On full calibration of hybrid local volatility and regime‐switching models

• Author: Song‐Ping Zhu,Xin‐Jiang He
• Date: 01 May 2018
• Published date: 01 May 2018
• DOI: http://doi.org/10.1002/fut.21901

Received: 23 May 2017

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Revised: 21 November 2017

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Accepted: 27 November 2017

DOI: 10.1002/fut.21901

RESEARCH ARTICLE

On full calibration of hybrid local volatility and

regime-switching models

Xin-Jiang He

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Song-Ping Zhu

School of Mathematics and Applied

Statistics, University of Wollongong,

Wollongong, New South Wales, Australia

Correspondence

Song-Ping Zhu, School of Mathematics and

Applied Statistics, University of

Wollongong, Wollongong, NSW 2522,

Australia.

Email: spz@uow.edu.au

Funding information

ARC, Grant number: DP140102076

Calibrating local regime-switching models is a challengingproblem, especially when

the volatilityfunctions are assumed to dependon both of the underlying price and time.

In this paper, the inverse problem of determining local volatility functions is firstly

established and thensolved through the Tikhonov regularizationto obtain the optimal

solution, whichis achieved iteratively through a newly designednumerical algorithm.

While our numericaltests with artificial data show thatour algorithm can provide quite

accurate and stable results, its performance with the involvement of real marketdata

have been further demonstrated using options written on the S&P 500 index.

KEYWORDS

inverse problem, local regime-switching model, Tikhonov regularization

JEL CLASSIFICATION

C51, C61, G13

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INTRODUCTION

In 1973, Black & Scholes made a great breakthrough in the area of option pricing by proposing an elegant model with the log-returns

of the underlying assumed to follow a standard normal distribution. Under this simple assumption, a closed-form pricing formula for

European options can be derived, and Zhu (2006) went even further and obtained an analytical pricing formula in infinite series form

for American options under the same model. However, it is this over-simplified assumption that has resulted in some mis-pricing

problems; plenty of empirical evidence shows that the log-returns of the underlying should be skew and fat-tailed (cf. Peiro [1999];

Rachev, Menn, & Fabozzi [2005]). Another main criticism for the Black-Scholes (B-S) model is the constant volatility assumption

since there exists the well-known phenomenon of the “volatility smile”exhibited by the implied volatility (cf. Dumas, Fleming, &

Whaley [1998]). As a result, different kinds of modifications have been proposed to improve the behavior of the B-S model.

In particular, one of the most popular methods is to incorporate non-constant volatility into the standard B-S model. Stochastic

volatility and local volatility are the two types that are mainly consideredin the literature. Specifically, stochastic volatility models make

volatility another random variable described by a particular process. For example, a mean-reverting OU (Ornstein-Uhlenbeck)process

was adopted by Scott (1987), while Heston (1993) introduced the CIR (Cox-Ingersoll-Ross) process for the volatility dynamic. In

contrast, local volatility models assume that the volatility be a deterministic function of the underlying price and time. In particular, Dupire

(1994) developed the so-called “Dupire equations”for the B-S model to extract the local volatility function from market option prices,

and the discrete version of the Dupire equation was taken into consideration by Rubinstein (1994) and Derman and Kani (1994).

It should be pointed out that regime-switching models, which can capture the essence that a financial market changes among

different states, have become very popular recently. The volatility in such models can jump from one state to another, controlled

by a Markov chain (cf. Hamilton [1989]), which clearly shows that regime-switching models are indeed included in the category

of stochastic volatility. As empirical evidence has suggested that regime-switching models can provide better model fitness to

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© 2018 Wiley Periodicals, Inc. wileyonlinelibrary.com/journal/fut J Futures Markets. 2018;38:586–606

market observed data (cf. Chernov, Gallant, Ghysels, & Tauchen [2003]; Eraker [2004]), they have already been introduced into

the area of financial derivative pricing by Naik (1993), and further studied by Herzel (1998), Zhu, Badran, and Lu (2012), and

other authors.

However, in their infancy, regime-switching models were kept in a relative simple form with the volatility in each state being

assumed to be constant (they are referred to as the standard regime-switching models hereafter, in order to distinguish them from

the local regime-switching models focused in this paper.). Again, like the constant volatility under the B-S model being

generalized to a calibratable volatility function that depends on time as well as the underlying in a local volatility model, the

constants associated with each state in a standard regime-switching model may also limit its adaptability to market data which

demand more flexibility in order to avoid mis-pricing problems. Thus, the concept of combining a local volatility model and a

regime-switching model to formulate a hybrid model, referred to as the so-called local regime-switching model in literature, is an

elegant idea, trying to take the advantage of some nice features associated with each of the models standing alone (cf. Elliott,

Chan, & Siu [2015]). On the other hand, once a model is proposed, its calibration should be considered too, because model

calibration is an extremely important aspect for any mathematical model to be useful in practice. In general, a key challenge

associated with calibration of local volatility models is that it is mathematically an inverse problem, which is characterized by the

ill-posedness nature as a result of insufficient option data available, trying to determine the whole volatility surface to produce

model prices, in order to best fit the corresponding market prices. Calibration of local regime-switching models is no exception.

In this area, He and Zhu (2017) pioneered the calibration problem of local regime-switching models. To overcome the difficulty

associated with the task, they decided to firstly work on pricing option contacts that are of relatively short tenor, in which case the

local volatility functions assumed to be independent of time (cf. He & Zhu [2017]), so that they could focus on establishing a new

closed system to realize a proper use of the Dupire-type formula proposed in local regime-switching models (cf. Elliott et al.

[2015]).

However, the simplified assumption that the volatility is independent of time is not always appropriate, and the volatility

values may change even for some short-tenor options (cf. Rubinstein [1985]), which demands an extension of the work presented

in He and Zhu (2017) to a more general case, where the local volatility functions are no longer assumed to be time independent.

This has motivated us to face the new challenge of trying to calibrate local regime-switching models with respect to both strikes

and maturities, so that the range of application of these models can be broadened, in order to suit more general cases in finance

practice. As expected, the Dupire-type formula for local regime-switching models can not be directly implemented since it may

not yield accurate results due to the denominator in the formula being directly affected by the second-order derivative of option

prices, similar to the case when the local volatility is assumed to be time-independent. Therefore, following Jiang and Bian

(2012) and He and Zhu (2017), the calibration problem for the local regime-switching model is formulated into an inverse

problem, which is ill-posed since observed option prices are always insufficient (cf. Coleman, Li, & Verma [1999]; Crépey

[2003]). However, a full calibration of local regime-switching models with the volatility functions being assumed to depend on

both of the underlying price and time poses some other challenges, which are to be addressed as the core of the present paper.

There are mainly two new challenges. The first one is that whenrecovering local volatility functions with respectto time, we

need all the price informationfor options with expiry less than a particularly given value,whereas there always exists a smallest

expiry associatedwith any set of data such that there is no market price information for optionswhose expiry time is less than this

criticalexpiry. This means that the values of the local volatility functionscorresponding to the smallest expiry timecontained in the

market data cannotbe easily and accurately determined numerically.On the other hand, another challenge is that we need to design

an accurate algorithm to recover local volatility functions for each expiry, with the local volatility functions within the two

consecutiveexpiry times being non-constant. Toovercome these difficulties, a two-stepapproach is developed and presentedhere.

In the first step, we assume that the local volatility is time independent when the expiry time is less than the smallest expiry

available, and recoverthe volatility functions with the technique developedin He and Zhu (2017). In the second step, a set of new

optimal controlproblems is developed with the Tikhonov regularization(cf. Tikhonov, Goncharsky, Stepanov, & Yagola [1995]),

and the volatility functionscan be recovered for every maturity of the observed optionprices step by step with a newly designed

numerical algorithm. It should be remarked here that in the process of determining the local volatility functions between two

consecutive expiry times, all the values of the volatility functions between the nodal values are needed for the computation as a

result of the volatilityfunctions being assumed to be time-dependent.Thus, linear interpolations are performedto obtain all values

between different expiry times. For the illustration purpose, we will only limit our discussion to the case of two states in a local

regime-switching model with a view that it is not difficult to extend our approach to the case of arbitrary but finite states.

The rest of the paper is organized as follows. In section 2, we briefly introduce a closed system for the calibration of local

regime-switching models. In section 3, the calibration problem will be formulated into an ill-posed inverse problem, which is

transformed into another inverse problem with a Dupire-type formula. In section 4, Tikhonov regularization is adopted, and this

particular inverse problem can be solved as an optimal control problem. Two sets of necessary conditions with respect to the

HE AND ZHU

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