Valuation of Long‐Maturity KIKO Options Under the Stochastic Volatility Model

• Published date: 01 August 2014
• Author: Junmo Song,Joon‐Haeng Lee
• DOI: http://doi.org/10.1111/ajfs.12056
• Date: 01 August 2014

Valuation of Long-Maturity KIKO Options

Under the Stochastic Volatility Model*

Joon-Haeng Lee

Department of Economics, Seoul Women’s University

Junmo Song**

Department of Computer Science and Statistics, Jeju National University

Received 23 February 2012; Accepted 5 January 2014

Abstract

This paper explores the valuation of KIKO currency options under the stochastic volatility

(SV) model. In particular, Heston’s (Review of Financial Studies, 6, 1993, 327) SV model is

adopted to price two types of KIKO options. The paper proposes the iterated feasible two-

step estimation method, a technique for estimating the parameters in Heston’s (Review of

Financial Studies, 6, 1993, 327) model. Given constant fluctuations in volatility over time and

the relatively long maturity of KIKO options, the SV model is more appropriate for evaluat-

ing long-maturity KIKO options, the value of which depends heavily on volatility estimates

for the constant-volatility (CV) model. The results indicate that KIKO options considered in

this paper are less attractive under the SV model than under the CV model, although it

remains unclear whether the options are significantly overpriced.

Keywords Currency option; Knock-In Knock-Out option; KIKO option valuation; Stochastic

volatility model; Heston’s model; Estimation

JEL Classification: G13, C13, C61

1. Introduction

Several hundred small- and medium-sized enterprises (SMEs) in Korea faced huge

losses from currency knock-in/knock-out (KIKO) options following the global finan-

cial crisis in 2008. Many Korean exporters use currency forward contracts to hedge

against foreign-exchange risk. It is somewhat surprising that structured products such

*Acknowledgements: We are grateful to the associated editor and anonymous referees for their

valuable comments. The work of the first author was supported by a research grant from

Seoul Women’s University 2013. The work of the corresponding author was supported by

Basic Science Research Program through the National Research Foundation of Korea (NRF)

funded by the Ministry of Science, ICT and Future Planning (NRF –2013R1A1A1008032).

**Corresponding author: Junmo Song, Department of Computer Science and Statistics, Jeju

National University, 102 Jejudaehakno, Jeju-si, Jeju 690-756, Korea. Tel: +82-64-754-3596,

Fax: +82-64-725-2589, email: jmsong@jejunu.ac.kr.

Asia-Pacific Journal of Financial Studies (2014) 43, 492–529 doi:10.1111/ajfs.12056

492 ©2014 Korean Securities Association

as KIKO options have been widely used as hedging alternatives to forwards since 2007.

All KIKO products are designed to provide issuers (banks) with huge gains when the

Korean won (KRW) depreciates sharply, whereas they provide holders (SMEs) with

limited gains when the KRW appreciates. After a decade-long appreciation, the KRW

has depreciated sharply since the second half of 2008. In the structured-product mar-

ket, there is a danger that an issuer may use technological and information advantages

to maximize profits at the expense of unsophisticated investors. As a result, this has

been a topic of intense debate in the contexts of investor protection.

Besides incomplete sales, the issue of whether the valuation of KIKO options is

reasonable has been controversial, and such pricing issues have been the subject of

legal proceedings. Here the key point is to determine whether the valuation of

KIKO options is unfavorable to investors, and therefore the estimation of exchange-

rate volatility has become important. Accordingly, the pricing of KIKO options has

attracted renewed research attention.

Lee (2009) raises the issue of valuing KIKO options and shows the extent to which

valuation is affected by the choice of the interest rate differential between two coun-

tries and changes in exchange-rate volatility. He assumes constant exchange-rate vola-

tility and emphasizes that KIKO option pricing depends heavily on volatility estimates

under a constant-volatility model. Khil and Suh (2010) analyze KIKO options from

the perspective of risk assessment and management and investigate pricing issues.

They draw risk management lessons from KIKO option transactions in the Korean

financial market and find that many KIKO contracts are not justifiable as a hedging

instrument at the time of purchase. They also investigate some pricing issues for KIKO

options and find that most KIKO options are not significantly overpriced. They use

the GARCH pricing model based on historical returns and the Black–Scholes model

with implied volatility for their estimation results. However, as indicated in Christof-

fersen and Jacobs (2004), GARCH parameters estimated based only on historical data

can lead to poor pricing results. In addition, the implied volatility of long-maturity

currency options cannot be considered as an appropriate measure for the Korean FX

market because these options are rarely traded. Therefore, it is unclear which volatility

estimates should be used to price long-maturity KIKO options.

The Korean capital market has been impacted by a similar problem in the

mark-to- market accounting of ELS (equity-linked security) products because

the volatility structure of underlying assets is not available. It is well known that the

valuation of long-maturity options embedded in ELS products is highly sensitive to

volatility estimates in the constant-volatility model (in this paper, a one-factor

model).

1

1

In practice, historical data used to estimate volatility have recently tended to cover longer

time periods. For example, some securities firms using 90-day moving-average estimates have

replaced them with 180-day estimates because the former are too volatile for pricing long-

maturity options.

Valuation of KIKO Options Under the SV Model

©2014 Korean Securities Association 493

Given that volatility varies over time and there is no consensus on estimates of

long-term volatility, it is troublesome to assume volatility to be fixed over the rela-

tively long life of options. Therefore, this paper valuates KIKO options by using a

two-factor model in which the dynamics of the exchange rate and volatility are

given by stochastic models.

Since Black and Scholes (1973), two main types of volatility models have been

developed to capture some stylized volatility features, including diffusion-type sto-

chastic volatility (SV) models as continuous-time models (see, e.g. Hull and White,

1987; Heston, 1993; Bates, 1996) and GARCH-type volatility models as discrete-

time models (see, e.g. Duan, 1995; Heston and Nandi, 2000).

Continuous-time SV models are ideal in theoretical aspects in that they have

more than two sources of randomness, which can serve as risk factors in the under-

lying asset’s market and its option market. On the other hand, in GARCH-type

models, only one source of randomness is used to describe the dynamics of the

underlying asset, indicating that option markets in these models are redundant.

GARCH-type models are somewhat limited in their ability to explain two markets

simultaneously, and therefore this paper considers a diffusion-type model, particu-

larly Heston’s (1993) SV model.

In terms of estimating parameters, continuous-time SV models have some diffi-

culty because volatility is not observable. In Heston’s model, parameters can be cali-

brated based on options data by using a closed-form solution for the price of the

vanilla option, and this method is widely used among practitioners. However, when

many options are considered, an optimization procedure can be computationally

burdensome. In addition, because the objective function to be optimized in He-

ston’s model is nonlinear and has many local minima it may not technically be easy

to find the global minimum. Nevertheless, Heston’s model has been preferred in

markets because it can successfully describe the mean-reverting property of volatility

and has a closed-form formula for plain options. Another merit is that it can pro-

duce a distribution of underlying assets more flexible than the log-normal distribu-

tion induced by the geometric Brownian motion. However, the Heston model does

not allow for jumps in returns.

Many empirical studies address the role of a jump component in terms of the

return distribution and option-pricing performance. In terms of distributional

aspects, most studies find that the stochastic volatility jump diffusion (SVJD) model

outperforms the SV model (see, e.g. Andersen et al., 2002; Pan, 2002; Jiang, 2002).

Several studies (Das and Foresi, 1996; Bakshi et al., 1997; Pan, 2002) report that the

incorporation of a jump component is necessary when pricing options that are close

to maturity.

This paper does not consider a jump component because the exchange-rate data

considered in this paper do not show very high kurtosis, which can be accommo-

dated by the SVJD model, and, on the contract day, most KIKO options consist of

several constituent contracts the maturity of which is longer than or equal to

1 month. Indeed, the present analysis (see Appendix C and D) provides support for

J.-H. Lee and J. Song

494 ©2014 Korean Securities Association