# On properties of solutions to Black–Scholes–Barenblatt equations

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## Abstract

This paper is concerned with the Black–Scholes–Barenblatt equation \(\partial _{t}u+r(x\partial _{x}u-u)+G(x^{2}\partial _{xx}u)=0\), where \(G(\alpha )=\frac{1}{2}(\overline{\sigma}^{2}-\underline{\sigma}^{2})|\alpha |+\frac{1}{2}(\overline{\sigma}^{2}+\underline{\sigma}^{2})\alpha \), \(\alpha \in \mathbb{R}\). This equation is usually used for derivative pricing in the financial market with volatility uncertainty. We discuss a strict comparison theorem for Black–Scholes–Barenblatt equations, and study strict sub-additivity of their solutions with respect to terminal conditions.

## Keywords

Black–Scholes–Barenblatt equation Strict comparison theorem Strict sub-additivity## MSC

35B05 60H30 91B28## 1 Introduction

*G*is a nondecreasing sublinear function defined on \(\mathbb{R}\).

*T*based on a single liquidly traded stock

*S*, which satisfies the Itô equation

*B*is a 1-dimensional Brownian motion and

*σ*is a non-anticipative function such that, for strictly positive constants \(\underline{\sigma}\) and

*σ̅*,

*r*is zero and \(S_{0}=1\). The results obtained in this paper can be easily generalized to the case for \(r>0\) and any \(S_{0}>0\).

*σ*satisfying (1.3). Denote the payoff of the derivative at maturity date by \(\varphi (S_{T})\). If there is no arbitrage opportunity and the assumption on volatility is correct, then the value of such derivative at time \(t\in [0,T]\) should lie between the bounds

*t*. As \(u^{\varphi }(S_{t},t)=-v^{-\varphi }(S_{t},t)\), we only discuss the properties of \(u^{\varphi }\) in this paper.

*σ*, by Krylov [6], we can obtain \(u^{\varphi }\) by solving the following BSB equation:

*S*is characterized by a diffusion process driven by

*G*-Brownian motion \(B^{G}\),

*G*-Brownian motion.

*φ*and

*ψ*such that the following strict properties hold:

The rest of this paper is organized as follows. In Sect. 2, we give some properties of solutions of the BSB equations. In Sect. 3, we provide a strict comparison theorem for the BSB equations and study strict sub-additivity of their solutions with respect to terminal conditions in Sect. 4.

## 2 Preliminaries

Since PDE (1.6) is the fully nonlinear partial differential equation of second order, the classical smooth solution may not exist. The notion of viscosity solution is that it allows merely continuous functions to be solutions of (1.6), that it provides very general existence and uniqueness theorems.

### Definition 2.1

*u*such that, for all \((t,x)\in (0,T)\times \mathbb{R}\), \(\phi \in C^{2}((0,T)\times \mathbb{R})\) such that \(u(t,x)=\phi (t,x)\) and \(u<\phi \) (resp. \(u>\phi \))on \((0,T)\times \mathbb{R}\setminus (t,x)\), we have

*φ*on \(\mathbb{R}_{+}\) satisfying

*φ*. We have the following solvability result for the viscosity solution of BSB equation (1.6) (see Theorem 7 in Gozzi and Vargiolu [4]).

### Theorem 2.2

*For each*\(\varphi \in C_{l,\mathrm{Lip}}(\mathbb{R}_{+})\), *BSB equation* (1.6) *has a unique viscosity solution *\(u^{\varphi }\).

Since the function \(G(\cdot )\) defined by (1.7) is nondecreasing and sublinear on \(\mathbb{R}\) with \(G(0)=0\), we immediately have the following properties for the functional \(V_{T}\).

### Proposition 2.3

*For*\(T>0\),

*the functional*\(V_{T}(\cdot )\)

*is a sublinear expectation on*\(C_{l,\mathrm{Lip}}(\mathbb{R}_{+})\),

*i*.

*e*., \(\forall \varphi ,\psi \in C_{l,\mathrm{Lip}}( \mathbb{R}_{+})\),

*the following properties hold*:

- (i)
*Monotonicity*:*If*\(\varphi \geq \psi \),*then*\(V_{T}(\varphi ) \geq V_{T}(\psi ) \); - (ii)
*Constant preserving*: \(V_{T}(c)=c\)*for all*\(c\in \mathbb{R}_{+}\); - (iii)
*Sub*-*additivity*: \(V_{T}(\varphi +\psi )\leq V_{T}(\varphi )+V _{T}(\psi )\); - (iv)
*Positive homogeneity*: \(V_{T}(\lambda \varphi )= \lambda V_{T}(\varphi )\)*for all*\(\lambda \geq 0\).

### Proposition 2.4

*Let*\(T>0\)

*and*\(\varphi \in C_{l,\mathrm{Lip}}(\mathbb{R}_{+})\).

*We have*

- (i)
\(V_{T}(\varphi )\geq \sup_{\underline{\sigma}\leq \sigma \leq \overline{\sigma}}V_{T}^{ \sigma }(\varphi )\);

- (ii)
\(V_{T}(\varphi )=V_{T}^{\underline{\sigma}}(\varphi )\)

*if and only if**φ**is concave*; - (iii)
\(V_{T}(\varphi )=V_{T}^{\overline{\sigma}}(\varphi )\)

*if and only if**φ**is convex*; - (iv)
*Let*\(\sigma \in (\underline{\sigma}, \overline{\sigma})\),*then*\(V_{T}(\varphi )=V_{T}^{ \sigma }(\varphi )\)*if and only if**φ**is linear*.

### Proof

*σ*coincides, i.e.,

To show (ii), first due to the preservation of convexity, we know that *φ* is concave if and only if \(\partial _{xx}u^{\sigma , \varphi }(x,t) \leq 0\), \(\forall (x,t)\in \mathbb{R}_{+}\times (0,T)\). This is to say: when \(\sigma =\underline{\sigma}\), (2.3) holds for all \((x,t)\in \mathbb{R}_{+}\times (0,T)\), which is equivalent to \(V_{T}(\varphi )=V _{T}^{\underline{\sigma}}(\varphi )\) by (2.2). The convex case (iii) can be proved similarly.

Now we turn to verify (iv). When \(\sigma \in (\underline{\sigma}, \overline{\sigma})\), it is easy to find that (2.3) holds if and only if \(\partial _{xx}u^{\sigma , \varphi }(x,t)=0\), \(\forall (x,t)\in \mathbb{R}_{+}\times (0,T)\), which is equivalent to that *φ* is linear. □

## 3 Strict comparison theorem for BSB equations

In this section, we shall look for conditions such that the strict comparison (1.8) holds. Recall that \(0<\underline{\sigma}<\overline{\sigma}\) in this paper.

### Theorem 3.1

*Fix*\(T>0\). *Let*\(\varphi ,\psi \in C_{l,\mathrm{Lip}}(\mathbb{R}_{+})\)*such that*\(\varphi (x)\leq \psi (x)\)*for all*\(x\in \mathbb{R}_{+}\). *Then*\(V_{T}(\varphi )< V _{T}(\psi )\)*if and only if there exists*\(x_{0}\in \mathbb{R}_{+}\)*such that*\(\varphi (x_{0})<\psi (x_{0})\).

### Proposition 3.2

*Let*\(\varphi \in C_{l,\mathrm{Lip}}(\mathbb{R})\)*such that*\(\varphi \leq 0\)*and there exists*\(x_{0}\)*such that*\(\varphi (x_{0})<0\). *Let**ũ**be the solution of PDE* (3.1) *with terminal condition*\(\tilde{u}(x,T)=\varphi (x)\). *Then*\(\tilde{u}(x,t)<0\)*for all*\((x,t)\in \mathbb{R}\times [0,T)\).

To prove Proposition 3.2, we need the following lemma.

### Lemma 3.3

*Fix*\(a\in \mathbb{R}\).

*Let*\(\tilde{u}_{m}\)

*be the solution of PDE*(3.1)

*with the terminal condition*\(\tilde{u}_{m}(T,x)=-\exp (-\frac{m \theta |x-a|^{2}}{2})\),

*where*\(\theta =T+\frac{1}{\underline{\sigma}^{2}}+1\)

*and*\(m\geq 1\).

*Then*,

*for any*\((x,t)\in \mathbb{R}\times [0,T)\),

*we have*

### Proof

Now we give the proof of Proposition 3.2, from which we can easily see that Theorem 3.1 holds.

### Proof of Proposition 3.2

*m*large enough such that

### Corollary 3.4

*Fix*\(T>0\). *Let*\(\varphi ,\psi \in C_{l,\mathrm{Lip}}(\mathbb{R}_{+})\)*such that*\(\varphi (x)\leq \psi (x)\)*for all*\(x\in \mathbb{R}_{+}\). *Then*\(V_{T}(\varphi )=V _{T}(\psi )\)*if and only if*\(u^{\varphi }(x,t)=u^{\psi }(x,t)\), \(\forall (x,t)\in \mathbb{R}_{+}\times (0,T)\).

### Remark 3.5

The strict comparison does not hold if \(\underline{\sigma}=0\). For example, let \(\varphi =\min(x, 1)\) and \(\psi \equiv 1\). However, when \(\underline{\sigma}=0\), \(V_{T}(\varphi )=u^{\varphi }(1,0)=1=u^{\psi }(1,0)=V_{T}(\psi )\).

### Remark 3.6

The price \(V_{T}(\varphi )\) obtained from BSB equation (1.6) is indeed the least possible initial cost to risklessly hedge a short position in the derivative security *φ* by self-financing portfolios. The properties above show that this type of pricing under volatility uncertainty is arbitrage-free in the sense: if *ψ* is “substantially” superior to *φ*, i.e., \(\varphi \leq \psi \) and there exists \(x_{0}\in \mathbb{R}_{+}\) such that \(\varphi (x_{0})<\psi (x _{0})\), then one could not hedge a short position of *ψ* with the initial value \(V_{T}(\varphi )\). In other words, if two options *φ* and *ψ* satisfy \(\varphi \leq \psi \) but they have the same optimal risk-averse ask price given by (1.6), then they generate the same path of their “present values”, i.e., \(u^{\varphi }(S_{t},t)=u ^{\psi }(S_{t},t)\), \(\forall t\in (0,T)\).

## 4 Strict sub-additivity for BSB equations

In the remainder of this paper, we consider the strict sub-additivity (1.9) of the functional \(V_{T}\).

### Theorem 4.1

*Fix*\(T>0\).

*For*\(\varphi ,\psi \in C_{l,\mathrm{Lip}}(\mathbb{R}_{+})\),

*we have*

*if and only if there exists*\((x_{0},t_{0})\in \mathbb{R}_{+}\times (0,T)\)

*such that*

### Proof

If (4.1) holds, which is equivalent to \(u^{\varphi +\psi }(1,0)< u ^{\varphi }(1,0)+u^{\psi }(1,0)\), then by the continuity of \(u^{\varphi + \psi }\), \(u^{\varphi }\), and \(u^{\psi }\), there exists \((x_{0},t_{0}) \in \mathbb{R}_{+}\times (0,T)\) such that (4.2) holds. □

### Remark 4.2

The strict inequality (4.1) could be explained as the diversification of volatility risk in the portfolio of European type derivatives (cf. [1]).

The theorem above shows that the optimal risk-averse ask price of the portfolio “\(\varphi +\psi \)” will be strictly lower than the sum of the individual prices for “*φ*” and “*ψ*” if the paths of “present values” \(u^{\varphi +\psi }(S_{t},t)\), \(u^{\varphi }(S_{t},t)\), and \(u^{\psi }(S_{t},t)\) are strictly sub-additive at some time \(t_{0}\in (0, T)\). In practice, (4.2) is very difficult to check, thus we look for sufficient but simpler conditions for the strict sub-additivity of \(V_{T}\) in what follows.

### Theorem 4.3

*Fix*\(T>0\).

*For*\(\varphi ,\psi \in C_{l,\mathrm{Lip}}(\mathbb{R}_{+})\),

*if there exists*\(x_{0}\in \mathbb{R}_{+}\)

*such that*

*then we have*

### Proof

Without loss of generality we can assume that there exists \(\varepsilon >0\) such that \(\varphi ''(x_{0})>\varepsilon \) and \(\psi ''(x_{0})<- \varepsilon \).

*φ*is twice differentiable at \(x_{0}\), we have

*G*, we can deduce that

The following corollary is straightforward.

### Corollary 4.4

*Fix*\(T>0\).

*For*\(\varphi ,\psi \in C_{l,\mathrm{Lip}}^{2}(\mathbb{R}_{+})\),

*if*\(V_{T}(\varphi + \psi )=V_{T}(\varphi )+V_{T}(\psi )\),

*then we have*

### Remark 4.5

Unfortunately, the inverse statement of Corollary 4.4 is not true, which means that (4.3) is merely a sufficient condition. We have the following counterexample: Let \(\varphi (x)=(\max \{x-2,0\})^{2}\) which is convex and \(\psi (x)=-(\min \{x-1,0\})^{2}\) which is concave. It is easy to find that \(V_{T}(\varphi +\psi )< V_{T}(\varphi )+V_{T}(\psi )\), but \(\varphi ''(x)\psi ''(x)=0\) for all \(x\in \mathbb{R}_{+}\).

## Notes

### Acknowledgements

The authors wish to thank the editor and the reviewers for their useful remarks on our paper.

### Authors’ contributions

All authors have equally contributed to this work. All authors read and approved the final manuscript.

### Funding

Xinpeng Li and Weicheng Xu are supported by NSFC (No. 11601281) and Shandong Province (No. ZR2016AQ11); Yiqing Lin is supported by NSFC (No. 11801365) and Shanghai Jiao Tong University (No. WF220507103).

### Competing interests

The authors declare that they have no competing interests.

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