T},\cdots ,\omega _{t_{m}\wedge T}),\quad \forall m\geqslant 1,\forall \varphi \in C_{l.\mathrm{Lip}({\mathbb {R}}^{m} )}\}. \end{aligned}$$

We also denote

$$\begin{aligned} L_{ip}(\Omega ):=\overset{\infty }{\underset{n=1}{\cup }}L_{ip}(\Omega _{n}). \end{aligned}$$

We now take the canonical space as example to show how to find a G-Brownian motion and conditional G-expectation. Set $B_{t}(\omega )=\omega _{t}$. For a given sublinear function $G\left( A\right) =\frac{1}{2}\underset{\gamma \in \Gamma }{\sup }\left\{ tr[A\gamma ]\right\}$, where $A\in {{S}}(d)$, $\Gamma$ is a given nonempty, bounded and closed convex subset of ${{S}}_{+}(d)$. By the following

$$\begin{aligned} \partial _{t}u(t,x)-G\left( D_{x}^{2}u\right) =0,\ u(0,x)=\varphi (x), \end{aligned}$$

Peng [33] defined G-expectation E as ${{E}} [\varphi (x+B_{t})]=u(t,x)$. For each $\mathrm {\beta }\geqslant 1$, $X\in \mathrm{L}_\mathrm{ip}(\Omega )$, $\Vert X\Vert _{\mathrm {\beta }}=\left( {{E}}[\left| X\right| ^{\mathrm {\beta }}]\right) ^{\frac{1}{\mathrm {\beta }}}$ forms a norm and E can be continuously extended to a Banach space, denoted by $L{_{G}^{\mathrm {\beta }}}(\Omega )$. [38] proved that $L{_{G}^{\mathrm {\beta }}}(\Omega )=\{X|\ X$ is ${\mathcal {B}}(\Omega )-$measurable and has a quasi-continuous version such that $\underset{n\rightarrow \infty }{\lim }{{E}}[\left| X\right| ^{\beta }1_{\left\{ \left| X\right| >n\right\} }]=0\}$. By the method of Markov chains, [33, 35] also defined corresponding conditional expectation, ${{E}}\left[ \cdot |\Omega _{t}\right] :L{_{G}^{\mathrm {1}}} (\Omega )\mapsto L{_{G}^{\mathrm {1}}}(\Omega _{t})$, where $\Omega _{t}:=\left\{ \omega ._{\wedge t}:\omega \in \Omega \right\}$. Under ${{E}}\left[ \cdot \right]$, the canonical process $B_{t}(\omega )=\omega _{t}$, $t\in [0,\infty )$ is a G-Brownian motion.

The following properties hold for ${{E}}\left[ \cdot |\Omega _{t}\right]$ q.s.

Proposition A.1

For $X,Y\in L{_{G}^{\mathrm {1}}}(\Omega )$, we have q.s.,

• (i) ${{E}}[\eta X|\Omega _{t}]=\eta ^{+}{{E}}[X|\Omega _{t}]+\eta ^{-}{{E}}[-X|\Omega _{t}]$, for bounded $\eta \in L{_{G}^{\mathrm {1}} }(\Omega _{t})$.

• (ii) If ${{E}}[X|\Omega _{t}]=-{{E}}[-X|\Omega _{t}]$, for some t, then ${E[}X+Y|\Omega _{t}{]}={E[}X|\Omega _{t} {]}+{E[}Y|\Omega _{t}{]}$.

• (iii) ${E[}X+\eta |\Omega _{t}{]}={E[}X|\Omega _{t}{]}+\eta$, $\eta \in L{_{G}^{\mathrm {1}}}(\Omega _{t})$.

For a partition of [0, T]: $0=t_{0}<t_{1}<\cdots <t_{N}=T$ and $\mathrm {\beta }\geqslant 1$, we set

${{\mathcal {M}}_{G}^{\mathrm {\beta ,0}}}(0,{ T })$: the collection of processes $\eta _{t}(\omega )=\sum _{j=0}^{N}\xi _{j}(\omega )\cdot 1_{[t_{j} ,t_{j+1}]}(t)$, where $\xi _{j}\in L{_{G}^{\mathrm {\beta }}}(\Omega _{t_{j} }),j=0,1,\cdots ,N$;

${\mathcal {M}}{_{G}^{\mathrm {\beta }}}(0,{ T })$: the completion of ${{\mathcal {M}}_{G}^{\mathrm {\beta ,0}}}(0,{ T })$ under norm $||\eta ||_{{\mathcal {M}}}=\left( {{E}}\left[ \int _{0}^{T}|\eta _{t}|^{\beta }\hbox {d}t\right] \right) ^{^{\frac{1}{\beta }}}$;

${\mathcal {H}}{_{G}^{\mathrm {\beta }}}(0,{ T })$: the completion of ${{\mathcal {M}}_{G}^{\mathrm {\beta ,0}}}(0,{ T })$ under norm $||\eta ||_{{\mathcal {H}}}=\left( {{E}}\left( \int _{0}^{T}|\eta _{t} |^{2}\hbox {d}t\right) ^{\frac{\beta }{2}}\right) ^{\frac{1}{\beta }}$. It is easy to prove that ${\mathcal {H}}{_{G}^{\mathrm {2}}}(0,{ T })={\mathcal {M}} {_{G}^{\mathrm {2}}}(0,{ T })$.

For any $\left( \eta _{t}\right) \in {\mathcal {M}}{_{G}^{\mathrm {2}}}$, G-Itô integral is well defined in [33, 35] and extended to ${\mathcal {H}}{_{G}^{\mathrm {\beta }}}$ by [56].

BSDE with Linear Generator and Driven by G-Brownian Motion

We define $D_{t}={\normalsize \exp }\left\{ -{\normalsize \int _{0}^{t}r_{s}\mathrm{d}s}\right\}$. Then $D_{t}$ satisfies

$$\begin{aligned} \mathrm{d}D_{t}=-D_{t}r_{t}\hbox {d}t\text {, and }D_{t=0}=1. \end{aligned}$$

Consider the following one-dimensional BSDE with linear generator and driven by one-dimensional G-Brownian motion:

$$\begin{aligned} \left\{ \begin{array} [c]{l} dY_{t} =\left( r_{t}Y_{t}-\phi _{t}\right) \hbox {d}t+Z_{t}\hbox {d}B_{t}-\hbox {d}K_{t} , \mathrm{q.s.}, \\ Y_{T} =\xi ., \end{array} \right. \end{aligned}$$

(B.1)

where $\xi \in L{_{G}^{\mathrm {\beta }}}(\Omega _{T})$, $\beta >1$, $r_{t}$ and $\phi _{t}$ are ${\mathcal {F}}_{t}$-measurable bounded processes belonging to ${{\mathcal {M}}_{G}^{\mathrm {\beta }}}$.

Definition B.1

A solution to BSDE (B.1) is a triple of adapted processes $(Y_{t},Z_{t},K_{t})$, where $(K_{t})$ is a continuous, increasing process with $K_{0}=0$ and $(-K_{t})$ being a G-martingale.

For BSDE (B.1), we have the following result.

Theorem B.1

There is a unique triple $(Y_{t},Z_{t},K_{t})$ satisfying (B.1) with $Y\in {{\mathcal {M}}_{G}^{\mathrm {\beta }}}$, $Z\in {\mathcal {H}}{_{G}^{\mathrm {\alpha }}}$ and $K_{T}\in L{_{G}^{\mathrm {\alpha }} }(\Omega _{T})$, $1\leqslant \alpha <\beta$, $\beta >1$. Furthermore we have q.s.,

$$\begin{aligned} Y_{t}=D_{t}^{-1}{{E}}\left[ D_{T}\xi +\int _{t}^{T}D_{s}\phi _{s}\mathrm{d}s|{\mathcal {F}}_{t}\right] .\ \end{aligned}$$

(B.2)

Proof

Consider the following BSDE under sublinear expectation E:

$$\begin{aligned} Y_{t}={{E}}\left[ \xi -\int _{t}^{T}\left( r_{s}Y_{s}-\phi _{s}\right) \mathrm{d}s|{\mathcal {F}}_{t}\right] . \end{aligned}$$

(B.3)

By the technique of contracting mapping principle employed in [37], Ch.V, Sect. 2, one can similarly prove that there is a unique solution $Y\in {{\mathcal {M}}_{G}^{\mathrm {\beta }}}$ to BSDE (B.3). Applying martingale representation theorem established in [56], there is a unique pair (Z, K) with $Z\in {{\mathcal {M}} _{G}^{\mathrm {\alpha }}}$ and $K_{T}\in L{_{G}^{\mathrm {\alpha }}}(\Omega _{T})$, $1\leqslant \alpha <\mathrm {\beta }$ such that

$$\begin{aligned} {{E}}\left[ \xi -\int _{0}^{T}\left( r_{s}Y_{s}-\phi _{s}\right) \mathrm{d}s|{\mathcal {F}}_{t}\right] =Y_{0}+\int _{0}^{t}Z_{s}\hbox {d}B_{s}-K_{t},\ \mathcal {P}\hbox {-} q.s. \end{aligned}$$

Hence

$$\begin{aligned} Y_{t}&={{E}}\left[ \xi -\int _{0}^{T}\left( r_{s}Y_{s}-\phi _{s}\right) \mathrm{d}s|{\mathcal {F}}_{t}\right] +\int _{0}^{t}\left( r_{s}Y_{s} -\phi _{s}\right) \mathrm{d}s\\&=Y_{0}++\int _{0}^{t}\left( r_{s}Y_{s}-\phi _{s}\right) \mathrm{d}s+\int _{0} ^{t}Z_{s}\hbox {d}B_{s}-K_{t},\ \mathcal {P}\hbox {-}q.s., \end{aligned}$$

or in backward form

$$\begin{aligned} Y_{t}=\xi -\int _{t}^{T}\left( r_{s}Y_{s}-\phi _{s}\right) ds-\int _{t}^{T} Z_{s}\hbox {d}B_{s}+\int _{t}^{T}\hbox {d}K_{s},\ \mathcal {P}\hbox {-}q.s. \end{aligned}$$

Thus the triple $(Y_{t},Z_{t},K_{t})$ constructed by above procedure is a solution of (B.1).

Applying Itô’s formula to $D_{t}Y_{t}$, we have

$$\begin{aligned} \mathrm{d}(D_{t}Y_{t})&=D_{t}\left[ \left( r_{t}Y_{t}-\phi _{t}\right) \hbox {d}t+Z_{t}\hbox {d}B_{t}-\hbox {d}K_{t}\right] -Y_{t}D_{t}r_{t}\hbox {d}t\\&=-D_{t}\phi _{t}\hbox {d}t+D_{t}Z_{t}\hbox {d}B_{t}-D_{t}\hbox {d}K_{t}. \end{aligned}$$

Note that $(D_{t}Y_{t}+\int _{0}^{t}D_{s}\phi _{s}\mathrm{d}s)$ is a G-martingale. Hence

$$\begin{aligned} D_{t}Y_{t}&={{E}}\left[ D_{T}\xi +\int _{0}^{T}D_{s}\phi _{s}\mathrm{d}s|{\mathcal {F}}_{t}\right] -\int _{0}^{t}D_{s}\phi _{s}\mathrm{d}s\\&={{E}}\left[ D_{T}\xi +\int _{t}^{T}D_{s}\phi _{s}\mathrm{d}s|{\mathcal {F}} _{t}\right] . \end{aligned}$$

Therefore the solution of (B.1) has the following unique form:

$$\begin{aligned} Y_{t}=D_{t}^{-1}{{E}}\left[ D_{T}\xi +\int _{t}^{T}D_{s}\phi _{s}\mathrm{d}s|{\mathcal {F}}_{t}\right] . \end{aligned}$$

Let $Y^{i}$ be the solution of (B.1) with parameters $\left( \xi ^{i},\phi ^{i}\right)$, $i=1,2$. It is interesting that ($Y^{1}+Y^{2}$) is no longer a solution of (B.1) with parameters $\left( \xi ^{1}+\xi ^{2},\phi ^{1}+\phi ^{2}\right)$, though BSDE (B.1) has a linear generator. All attributes to the sublinearity. $(-K_{t}^{1}-K_{t}^{2})$ is no more a G-martingale. We have the following:

Corollary B.1

Let ${\widetilde{Y}}$ be the solution of (B.1) with parameters $\left( \xi ^{1}+\xi ^{2},\phi ^{1}+\phi ^{2}\right)$. Then

$$\begin{aligned} Y^{1}+Y^{2}\geqslant {\widetilde{Y}}. \end{aligned}$$

Proof

It is just a sequence of (B.2) and the sublinearity of G-expectation E.

This property reflects that if two agents cooperate with each other, then superhedging the whole might yield less pricing error.