Valuation of reverse convertibles in the variance gamma economy
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Abstract
Prior research on structured products has demonstrated that equitylinked notes (ELNs) sold to retail investors in initial public offerings are typically issued at above their fair market value. A particular type of ELN – reverse convertibles – embed downandin put options and offer investors relatively high coupon payments in exchange for bearing some of the downside risk of the equity underlying the note. We analytically study the magnitude of the overpricing of reverse convertibles – one of the most popular structured products on the market today – within a stochastic volatility model.We extend the current literature to include analytical valuation formulas within a model of stochastic volatility – the variance gamma (VG) model. We show that these complex notes are even more overpriced than previously estimated when stochastic volatility is taken into account. As a result of their complex payoffs and the lack of a secondary market to correct the mispricing, reverse convertible notes continue to be sold at prices substantially in excess of their fair market value.
Keywords
stochastic volatility structured products barrier options reverse convertiblesINTRODUCTION
In this article we present new valuation formulas for structured products with pathdependent payoffs incorporating stochastic volatility. The methodology used in this article could be applied to any structured product with an embedded barrier option to determine the impact of volatility skew on product valuation. We study reverse convertibles mainly because they are among the most popular structured products and because they provide insight into the pricing and sales practices of other equitylinked structured products.
Reverse convertibles are shortterm notes whose principal repayment is linked to a stock, an index or a basket of stocks. If the reference security’s price or level falls below a prespecified level – called the ‘trigger price’ or ‘protection level’ – during the term of the note, investors may receive substantially less than the facevalue of the notes. Reverse convertibles are fundamentally notes composed of a coupon paying bond and an embedded short put option.
Thus, while buy and hold investors in traditional shortterm notes are only exposed to the issuer’s credit risk, investors in reverse convertibles are also exposed to the risk of a decline in the price of the reference security. Investors in reverse convertibles are partially compensated for the risk of the embedded short put options with higher periodic coupons. The risk of these embedded options was realized by many investors in late 2008 and early 2009 as some notes matured after substantial stock market declines.
By valuing over 2000 reverse convertibles issued between 2001 and 2010, Deng et al (2010) report that the average fair value of the products was just 93 per cent of the offering price. Szymanowska et al (2009) and Hernández et al (2007) find a similar level of mispricing for this type of structured product. Henderson and Pearson (2010) estimate that investors who purchased an aggregate of $2 billion of shortterm Stock Participation Accreting Redemption Quarterly Securities (SPARQS) reverse convertibles from Morgan Stanley in 69 offerings from 2001 to 2005 paid on average 8 per cent more than the securities’ true value.
Regulators have been paying attention to these products as well. The Financial Industry Regulatory Authority (FINRA) Chairman Richard Ketchum commented, ‘Reverse convertibles are complex investments which, like many structured products, often entail significant risk of loss. For the typical retail investor, for instance, it would be unwise to put a significant portion of life savings into riskier structured products such as reverse convertibles’.^{1} FINRA followed by issuing an ‘Investor Alert’^{2} and a ‘Regulatory Notice’^{3} to highlight these concerns.
Reverse convertibles accounted for approximately 6.0 per cent ($2.34 billion) of SECregistered structured notes in 2012, 12.0 per cent ($5.46 billion) in 2011 and 13.7 per cent ($6.76 billion) in 2010 according to Bloomberg Briefs. Reverse convertibles stand as third most popular type of SECregistered notes, behind equitylinked notes (ELNs) and securities tied to rates.^{4} Much like reverse convertibles, ELNs are debt instruments that are tied to the the underlying equity and expose investors to the issuer’s default risk. Typically ELNs are principalprotected and the payoff at maturity depends upon the return of the underlying equity and a participation rate.^{5} Ratelinked notes have a payoff that depend upon interest rate(s). These notes could be tied to the value of a particular interest rate, the difference of two interest rates or on the steepness of a given yield curve.
Asset return volatilities are stochastic (Engel, 2004). Analysis of asset return distributions suggests that infiniteactivity jump specifications are well suited to capture the daily market value fluctuations of many financial assets (Wu, 2007). The VG model that we consider in this article is an infiniteactivity pure jump process.^{6} Though there are many models of stochastic volatility, we chose to work with the VG model because of its analytic tractability and the parsimony of its underlying model. The VG process can be thought of as a geometric Brownian motion with a gamma timechange and therefore could be seen as a generalization of the Black–Scholes (BS) model. The gamma timechange is also the most simple continuous statespace analog of the Poisson process (Jaimungal, 2004). The VG model is minimal as it leads to a characterization of the two most important inherent biases in the BS formula – lack of skewness and excess kurtosis – using only one extra parameter (see Black and Scholes, 1973 and Merton, 1976).
The VG model was first introducted in Madan and Seneta (1990) and was later generalized to the Carr, Geman, Madan, Yor (CGMY) model in Carr et al (2002) to include a set of models within a convenient parameterization. Recently, the VG model has been extended and applied to study corporate defaults (Fiorani et al, 2010). Closedform solutions for European options in other stochastic volatility models have been presented in the literature. See, for example, a Heston stochastic volatility model (Heston, 1993), which assumes dynamic variance follows a Cox, Ingersoll, Ross (CIR) process. Madan and Milne (1991) price European call options in the symmetric VG economy for individuals with varying degrees of risk aversion. Fourierbased valuation methods on European call options have been applied to the VG model by Carr and Madan (1999).
Hernández et al (2007) present valuation formulas for several types of reverse convertibles with embedded barrier options within the BS model and find significant underpricing of these instruments. Baule and Tallau (2011) studied the valuation of a related type of structured product – bonus certificates – in the context of the Heston model. Bonus certificates are similar in structure to reverse convertibles. The main difference is that bonus certificates embeds a downandout put option rather than the downandin put option embedded in reverse convertibles. Baule and Tallau (2011) used simulations to price the bonus certificates under the Heston model. Lipton (2001) presents a closedform valuation of barrier options using a Green’s function approach within a special case of the Heston model where the asset price and volatility are assumed to be uncorrelated. There is no closedform solution for barrier options in the generic Heston model. Several approaches to the valuation of barrier options can be found in the literature, including Wiener–Hopf factorization theory Kudryavtsev and Levendorskiĭ (2009), finitedifference methods Itkin and Carr (2010) and Fourierbased approaches (Fang and Oosterlee, 2011). For an overview of the valuation of exotic options within Lévy models in general, see Schoutens (2006).
The main contributions of our article are as follows. We derive the first closedform valuation of reverse convertibles within the VG model. More precisely, we provide a valuation formula for the downandin (barrier) put option embedded within the reverse convertible. We first derive a reflection lemma specific to the stochastic volatility model being considered. The reflection lemma maps the pathdependent payoff for a barrier option to a pathindependent payoff, conditional on a fixed gamma time. Using this reflection lemma, we derive a solution for the barrier option within VG stochastic volatility model. We isolate the pricing effect of the stochastic volatility assumption by comparing the results of the VG model and the BS model. We priced approximately 1800 reverse convertibles issued by several large investment banks in 2010 and 2011. We calibrated each model using options data collected for the underlying asset and analyzed the impact of volatility skew on the value of reverse convertibles. We find that estimates of the issue date overpricing in reverse convertibles increases when we incorporate stochastic volatility.
This article is organized as follows. We begin with a section introducing the VG model in detail, including the valuation of European call options. We then present our analytic valuations, based upon a new lemma we present, of pathdependent options within the VG model. We calibrate the model with options data and then price reverse convertibles using the calibrated model. The final section is reserved for our conclusions.
VG MODEL
Model specifications
The VG model is a purejump process that can also be written as a Brownian motion with a gamma timechange. The first author to consider the effect of discontinuous evolution of stock prices was Merton (1976). Monroe (1978) showed that any semimartingale has a representation as a timechanged Brownian motion. Monroe’s result implies that the logreturns of the underlying asset are normally distributed with respect to financial time (which is positively correlated to businessactivity time). The financial clock ticks more quickly than the observable clock in times of high business activity and ticks more slowly than the observable clock in times of low business activity (Jaimungal, 2004).
where Γ is the usual gamma function. Conditioning on g, the distribution becomes a normal distribution with mean θg and variance σ^{2}g.
In the limit ν approaches zero, one recovers the characteristic function corresponding to the BS probability distribution.
The proof of these equations is trivial once one derives the characteristic function for the VG process. For an explicit proof, see Madan et al (1998). These formulae imply positive and negative jumps in the price of an underlying asset have separate distributions (arrive at different rates).
Notice that the sign of the skewness of the returns distribution is determined by θ. Furthermore, the skewness is only nonzero for ν > 0.^{7} Kurtosis beyond that which is present in the BS model can also be accounted for through this model as shown above.
European call option valuation
and Φ is the standard normal cumulative distribution function.
where, again, Ω=(1−θν−σ^{2}ν)/2. Madan et al (1998) show that the function Ψ can be written in terms of degenerate hypergeometric functions of two variables and modified Bessel functions of the second kind. We refer the reader to their paper for explicit formulae for Ψ in terms of these special functions of mathematics.
REVERSE CONVERTIBLES IN THE VG ECONOMY
Reverse convertibles can be decomposed into three investments: a long position in a zero coupon bond, a long position in a series of coupon payments and a short position in a downandin (barrier) put option. The formula for a reverse exchangeable decomposed into these three components is presented in Hernández et al (2007).^{9}
where Open image in new window is the CDS rate of the issuer and we have made the assumption that the strike of the downandin put option is S_{0}. Following Baule and Tallau (2011), we include the issuer’s CDS rate in the pricing formula to reflect credit risk of the issuer.
Lemma 1:

In a VG economy, suppose that X is a portfolio of European options expiring at time T, conditional on the gamma timechange, with payoff:
For H>0, let Y be a portfolio of European options with maturity T and payoff:
where the power p(g) is given by
and the parameters {θ, σ, ν} characterize the VG economy, r is the constant continuously compounded riskfree rate and q is the constant continuously compounded dividend yield. Then X and Y have the same payoff whenever the spot equals H.
For a proof of the Lemma 1, see the Appendix A. This lemma allows us to write a dynamic payoff in terms of an equivalent static payoff. This new result facilitates the efficient valuation of barrier options within a model of stochastic volatility.
As reverse convertibles are structured products with an embedded downandin put option, we present the valuation formula for this type of barrier option. The value of any other type of barrier option, including double barrier options, can in principle be calculated based upon the methodology we develop below.
Each term in the series is a linear combination of hypergeometric functions and modified Bessel functions as given by Madan et al (1998).
This is an analytic solution for the value of a pathdependent option within the VG model.
VALUATION OF REVERSE CONVERTIBLES
Model calibration
Our calibration of the VG model follows closely that of Madan et al (1998). For the issue date of each reverse convertible, we calibrated the parameter σ in the BS model and the parameters {σ, θ, ν} of the VG model based upon contemporaneously traded atthemoney put options of the underlying asset. To be precise, we used the traded European put options that were close to atthemoney. For example, if the underlying asset had a price of S_{0} at closing the day before the valuation date, then we used options with strike prices in the range of 0.75S_{0} and 1.25S_{0}. We considered options of all available maturities.
Fit results for parameter values within the BS and VG models. We present the mean value for each parameter, along with the standard deviation
2*Parameter  Mean value  SD 

Black–Scholes model  
σ  0.453  0.096 
Variance gamma model  
σ  0.425  0.125 
θ  −0.705  0.653 
ν  0.108  0.077 
VG and BS valuations
We collected 1817 reverse convertibles issued in 2010 and 2011 by four large investment banks. Barclays issued 1324, JP Morgan issued 404, UBS issued 74 and Morgan Stanley issued 15 of the notes. The average issue size for those notes is $1 185 163.
After that we calibrated the parameters of the models and we obtained each issuer’s CDS rates. As a proxy for riskfree rates, we used treasury yields of approximately the same maturity as the reverse convertible. We also obtained the dividend yield for each of the underlying assets as of the offering dates from Bloomberg.
In the BS model, the return distribution is normal. In the VG model, the returns distribution exhibits higher skewness and kurtosis (fatter tails).
Results of reverse convertible evaluation by issuer for the Black–Scholes model and variance gamma model.The average BS and VG values are normalized to notes in units of $1000.
Issuer  Number issued  Average Black–Scholes value ($)  Average variance gamma value ($) 

Barclays  1324  935.06  934.44 
JP Morgan  404  960.56  957.65 
UBS  74  959.63  958.46 
Morgan Stanley  15  960.07  956.80 
CONCLUDING REMARKS
We have directly integrated the probability distribution function with an equivalent payoff function to value the downandin put options embedded within reverse convertibles. An alternative method is a Fourierbased approach using readily available Lévy measures.^{13} The benefit of using the Fourierbased approach is the applicability to many different stochastic volatility models. We have included a detailed derivation of the valuation of a European put option in the Appendix following closely the approach of Carr and Madan (1999). The Fourierbased approach has two limitations in this context. The first limitation is that the valuation of barrier options requires the derivation of ‘incomplete’ characteristic functions owing to the truncation of the payoff function at the trigger price. The second limitation is that we have only presented the reflection lemma for the VG economy, and would therefore need to derive a similar result for other stochastic volatility models. We see this as a direction for future research.
In this article, we studied the valuation of one of the most popular varieties of equitylinked structured products – reverse convertibles. We studied these products in the context of a particular model of stochastic volatility. Within the VG model, we derived a reflection lemma similar to that found in the BS model. We used this reflection lemma to convert the pathdependent payoff of the reverse convertible to a static payoff. We outlined a procedure for the valuation of barrier options commonly embedded in structured products and presented a closedform valuation formula for downandin put options.
Using our new formula, we compared the value of reverse convertible notes issued between 2010 and 2011 predicted by the BS model and the VG model. The overall effect of stochastic volatility is to increase the value of the embedded put option and therefore decrease the value of the note. The average value of a $1000 facevalue reverse convertible note issued during this time period was $941.90 in the BS model and $940.70 in the VG model.
Footnotes
 1.
 2.
 3.
FINRA Regulatory Notice 10–09, ‘Reverse Convertibles’, February 2010.
 4.
Bloomberg Brief: Structured Notes, 5 January 2012, 3 January 2013 and 10 January 2013.
 5.
‘Principalprotected’ in this context means, at a minimum, the security returns the amount invested (unless the issuer defaults).
 6.
These processes generate an infinite number of jumps within any finite interval.
 7.
The parameter v is positive semidefinite. As v tends to zero, the return distribution should match the normal distribution. In agreement with expectations, in this limit the distribution has vanishing skewness.
 8.
The formula for the European call option valuation takes the same form in most of the literature; however, the dependence upon the model parameters varies widely. For three explicit examples of the discrepancies, see Madan et al (1998), Ballotta (2010)and Jaimungal (2004).
 9.
For a discussion of barrier options in the Black–Scholes model, see Hull (2011).
 10.
For a review of the use of static portfolios for the replication of portfolios with pathdependent payoffs, see Carr and Chou (1997).
 11.
Usually a reverse convertible has an embedded downandin put option with strike K=S_{0}. To keep our results general, we do not impose such a restriction.
 12.
 13.
For a veritable catalog of Lévy measures, see Wu (2007).
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