Identifying the implied volatility using the total variation regularization
- Yun Liu^{1} and
- Xijuan Liu^{1}Email author
Received: 27 June 2017
Accepted: 2 January 2018
Published: 15 January 2018
Abstract
This paper studies an inverse problem of determining the implied volatility in the financial products linked with gold price, which has important application in financial derivatives pricing. Based on the total variation regularization strategy, the existence and necessary condition of the minimum for the control function are addressed, and the local uniqueness of the solution is also given by a modified case. Furthermore, the stability and convergence for the regularized approach are discussed. The results obtained in this paper may be useful for those who engage in hedging or risk management.
Keywords
1 Introduction
In this work, we study an inverse problem of determining the implied volatility in the financial products linked with gold price. This problem was issued by the bank of East Asia in Guangzhou (China) on February 20, 2006. The profits of this product not only depend on some cumulative indexes of gold price but also on some trigger indexes. Thus this problem can be regarded as a double barrier option question (see [1, 2]). The mathematical model can be stated in the following form:
- (1)Gold price \(S_{t}\) satisfies a geometric Brownian motionwhere μ is the expected rate of return, \(\sigma(S)\) is the volatility, and \(W_{t}\) is the standard Brownian process which satisfies$$\frac{dS_{t}}{S_{t}}=\mu \,dt+ \sigma(S) \,dW_{t}, $$$$E(dW_{t})=0,\qquad \operatorname{Var}(dW_{t})=dt. $$
- (2)
The limit range of the gold price is \([S_{a},S_{b}]\), and the initial price is \(S_{0}\), which satisfies \(S_{a}\leq S_{0}\leq S_{b}\). The upper bound \(S_{a}\) and the lower bound \(S_{b}\) are determined by the estimation of expectation of gold price, which are made by the bank.
- (3)
The risk-free rate of the dollar market is a nonnegative constant r, and the annual rate of gold price is \(r_{0}\) when the price does not hit the interval \([S_{a},S_{b} ]\). Otherwise, it is denoted by \(r_{1}\).
Volatility is a very important parameter in the Black-Scholes equation, it is necessary to accurately measure it in portfolio selection, underlying financial products pricing and hedging or risk management. Determining the implied volatility is a typical PDE inverse problem. It was first considered by Dupire in [4] where he obtained a formula of the local volatility with all possible strikes and maturities. However, the formula was ill-posed and could not be used in practice, it had to be modified. Some detailed treatments of problems in these areas can be found in [4–16]. In [5, 9, 15], Bouchouev and Isakov reduce the identification of volatility to an inverse problem with the final observation, and the local uniqueness and stability of volatility are proved under certain assumptions. Lu and Yi in [12] obtain a Fredholm integral equation from the Dupire equation. In [16], Mishura consider the diffusion approximation for the recurrent schemes of financial markets and generalize a classical scheme of weak convergence for discrete-time markets to the Black-Scholes model. Based on the optimal control framework, in [10, 11], Jiang and Tao consider the inverse problem of determining the implied volatility \(\sigma=\sigma(S)\) using the Tikhonov regularization strategy and analyze the existence and uniqueness of \(\sigma=\sigma(S)\), and a new well-posed algorithm is presented. Similar results are obtained in [7], where a new extra condition (i.e., the average option premium) is assumed to be known. In [17], the authors also consider an optimal control problem to a mathematical model of drug treatment, a cost functional of the system is minimized and the total amount of drugs is given.
Considering the advantages and great success of the total variation regularization in image processing and the work of [13, 18, 23, 24], in this paper, we would like to reconstruct the unknown volatility function \(\sigma=\sigma(S)\) in (1.1) by using the total variation regularization strategy, based on the optimal control method (see [6, 7, 13, 17, 22]). Being different from the common inverse volatility problems arising in the area of option pricing (see [6, 7, 11, 13, 16, 18, 20, 22]), our inverse problem has the following unusual features. First, in [6, 7, 11, 13, 18, 20, 22] the underlying mathematical model is a semi-infinite initial value problem, whereas our model is an initial-boundary value parabolic equation defined on a bounded domain. Secondly, to reduce the inverse option pricing problem to an inverse parabolic problem with terminal observations, the conjugate theory (see [4]) should be adopted. For our problem, since the underlying asset has the property of double barrier option and the measurement data only depend on the price of gold, it is not necessary to use the Green function theory. Finally, the authors in [6, 7, 10, 11, 22] use the Tikhonov regularization to identify the implied volatility, while it may over-smooth the exact solution and may not preserve the singularities of the solution well when the exact solution has some singularities. However, the total variation regularization model might be able to characterize the properties of the volatility better. Therefore, using our model to identify the implied volatility is a question worth thinking about. The inverse problem considered in the paper and the obtained results can be regarded as the beneficial supplement of the inverse option pricing problem.
The paper is organized as follows. In Section 2, we transform the parabolic problem (1.3) into an optimal control problem. In Section 3, the existence of the minimum for the control functional is given. The necessary condition which must be satisfied by the minimum is deduced in Section 4. In Section 5, we investigate the uniqueness of the minimum under some assumptions. The stability and convergence for the regularized approach are also presented in the last section.
2 Optimal control problem
Consider the following optimal control problem P:
3 Existence
Theorem 3.1
Proof
4 Necessary condition
Theorem 4.1
Proof
5 Uniqueness
It is well known that the optimal control problem is nonconvex. In general, one may not expect a unique solution. However, the local uniqueness of the minimizer for the control functional can be obtained if \(\tau^{\ast}\ll1\).
Suppose that \(U_{1}^{*}(x)\) and \(U_{2}^{*}(x)\) are two given functions which satisfy condition 4.1. Let \(a_{1}(x)\) and \(a_{2}(x)\) be two minimizers of the control problem (2.1) corresponding to \(U_{i}^{*}(x)\) (\(i=1,2\)), respectively, and \(\{U_{i},\varphi_{i}\}\) (\(i=1,2\)) be the solutions of system (4.1)/(4.2) in which \(\bar {a}=a_{i}\) (\(i=1,2\)), respectively.
Lemma 5.1
Lemma 5.2
Proof
Lemma 5.3
Proof
The proof of Lemma 5.3 is thus completed. □
Lemma 5.4
Proof
Theorem 5.1
Proof
6 Stability
In the previous section, we have obtained the existence and uniqueness of the optimal solution. In this section, we will discuss the stability of the solution.
Theorem 6.1
Proof
7 Concluding remarks
A lot of research works have been made to identify the implied volatility by regularization methods. In this paper, we propose the total variation regularization strategy for solving the implied volatility in the Black-Scholes model. Based on the optimal control framework, the inverse problem of determining the implied volatility in financial products linked with gold price is discussed, which is still an interesting issue in financial mathematics. Since the profit of the derivative product depends not only on some cumulative indexes of gold price but also on some trigger indexes, such kind of product is, in a sense, similar to the double barrier option. So, it is quite meaningful to reconstruct the volatility function by the information obtained from the financial market.
The difficulty is due to the lack of conventional stability, nonlinearity and nonconvexity. Based on the optimal control framework, the inverse problem is reduced to an optimization problem, then we propose the total variation regularization for identifying the implied volatility, and the existence, uniqueness and stability of the minimizer are proved. The results obtained in the paper are interesting and useful and may be applied to a variety of derivatives pricing problems.
Declarations
Acknowledgements
We would like to thank the editor and anonymous referees for their valuable comments and helpful suggestions, which improved the earlier version of the manuscript. This work is supported by the National Natural Science Foundation of China (No. 11261029, 11461039).
Authors’ contributions
All authors read and approved the final version of the manuscript.
Competing interests
None of the authors has any competing interests as regards the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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