Time‐changed Lévy jump processes with GARCH model on reverse convertibles

• Author: Xiaoli Wang,Wei W. Simi
• Date: 01 November 2013
• Published date: 01 November 2013
• DOI: http://doi.org/10.1016/j.rfe.2013.04.005

Time-changed Lévy jump processes with GARCH model on

reverse convertibles

Wei W. Simi

a,

⁎, Xiaoli Wang

b,1

a

Economics/Finances and Risk Management, City University of New York, United States

b

School of Management, Marist College, United States

abstractarticle info

Available online 10 May 2013

Keywords:

Lévy jump process

Fourier transforms

Exotic options

Reverse convertible

GARCH

For decades, financial institutions have been very motivated in creating structured high-yield financial prod-

ucts, especially in the economic environment of lower interest rates. Reverse convertible notes (RCNs) are

the type of financial instruments, which in recent years first in Europe and then in the US –have become

highly desirable financial structured products. They are complex financial structured products because they

are neither plain bonds nor stocks. Instead, they are structured products embedding equity options, which

involve a significant amount of asset returns' uncertainty. Given this fact, pricing of reverse convertible

notes becomes a really big challenge, where both the general Black–Scholes option pricing model and the

compound Poisson jump model which are designed to catch large crashes, are not suitable in valuing these

kinds of products. In this paper, we propose a new asset-pricing framework for reverse convertible notes

by extending the pure Brownian increments to Lévy jump risks for the underlying stock return movements.

Our framework deals with time-changing volatilities of stock options with Lévy jump processes by consider-

ing the stocks' infinite-jump possibilities. We then use a discrete-time GARCH with time-changed dynamics

Lévy Jump processes in order to derive the assets' valuations. The results from our new model are close to the

market's valuations, especially with the normal-inverse-Gaussian model of the Lévy jump family.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

This paper deals wit h a pricing methodology for stru ctured prod-

ucts and more precisely, f or the valuation of reverse c onvertible

notes (bonds). For dec ades, investors have been diligent ly searching

for high-yield financ ial investment instruments in an economic e n-

vironment of low inter est rates. Thus, the incre asing demand for

high-yield produc ts has given financial instituti ons plenty of motiva-

tion and opportunit y to create financial st ructured produc ts. Since

the financial crisis and economic doldrums of 2008, the United States

short-term interes t rates have remained at the low level of 0–1% for

a long and extended period of time. Chart-1in theAppendix A, which

demonstratesthe historicalyield of the 13-week United StatesTreasury

Bills,confirms this. Our data furtherindicates that duringthe mid-2011,

the rateseven became negative. These negativeyields mean that inves-

tors wouldlose some of their investmentsby investing the assetsin real

term. This low-yield economic environment motivatedmany investors

to look for those risky, high-yield driven structured products. A struc-

tured product is a financial asset in which the attributes of several

other financial assets arecombined in one. A reverse convertible note

(RCN) or reverse convertible bond (RCB) is one of these structured

products. It is a combination of a fixed-income note and an equity

option. It's also an over-the-counter product. In general, an RCN is a

short-term investment vehicle with a one or two year maturity date.

However, the realattraction of an RCN is that it offers generousyields.

Some of the products even offer a double-digit yield. This is highly desir-

able for such a short investmentperiod, even when compared to 10-year

US Treasury bonds, which offer much less yield during the same period.

Table 1 in the Appendix A shows a list of sample reverse convertible

notes traded in the markets as of May 2011. From the details of this

list, we can see the attractiveness of the coupon rates of reverse convert-

ibles. Based on the data we obtained that there are more than 94% of

RCNs have coupon rates with two digits during the period of year 2008

through 2012. In addition, since the US Central Bank (the Federal Re-

serve) started the program of open market purchasing of the United

States Treasuries, in order to stimulate the US economy –also called

quantitative easing (QE) –and decided to purchase $40 billion mortgage

backed securities on a monthly basis, beginning in the summer of 2012,

the Treasury yields are going to be driven even lower across the Treasury

yield curve. These kinds of economic environments have pushed inves-

tors to take risks in chasing high yields and more broadly demanding

risky products, such as RCNs. Despite the attractive yield an RCN can pro-

vide, however, its risks and complexities can be a real challenge to most

investors, and especially to retail investors. Chart-6in the Appendix A

showsthechangesinanRCNduetothechangesofitsunderlyingstock.

When the underlying stock of an RCN drops to the pre-determined trigger

value, the RCN will loss a portion or all principal. Therefore, an RCN is not

principalprotected,ingeneral.Aninvestorcould,therefore,losehis/her

entire investment, even though the note's issuer does not default. This is

Review of Financial Economics 22 (2013) 206–212

⁎Corresponding author. Tel.: +1 718 997 5241.

E-mail addresses: wei.simi@qc.cuny.edu (W.W. Simi), xiaoli.wang@marist.edu

(X. Wang).

1

Tel.: +1 732 648 3326.

1058-3300/$ –see front matter © 2013 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.rfe.2013.04.005

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