^{1,2}

^{1,2}

^{3}

^{1}

^{2}

^{3}

In market transactions, volatility, which is a very important risk measurement in financial economics, has significantly intimate connection with the future risk of the underlying assets. Identifying the implied volatility is a typical PDE inverse problem. In this paper, based on the total variation regularization strategy, a bivariate total variation regularization model is proposed to estimate the implied volatility. We not only prove the existence of the solution, but also provide the necessary condition of the optimal control problem—Euler-Lagrange equation. The stability and convergence analyses for the proposed approach are also given. Finally, numerical experiments have been carried out to show the effectiveness of the method.

Volatility is a very important risk measurement in financial economics. The estimation of it is critical for option pricing and management of the derivative positions. In order to estimate the volatility effectively, two main classes of parametric approaches have been developed: discrete-time models and continuous-time models.

There are numerous literatures on the discrete-time models and here we provide only a partial overview related to our studies. The ARCH model developed by Engle [

However, those models generally suffer from a curse of dimension that severely constrains their practice and the coming of high frequency financial data makes it worse. Nowadays, the availability of intraday data has facilitated the use of the so-called Realized Volatility (RV) which was introduced in the literature by Taylor and Xu [

There is also a common practice to infer the volatility from quoted option prices based on the Black-Scholes theoretical framework [

The stochastic process of the asset price

An option is classified either as a call option or a put option. A call (put) option is a contract which gives the buyer (the owner) the right, but not the obligation, to buy (or sell) an underlying asset or instrument at a specified strike price on or before a specified date.

Suppose

Consider a portfolio that involves short selling of one unit of a European call option and long holding of

The option prices obtained from the Black-Scholes pricing model are functions of five parameters:

A natural question then arises: how can we get the implied volatility of the future underlying asset by option quotes? This is the typical IPOP (inverse problem of option pricing).

The PDE inverse problem of option pricing was first considered by Dupire in [

The total variation regularization might be able to characterize the properties (the jump, overnight, weekend effect, etc.) of the volatility better. So whether the total variation regularization strategy could be applied to identify the implied volatility is a question worth pondering.

This paper is organized as follows. Section

In [

Based on their work, Chiarella et al. [

Tikhonov regularization strategy may oversmooth the solution, so it may not preserve the singularities of the solution well. We adopt total variation regularization strategy proposed by Rudin et al. [

Set

Consider the following bivariate total variation regularization problem:

The term

Our total variation regularization strategy has two advantages compared with Tikhonov regularization strategy proposed by Lagnado and Osher: one is that it contains no terms involving the Dirac delta function [

The minimization problem (

Under the constraints of the total variation regularization problem (

This lemma can easily be similarly proved like proposition A.3 in [

The total variation minimization problem (

The weak lower semicontinuity of the norm and weakly continuity of the operator

We can calculate approximate solutions by solving the Euler-Lagrange equation. Generally speaking, the total variational regularization problem (

Set

Necessary optimality condition: let

By using the variational method, the corresponding Euler-Lagrange partial differential equation is

The next theorem states well posedness of the regularized problem.

Under the constraints of the total variation regularization problem (

In the next theorem, we show that under the same conditions on

Under the constraints of the total variation regularization problem (

Moreover, assume that

Let

From the lower semicontinuity of

Using this and (

If the minimizing solution of (

Next we will discretize the term

As in Figure

Gridding.

Let

To obtain the local optimal solution, we have to handle the problem of calculating the partial derivative

Let

An important issue in practice is the choice of the regularization parameter

Total variation for solving the implied volatility.

Choose a function

Determine

Compute

Compute

Adopt the Gauss-Jacobi iteration scheme:

If

In this section, we present numerical experiments to illustrate the theory and algorithm presented in above sections. First we assume that the true volatility function,

In numerical experiments, the interest rate

Volatility function

We solve the volatility by using Algorithm

The error between

If we fix

According to Figures

A lot of research works have been made to determine the implied volatility by regularization strategies. Based on the advantages and great success of the total variation regularization strategy in image processing, we propose the total variation regularization strategy to estimate the implied volatility under the framework of the Black-Scholes model. We identify the implied volatility by solving an optimal control problem and investigate a rigorous mathematical analysis. Not only the existence is discussed, but also the stability and convergence for this regularized approach are given. We also deduce the Euler-Lagrange equation. Furthermore, the results of numerical experiments are presented.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the NNSF of China (nos. 60872129, 11271117) and Science and Technology Project of Changsha City of China (no. K1207023-31).