The effect of stochastic dividend yield on the pricing of convertible bonds

(1)

on the pricing of convertible bonds

Thijs van der Groen

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: Thijs van der Groen

Student nr: 10244123

Email: thijsvdgroen@gmail.com

Date: April 21, 2018

Supervisor: Prof. Dr. Ir. Michel Vellekoop Second reader: Prof. Dr. Roger Laeven

(2)

Statement of Originality

This document is written by Student Matthijs van der Groen who declares to take full responsi-bility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

(3)

Abstract

This thesis investigates if convertible bond pricing models produce bet-ter results if the dividend yield is assumed to be stochastic. Dividend yield is known to be not constant in the long term, but most model assume a constant dividend yield. An PDE and Monte Carlo method are used to test the hypothesis of a better forecast. Early results sug-gest that modeling stochastic dividend yield produces better forecasts for bonds with a long maturity.

Keywords Convertible bonds, Least squares Monte Carlo, Partial differential equation, Stochastic dividend yield, Stochastic processes

(4)

Introduction

A company can raise capital in financial markets either by issuing equity, bonds, or hybrids, instruments that possess features of both equity and debt. A convertible bond is such a hybrid instrument and refers to a bond which can be converted into a firm’s common shares at a predetermined conversion rate at the bondholder’s decision. This kind of bond is a financial instruments with complex features as it possesses character-istics of both debt and equity. Usually several equity options are embedded in contracts of convertible bonds.

The motivation for a firm to issue a convertible bond can be explained by two hypotheses: the risk-shifting hypothesis (Green,1984) and the backdoor-equity hypoth-esis (Stein,1992). Green’s risk-shifting hypothesis argues that a firm issues convertible bonds as a substitute for straight bonds to ease potential bondholder-stockholder con-flict. Once straight bonds are issued, the presence of restrictive covenants imposed by the firm’s creditors may encourage the firm to undertake low-risk investments. This is opposite to the incentive of the stockholders. Alternately, undertaking high-risk in-vestments may increase the default probability of the firm and thereby penalize bond holders through falling bond prices and higher credit spreads. These conflicts can be resolved by issuing convertible bonds as the equity components reduces the expropria-tion of wealth, whereas the debt component is likely to possess less restrictive covenants than a straight bond.

The backdoor-equity hypothesis developed by Stein (1992) argues that firms is-sue convertible bonds as a substitute for equity. Firms having significant informational asymmetries between managers and investors, while also facing high expected costs of financial distress, are more likely to issue convertible bonds. High growth firms fall into this category. Firms may be reluctant to issue additional equity since it may dilute ex-isting shareholders claims while information asymmetries mean that the current stock price does not reflect the firm’s growth opportunities.Mayers(1998) adds that convert-ible bonds are also a preferred security choice when an issuer faces a sequential financing problem.

From an investor’s perspective, convertible bonds with embedded optionality offer certain benefits of both equities and bonds. Like equity, they have the potential for capital appreciation and like bonds, they offer interest income and safety of principal. Convertible bonds can be thought of as normal corporate bonds with various embedded options. The conversion right enables the holder to exchange the bond asset for the issuer’s stock. A put feature gives the investor the right to return the bond to the issuer, whereas a call feature gives the issuer the right to call the bond. Given its hybrid nature, the convertible bonds pose modeling challenges. Further complications arise due to the presence of complex contractual clauses, which are generally path-dependent.

There are no analytical solutions to contracts of such complexity, so numerical meth-ods are used to approximate the solution. This can be done using various approaches. The lattice approach uses a discrete-time model of the underlying financial instruments’

(6)

2 Thijs van der Groen — CB pricing using stochastic dividend yield

prices to price the convertible bond. PDE approaches use an approximate of the par-tial differenpar-tial equation that describes how the value of the convertible bond evolves over time by a set of difference equations. Monte Carlo valuation can also be used, this method generates price paths for the underlying risk drives and calculates the exer-cise value of the convertible bond for each path. These values are then averaged and discounted to today to value the convertible.

Closed form solutions for the pricing of a convertible bond have been obtained by

Ingersoll (1977b) for when the convertible bond has no call features embedded. Intro-ducing more realistic specifications, like discretely payable coupons, dividends on the underlying stock, hard call constraints and soft call provisions, prevents the derivation of explicit pricing formulas. For these reasons, various techniques have been considered in the literature such as numerical solutions to partial differential equations by, e.g. by Brennan and Schwartz (1977) for deriving the optimal call and converse strategies,

Brennan and Schwartz(1980) value the convertible allowing for stochastic interest rates and the possibility of senior debt,Tsiveriotis and Fernandes(1998) take credit risk into account in the fixed income part of the convertible, Takahashi et al. (2001) take de-fault risk into account for the fixed income and contingent claim part of the convertible,

Barone-Adesi et al.(2003) implement the Hull-White model for stochastic interest rates, and Kovalov and Linetsky(2008) allows the equity, volatility, interest rate, and default risk to be stochastic.

Lattice methods can also be used for approximating the solution, these methods were used in the context of convertible pricing by Bardhan et al. (1994) that models the convertible as a derivative while taking equity as its only risk driver, andChambers and Lu(2007) taking equity, interest rate, and default risk as underlying risks. Another numerical method is Monte Carlo simulation, which is used byLvov et al.(2004), taking credit risk into account and employing the Longstaff and Schwartz(2001) least squares algorithm.

Closed form solutions are available primarily for European options and affine models. Finite difference methods are important for products with early exercise features, which is the case for convertible bonds. For path dependent products these methods are of limited use due to the high dimensionality of the state space. Monte Carlo simulation is a flexible approach not limited to specific products and models. It is therefore an essential numerical method for pricing of equity derivatives beyond vanilla options. Therefore these two approaches are the preferred approaches to modeling convertible bonds when the dimensionality of the problem is kept low.

Convertible bonds are subject to a number of market risks, such as equity and interest rate risk, but are also exposed to default risk. Convertibles originate as debt and therefore default risk may play a significant role. As a result, the interplay of equity and default risk must be modeled explicitly.

Default risk models can be categorized into two classes. The first class is the struc-tural model, where the firm’s value is modeled as a stochastic process which indirectly lead to default. The default risk in this approach depends on the stochastic evolution of the equity value and default occurs when the random variable of the equity value is insufficient for repayment of debt. In the second class, the reduced form model, the default process is modeled directly. This is done by modeling the intensity of the default process exogenously by using both market-wide and firm-specific factors, such as stock prices. The default intensities cannot be be observed directly, but by imposing absence of arbitrage conditions, explicit pricing formulas and/or algorithms can be inverted to find estimates for them. As the firm’s asset value is not observable in the market, reduced form models are preferred to value or hedge fixed income securities. This is because the parameters of the stochastic processes underlying the structural model cannot be based on the market’s information.

(7)

con-stant dividend yield. Harvey and Whaley (1992) show that this assumption leads to large pricing errors. Observations of the dividend yield show that the process is not constant in the long term. Valuation without this assumption is challenging as the divi-dend yield was at the time non-tradable and its changes are imperfectly correlated with the stock return, which causes market incompleteness. Today, several products to trade this risk exists, such as dividends swaps. The dividend yield risk premium must be in-cluded in pricing formulas, but cannot be estimated with precision. This has delayed the development of models that take the randomness of the dividend yield explicitly into account.

Dividends are comparable to coupons of a bond since they provide the equity holder with a stream of income. The main difference is that the company can change the amount of dividend paid. Therefore, the dividend amounts are a function of the company’s performance. Dividends can lead to call delay for a number of reasons.Ingersoll(1977b) recognizes that if there are bondholders who are not converting but should be because of large dividends, the management should not call and wake these ”sleeping investors”.

Constantinides(1987) argues that dividends delay the call when firms prefer voluntary conversion over the forced conversion of a call.

The total number of randomness sources quickly adds up to a total of four: equity price, interest rate, credit spread, and dividend yield. As more sources of risk are con-sidered, the issue is to find a balance between a theoretically sound and numerically feasible model. Convertible bonds are one type of instrument whose pricing poses a substantial set of complications, both with regard to the theoretical model, but also with respect to obtaining a numerical solution that should be stable, flexible, fast, and commercially usable. Practitioners tend to avoid models with more than two factors. Therefore a legitimate question is how to reduce the number of factors or which factors are most important.

This thesis investigates whether the modeling of stochastic dividends affects the pricing of convertible bonds, by answering the following research question:

”Does incorporating stochastic dividend yield in convertible bond pricing models in-crease the accuracy of the calibration?”

To answer this question the pricing of a number of bonds is studied using two different pricing methods. First the convertibles are priced using a model assuming constant dividend yield. Different components of the pricing model are dependent on the dividend, such as the conversion ratio, the conversion decision, and the underlying stock dynamics. Adding a stochastic dividend yield process might yield better pricing results and is therefore added to both of these approaches. In order to test the accuracy of the methods, the parameters are estimated on a training sample and a period is forecasted using these parameters. These forecasting results are then compared using three pricing error measures.

The literature on convertible bond pricing is discussed in the next chapter. Section 3.1 describes the valuation of convertible. In the rest of Chapter 3 the underlying pricing methods used in this thesis are reported. Chapter 4 contains a description of the data set used, an analysis of the pricing models using numerical examples, and the empirical results. A conclusion is drawn in Chapter 5.

(8)

Chapter 2

Literature review

Given the many challenges in the pricing of convertible bonds and the increasing market size of convertible debt, convertibles have been a subject of active academic research. This chapter contains the literature review on the pricing of convertible bonds. The next section describes multiple convertible bond pricing models, the second section discusses different theories on optimal conversion strategies, and the last section gives some empirical results.

2.1 Convertible bond pricing models

Pricing models of convertible bonds are similar to contingent claim pricing models as the features embedded in the conversion option are similar to equity options. Contingent claim pricing models price an asset where its expected value is dependent on another security. However, the valuation of a convertible bond is more complicated due to its hybrid feature. The academic literature on the convertible bond pricing begins with the papers of Ingersoll (1977b) and Brennan and Schwartz(1977,1980). These papers view the firm’s debt and equity as contingent claims on the firm’s value and use the Black-Scholes contingent claim model to price convertible bonds. Pricing models prior to these papers were either incomplete or misspecified.

The framework used by these papers noted before was developed by Black and Scholes (1973) and Merton (1973), by assuming the price of the underlying asset as a lognormal process with constant volatility and constant drift. Using a hedging portfolio consisting of the underlying stock and the contingent claim they derive a closed form formula for pricing European options on equity.

In order to value corporate debt, which can be seen as a contingent claim on firm value, Merton(1974) further expands this framework. Under this approach, the capital structure of the firm consists of only a single, homogenous class of debt, and equity. It is assumed that the bondholders receive the entire firm’s assets upon default and the shareholders receive nothing. Default occurs when the total firm value falls below the total redemption value of debt at maturity. The degree of solvency is then a function of the firm’s capital structure and may be measured as the ratio of the market value of the assets and the book value of debt. Merton (1974) uses a standard partial dif-ferential equation (PDE) that can be applied to value any financial security depending on boundary specifications. The PDE is solved to discount corporate debt with some default probability. This approach is called the structural approach.

Using the structural approach has some drawbacks as Zabolotnyuk et al. (2010) notes. The writers state that the PDE becomes increasingly complicated when the model considers more complex capital structure. McConnell and Schwartz (1986) add that parameter estimation is difficult as the firm’s value is not observable or directly tradable in the market. The reduced form approach is introduced in this paper to overcome these limitations. This framework views the security as a contingent claim on the underlying

(9)

stock and estimates the credit risk exogenously by adding a constant credit spread to the risk free rate. Jarrow and Turnbull (1992, 1995) expand the framework for credit risk by modeling the default event as a Poisson process with constant intensity.

In order to improve the reduced form approach for convertible bonds, Tsiveriotis and Fernandes (1998) claim that a convertible bond can be decomposed into two com-ponents. One component is the fixed income component, which is exposed to credit risk. The other component is the stock component, assumed to be risk free as the issuer can deliver its own stock. The fixed income element is discounted at the risk adjusted rate and the stock element at the risk free rate. Grimwood and Hodges (2002) andAyache et al. (2003) argue that this model is incoherent as it assumes bonds are affected by credit risk and equities are not. These two papers assume that both debt and equity are affected by the same credit risk. Even more extensions are possible in the modeling such as probability of default, hazard rate, fractional loss, and recovery upon default (Ayache et al.,2003).

2.1.1 Extensions

In order to reduce the number of factors in the valuation model a sensitivity analysis is conducted by Grimwood and Hodges (2002). This paper concludes that modeling the equity process is crucial. The valuation model usually assumes the volatility to be constant (McConnell and Schwartz,1986;Ammann et al.,2003).

Another factor that might be of importance due to the long maturity of a convertible bond is the interest rate process. In the same study, Grimwood and Hodges(2002) find that modeling the interest rate process is of second order importance to the equity process. Brennan and Schwartz (1980) find that it is computational more efficient to assume a flat term structure. This claim is further investigated by Ammann et al.

(2008). They investigate the effects of stochastic interest rates by comparing convertible bond prices under the assumption of constant interest rate and under the assumption of a Cox-Ingersoll-Ross term structure model. The authors find no significant difference between these two assumptions.

Grimwood and Hodges (2002) and Ammann et al.(2008) state that capturing the contractual features of a convertible bond is as crucial as accurately modeling the equity process. These features include specific factors as moneyness, maturity, coupon, and conversion ratio. In addition, Ammann et al. (2008) examines more complex features, such as call provisions. This feature tends to lead to estimation biases as the issuers are assumed to act optimally by calling the bond as soon as possible whereas in practice call restrictions prevent this.

Another issue specific factor influencing the pricing of the convertible bond is the dividend (Ammann et al.,2008;McConnell and Schwartz,1986). They find that when the dividend yield is close to zero the investor has little incentive to convert the bond. Apart from the sensitivity of the convertible bond price to the current dividend yield, the bond price also depends on the expected dividend level. The covenants of the bond state that investors cannot receive the coupon and dividend payment in the same accounting period. In this case, the dividend will be received the following period. Due to this, the future dividend level must be predicted for an optimal conversion decision.

The effect of randomness of the dividend yield on contingent claims has been stud-ied by Geske (1978). The author uses an equilibrium argument based on the CAPM to obtain the dividend yield risk premium and derive an option formula. By means of a numerical example it is made clear that incorporating stochastic dividend yield may be more important for longer lived options. Lioui (2006) investigates pricing deriva-tives with stochastic dividend yield under complete markets. This paper states that, even under market completeness, assuming stochastic dividend yield complicates the implementation of option formulas, as the risk premium must be computed. This paper developed a European option pricing formula using stochastic dividend yield and using

(10)

stochastic market price of risk.

2.2 Conversion strategies

Apart from technical specifications, the valuation of convertible bonds also ask for spec-ifications on the behavior of the bondholder and the issuer, as many convertible bonds posses a dual option. The bondholder can convert the bond into common stock and the firm can call the bond for redemption. When the bond is called the bondholder can choose between converting the bond or to redeem it. The optimal strategy for conver-sion depends on the call strategy of the issuer and vice versa.Ingersoll(1977a) studies the optimal conversion and call policies and show that it is never optimal to convert a callable convertible bond. Exceptions are immediately prior to a dividend date, an adverse change in the conversion ratio, constrained by an early exercise of the call fea-ture, or at maturity. The same paper states that for any conversion prior to a dividend payment the actual conversion premium must be lower than the difference between the received dividend and coupon payments. The conversion process, however, must be immediate and the costs must be negligible.

Calling the bond is found to be optimal for the issuer as soon as the market price of the convertible bond reaches the call price. However,Asquith (1995) finds that firms are postponing this call decision. This paper state that the presence of a notice period actives a rational call delay, as firms may want to delay the calling to receive a cashflow advantage. This is likely when the after tax interest rates are lower than the dividend payments.

Harris and Raviv (1985) states that the investors reactions are rational given the firm’s call policy. The idea is that managers decide given their private information whether or not to call the outstanding convertible bonds. The markets infers the private information by observing the call decisions. Therefore when the decision is made to call the bond, this is perceived as a signal of unfavorable private information. This may also occur when the credit rating of a firm improves or when yields fall considerably. Under these circumstances the firm should act in the best interest of the shareholders by calling the bonds immediately. The value of equity will then be maximized and the bond value minimized by this decision.

2.3 Empirical results

The research ofMcConnell and Schwartz(1986) is the first of the reduced form approach. Extensions of their model have been developed to incorporate real world features into the pricing model.McConnell and Schwartz(1986) reports an underpricing of the model prices. Ammann et al. (2003) also reports an underpricing of 3.24%. The overpricing varies from 0.36% inAmmann et al.(2008) to 5% inBarone-Adesi et al.(2003).Ammann et al. (2003) and Zabolotnyuk et al. (2010) find that in the money bonds tend to be overpriced, where out of the money bonds are more likely to be underpriced. Pricing errors like these can lead to arbitrage opportunities. When the convertible bond is underpriced, the arbitrageur buys the bond and shorts the underlying stock at the current delta.

Zabolotnyuk et al.(2010) compares the efficiency of three pricing models byBrennan and Schwartz(1980),Tsiveriotis and Fernandes(1998), andAyache et al.(2003). These three models treat credit risk differently. The highest mean absolute deviation (3.73%) is given by theBrennan and Schwartz(1980) model, the Tsiveriotis and Fernandes(1998) gives an error of 1.94%, and the Ayache et al. (2003) model gives the lowest error of 1.86%.

The value of the convertible bond can be solved by either a closed form or a numerical solution, depending on the underlying assumptions regarding the pricing model. When

(11)

adding stochastic variables to the pricing model, the efficiency would ideally increase and the computational time would be reduced. Ingersoll (1977b) developed a closed form solution for pricing convertible bonds under perfect market assumptions. Using this model it is difficult to generalize the solution. Therefore numerical methods must be used.

The finite difference approximation is obtained by replacing the partial derivatives with finite differences after building a grid of points. This method is appropriate in solving early exercise features and is used byAyache et al.(2003);Brennan and Schwartz

(1977,1980) andTsiveriotis and Fernandes(1998). An alternative to the finite difference method is the finite element method. It approximates the partial differential equation differently. Barone-Adesi et al. (2003) investigates this method on convertible bond pricing and concludes that the difference between the finite difference and finite elements method is only relevant on a theoretical level.

Another method to price convertible bonds are simulation methods. These methods better are better capable of capturing the underlying state variables’ dynamics and any complex features of the convertible bond.Longstaff and Schwartz (2001) find that these methods are more efficient in solving pricing models with one or two sources of uncertainty as the convergence rate is exponential in the number of state variables. In this method the simulation model is used to generate a random path of the underlying dynamics. The next step is to determine the optimal conversion strategies and issuer at any date. The bond is conversed when the continuation value is higher than the payoff from exercising. The continuation value can be estimated via the least squares Monte Carlo simulation model of Longstaff and Schwartz(2001) that uses a backward induction procedure by means of a simple regression.

(12)

Chapter 3

Valuation framework

This chapter sets out the framework for the valuation of a convertible bond. First the contract features of a convertible bond are given. In the second section the underlying dynamics are explained. The PDE methods for valuation are given thereafter. Lastly, the Monte Carlo algorithm is given.

3.1 Contract features

The basic contract features of convertibles bond are laid out in this section. A convertible bond issued by a firm at time t = 0 with a maturity date T > 0 has a par value of X > 0 to be repaid at maturity. The cashflow of the bond feature can be summarized by the following equation:

Ci= cδX for ti = iδ, i = 1, 2, . . . , N − 1 (3.1)

X + CN = (1 + cδ)X for tN = T. (3.2)

where Ci denotes the coupon payment, ti denotes the coupon payment dates, and δ > 0

the time interval between two coupons. The accrued interest at time t ∈ (iδ, (i + 1)δ] between two coupons is given by A(t) = c(t − iδ)X.

The holder of the bond has an option to convert the bond into η > 0 common stock shares of the issuer prior to the maturity date. η is called the conversion ratio and it specifies the number of shares of stock to be received. The conversion feature is usually conditional on a number of factors, such as that the stock’s trading price must be above a percentage of the conversion price. In the U.S. convertible bond market, upon conversion no accrued interest is paid. The conversion ratio is specified at issuance and is initially fixed. This ratio can change when the nominal value of the underlying stock changes (such as stock splits or consolidations) or other financial operations that affect the stock price (Ammann et al.,2008).

Another event that can change the conversion ratio is when dividend payments are made. If the firm increases its dividend on the common stock, the stock price decreases and consequently the conversion value of the convertible bond decreases. In order to keep the convertible bond attractive to investors, a dividend protection provision is added to the contract. This provision gives the bond holder insurance against the decline of the stock price caused by the dividend payment. Dividend protection can be given by either a conversion ratio adjustment or a dividend pass through.

The conversion ratio is adjusted in the following manner. Let the current stock spot price be S0 with an indicated annual dividend D0, which makes the dividend

yield q0 = D0/S0. Suppose that the issuer announces a new dividend D1, then the

new dividend yield is q1 = D1/S0. The excess dividend is given by D1 − D0. At the

announcement the holder wants to be compensated for the change in stockprice, which is done by changing the conversion ratio. The holder wants the following to hold if he

(13)

wants to convert: η0(S0− (D1− D0)) = ηS0 (3.3) η0 = η S0 S0− (D1− D0) = η 1 1 − (q1S_S1₀ − q0) (3.4)

The excess dividend payout is reflected in the new conversion ratio. It represents the change of dividend received under the new ratio as opposed to the old ratio.

Some constraints can be imposed on the dividend protection provision. Three types of thresholds are possible. The first prescribes that the ratio is not adjusted if the change in conversion price due to the dividend payment is less than a specified threshold. There can also be a threshold on the stock dividend yield. The third type is a threshold on the stock dividend amount, where the dividend must be greater than a threshold dividend amount.

Apart from the conversion feature, convertibles may also have put and call features. A put feature gives the investor the right to return the bond to the issuer at a predeter-mined price, which is typically the bond face value plus accrued interest. These options are usually Bermudan and can therefore only be exercised at predetermined dates.

A call feature gives the issuer the right to call the bond for redemption at the call price. These embedded call options are of American type and may be exercised at any time during the life of the bond. The feature creates greater flexibility in the capital structure of the company (Ingersoll, 1977b). There is often a period after issue called the hard non call protection where the issuer cannot call the bond. Furthermore, there exist soft call provisions where the issuer may call the bond only if it trades for more than a trigger price for a period of time. Similar is the call bond lag and the call parity lag. In the former case the call is delayed while the bond value is less than the bond lag times the call price plus the accrued interest. The call date is often allowed to vary in time and the prices and dates are set out in the call schedule.

The issuer has an option to call the bond after a call protection period [0, tc), tc ∈

[0, T ], tc is called the first call date. At the time of the call t ∈ [tc, T ), the issuer

re-purchases the bond from the bondholder for a dirty call price Cd(t). In the prospectus

of the bond the call schedule is specified which provides the clean call price Cc(t) in

effect during each year of the bond’s life when the bond is callable. If the bond is called at some time t during the year, the actual (dirty) call price Cd(t) to be paid to the

holder is the clean call price plus the accrued interest A(t) since the last coupon paid, Cd(t) = Cc(t) + A(t). The conversion option of the holder has a precedence over the

call option of the issuer. If the issuer calls a bond at t, the bondholder has the right to either convert the bond at t and receive η shares of the underlying stock worth ηSt or

receive the dirty call price Cd(t).

A convertible bond, as any corporate debt, is subject to a default by the issuer firm. If a company files bankruptcy, both bonds and stocks go into a default status. The default probabilities are the same for both, but the recovery rates are different as stockholders have a lower priority on the list of the stakeholders in the company than bondholders. The default proceedings provide a justification for modeling different recovery rates for stock and bonds with the same default probability.

3.2 Dynamics

The valuation of the convertible bond is subject to a number of processes. In this section the underlying processes are given for equity risk, and dividend yield risk.

The default possibility on the underlying firm is assumed to be the first jump of a Poisson process. The default event is modeled as one jump to default in period [0, t] under the default probability. This inhomogeneous Poisson process N with intensity

(14)

function λ(t) > 0 is a non decreasing, integer valued process with independent incre-ments, N0= 0. The probability of n jumps in [s, t] is denoted by

P [Nt− Ns = n] = 1 n!( Z t s λ(u)du)nexp{− Z t s λ(u)du}. (3.5)

So therefore the probability of no jumps in [s, t] equals

P [Nt− Ns= 0] = exp{−

Z t

s

λ(u)du}. (3.6)

The first jump time of N is given by τ = inf t ≥ 0 : Z t 0 λ(u)du ≥ E1 , (3.7)

where E1 is an exponentially distributed random variable with parameter 1.

The compensated Poisson process Mt, where 0 ≤ t ≤ T

Mt:= Nt−

Z t

0

λ(u)du, t ≥ 0 (3.8)

is a martingale with respect to the filtration (F_tN)t∈[0,T ]generated by the process Nt, 0 ≤

t ≤ T . Conditional on that a firm survives until time t, its default probability within the next small time interval ∆t equals

Q[Nt+∆t− Nt= 1|Ft] = λ(t)∆t + o(∆t) (3.9)

and the survival probability of a firm equals

Q[τ > t] = exp − Z t 0 λ(u)du . (3.10)

Following Zabolotnyuk et al. (2010), the probability of default is assumed to be a decreasing function of stock price λ(S) = λ0₀(_SS

0)

α_{. The symbols S}

0, λ00, and α represent

constants for a given firm. λ0₀ is the probability of default when the stock price is So.

Introducing γ = λ00

Sα

0 , the hazard function can be rewritten as λ(S) = γS

α_{. α is a negative}

number to represent a decreasing function in the stock price. When the stock price is high the probability of default will decline.

Under the risk neutral measure Q the stock price is assumed to follow the following stochastic differential equation:

dSt= (rt− qt+ nλt)Stdt + σStdWt− nSt−dNt, (3.11)

where dWt is a standard Brownian motion, rt is the spot interest rate, and qt the

dividend yield. S_t− is the stock price prior to the default event and λt= λ(S, t) = γStα

denotes the arrival rate of the default event. When default occurs, the stock price will drop to the recovery value of (1−n). The default intensity λtin the drift compensates for

a possibility of a jump, so that the discounted total return process remains a martingale in the risk neutral economy. Its expectation is given by

E(dSt|Ft) = (rt− qt+ nλt)Stdt − nStλtdt = (rt− qt)Stdt (3.12)

using E[dNt|Ft] = λtdt and E[dWt|Ft] = 0.

Observations of the dividend yield show that the process is not constant in the long term. As some convertible bonds have high maturities, modeling the dividend yield as a

(15)

stochastic process may yield better pricing results. FollowingLioui(2006), the stochastic dividend yield is represented as:

dqt= θ(ω − qt)dt − λqdt + νdZt+ φ

dSt

St

. (3.13)

θ represents the speed of adjustment and is assumed to be positive, ω is the nonnegative equilibrium value, Zt is a standard Brownian motion, and ν is the volatility of the

process. φ captures the relation between stock returns and dividend yield, which are negatively correlated. λq denotes the market price of dividend yield risk, explained in

the next subsection.

3.2.1 Market price of risk

Since the dividend yield is not traded in the market, the assumed financial market is incomplete and it cannot be hedged. An investor cannot avoid bearing pure dividend yield risk, therefore it has a market price of risk λq. Consider the stochastic differential

equation for the stock price without credit risk under the real world measure P: dSt= (µt− qt)Stdt + σStdWtP, (3.14)

The expected return µ can be written as µt= 1 dtE[ dSt St ] = rt+ b (3.15)

where rtis the risk free rate and b the stock risk premium. Plugging this expression into

Equation 3.14 yields: dSt= ((rt+ b) − qt)Stdt + σStdWtP = (rt− qt)Stdt + σSt( b σdt + dW P t) = (rt− qt)Stdt + σStdWt. (3.16) By Girsanov’s theorem Wt= Rt 0 bs

σds+dWtP is a Brownian motion under the risk neutral

measure Q. For the dividend yield process this yields: dqt= θ(ω − qt)dt + νdZtP+ φ dSt St = θ(ω − qt)dt + νdZtP+ φ((µt− qt)dt + σdWtP) = θ(ω − qt)dt + νdZtP+ φ((rt− qt)dt + σdWt) = θ(ω − qt)dt − λqdt + ν( λq ν dt + dZ P t) + φ dSt St = θ(ω − qt)dt − λqdt + νdZt+ φ dSt St (3.17) λq is the risk premium for bearing dividend yield risk and is assumed to be constant.

3.3 Valuation using PDEs

This section describes the derivation of the PDE methods. First the constant dividend yield PDE approach is given and after that the PDE assuming stochastic dividend yield is described.

(16)

3.3.1 PDE based benchmark without stochastic dividend yield

This section follows the derivation ofAyache et al.(2003) and is a modified reduced-form model that assumes the default process follows a Poisson process.

We start with the assumption that the stock process under Q without any credit risk follows the next equation, where r(t) denotes the risk free rate, q the continuous dividend yield, and σ the volatility,

dSt= (r(t) − q)Stdt + σStdWt. (3.18)

The no arbitrage value of any contingent claim G(S, t) based on St is given by, using

Ito’s Lemma, dGt= (r(t) − q)St ∂G ∂S + ∂G ∂t + 1 2σ 2_S2 t ∂2G ∂S2 dt + σSt ∂G ∂SdWt. (3.19) The Wiener process underlying S and G are equal, so therefore constructing the follow-ing portfolio eliminates this Wiener process:

Π = G − S∂G

∂S. (3.20)

A small change in Π leads to: dΠ = dG − ∂G ∂SdS = ∂G ∂t + 1 2σ 2_S2 t ∂2G ∂S2 − qSt ∂G ∂S dt. (3.21)

Under Q the portfolio earns the riskfree rate: r(t)(G − ∂G ∂SSt)dt = dΠ = −qS_t∂G ∂S + ∂G ∂t + 1 2σ 2_S2 t ∂2G ∂S2 dt. (3.22)

Simplifying the equation gives the PDE for any contingent claim with no credit risk on St: ∂G ∂t + 1 2σ 2_S2 t ∂2G ∂S2 + (r(t) − q)St ∂G ∂S − r(t)G = 0. (3.23)

In order to incorporate the probability of default Ayache et al. (2003) argues the following. Let the value of a convertible bond be denoted by V (S, t), the stock price immediately after default S+, and S− the stock price right before default. Assuming the in the event of a default with probability λ the stock price follows the following process, with 0 ≤ n ≤ 1,

S+= S−(1 − n). (3.24)

This notation is equal to Formula 3.11. The hedging portfolio with G denoting the convertible bond is given by

Π = G − αS. (3.25)

Choosing α = ∂G_∂S would suffice if no credit risk is present. The standard arguments would give dΠ = ∂G ∂t + σ2S2 2 ∂2G ∂S2 − qSt ∂G ∂S dt + o(dt). (3.26)

Let λ denote the non zero hazard rate. Upon default the bond holder has the option of receiving either an amount RX, where 0 ≤ R ≤ 1 is the recovery factor and X is the bond’s face value, or receive shares worth ηS(1 − n). According toAyache et al.(2003)

(17)

X can take many forms, but Zabolotnyuk et al.(2010) chooses X to be the face value of the bond. R is therefore the proportion of the bond’s face value that is recovered immediately after a default.

Under these additional assumptions, the change of value of the hedging portfolio is

dΠt= (1 − λdt) ∂G ∂t + σ2S2 2 ∂2G ∂S2 − qSt ∂G ∂S dt − λdt(G − αSn) + λdtmax(ηS(1 − n), RX) + o(dt) = ∂G ∂t + σ2S2 2 ∂2G ∂S2 − qSt ∂G ∂S dt − λdt(G − αSn) + λdtmax(ηS(1 − n), RX) + o(dt). (3.27) The portfolio must earn the riskfree rate dΠt= r(t)Πdt:

r(t)(G − S∂G ∂S)dt = ∂G ∂t + σ2_S2 2 ∂2_G ∂S2 − qSt ∂G ∂S dt − λdt(G − ∂G ∂SSn) +λdtmax(ηS(1 − n), RX) + o(dt), (3.28) implying ∂G ∂t + (r(t) + λn − q)S ∂G ∂S + σ2S2 2 ∂2G ∂S2 − (r(t) + λ)G + λmax(ηS(1 − n), RX) = 0. (3.29) Along with the PDE the following assumptions are made for the following contractual features:

• The bond can be converted at any time to η shares.

• The issuer can call the bond for price B_c. The holder can convert the bond if it is called.

As noted in the literature review, it is optimal to convert as soon as the current con-version value is equal to or larger value of the convertible. The optimal strategy for the issuer is to call the bond as soon as the current call value is equal to or smaller than the convertible. Therefore, the value of the convertible bond is given by the solution to Equation 3.29, subject to the following boundary conditions:

G(S, T ) =max(ηS, F + CT), (3.30)

G ≥ηS, (3.31)

G ≤max(Bc, ηS). (3.32)

3.3.2 PDE method with stochastic dividend yield

The PDE method assuming stochastic dividend yield is described here. Having the following two processes for equity and dividend yield without credit risk:

dSt= (r(t) − qt)Stdt + σStdWt, (3.33)

dqt= θ(ω − qt)dt − λqdt + νdZt+ φ

dS

(18)

Any contingent claim dependent on S, q, and t is denoted by G(S, q, t). Using Ito’s Lemma, this is given by

dG = ∂G ∂tdt + ∂G ∂SdS + 1 2 ∂2G ∂S2dS 2 t + ∂G ∂qdq + 1 2 ∂2G ∂q2dq 2 t + ∂2G ∂S∂qdqdS = ∂G ∂tdt + ∂G ∂SdS + 1 2 ∂2G ∂S2σ 2_S2_{dt +}∂G ∂qdq + 1 2 ∂2G ∂q2 (ν 2_{+ φ}2_σ2_{)dt +} ∂2G ∂S∂qσ 2_φdt = ∂G ∂tdt + ∂G ∂S((r(t) − qt)Stdt + σStdWt) + 1 2 ∂2G ∂S2σ 2_S2_{dt +}1 2 ∂2G ∂q2 (ν 2_{+ φ}2_σ2_)dt +∂G ∂q(θ(ω − qt)dt − λqdt + νdZt+ φ dS S ) + ∂2G ∂S∂qσ 2_φdt = ∂G ∂tdt + ∂G ∂S((r(t) − qt)Stdt + σStdWt) + 1 2 ∂2_G ∂S2σ 2_S2_{dt +}1 2 ∂2_G ∂q2 (ν 2_{+ φ}2_σ2_)dt +∂G ∂q(θ(ω − qt)dt − λqdt + νdZt+ φ((r(t) − qt)dt + σdWt) + ∂2G ∂S∂qσ 2_φdt. (3.35) Assume the following hedging portfolio, consisting of two contingent claims and one stock:

Πt= G(S, q, t) + ∆St+ βH(S, q, t), (3.36)

for which the change of value is given by

dΠt= dG(S, q, t) + (∆dSt+ ∆qtStdt) + βdH(S, q, t)

= dG(S, q, t) + ∆σStdWt+ ∆rStdt + βdH(S, q, t). (3.37)

As G(S, q, t) and H(S, q, t) have the same underlying PDE, combining these two expres-sions yields: dΠ = ∂G ∂t + 1 2 ∂2_G ∂S2σ 2_S2₊1 2 ∂2_G ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2G ∂S∂qσ 2_{φ +} ∂G ∂S(r(t) − qt)St +∂G ∂q(θ(ω − qt) − λq+ φ(r(t) − qt)) dt + β ∂H ∂t + 1 2 ∂2H ∂S2σ 2_S2₊1 2 ∂2H ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2H ∂S∂qσ 2_{φ +} ∂H ∂S(r(t) − qt)St +∂H ∂q (θ(ω − qt) − λq+ φ(r(t) − qt)) dt + ∆r(t)Stdt + ∂G ∂S + β ∂H ∂S + ∆ σStdWt+ ∂G ∂q + β ∂H ∂q (νdZt+ φσdWt). (3.38) In order for the portfolio to be hedged against movements in stock and dividend yield, the last two terms involving dWt and dZt must be zero. This implies that:

β = − ∂G ∂q ∂H ∂q ∆ = −β∂H ∂S − ∂G ∂S. (3.39)

(19)

dΠ with the found values of β and ∆ becomes: dΠ = ∂G ∂t + 1 2 ∂2G ∂S2σ 2_S2₊1 2 ∂2G ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2G ∂S∂qσ 2_{φ +} ∂G ∂S(r(t) − qt)St +∂G ∂q(θ(ω − qt) − λq+ φ(r(t) − qt)) − r(t)St ∂G ∂S dt + β ∂H ∂t + 1 2 ∂2H ∂S2σ 2_S2₊1 2 ∂2H ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2H ∂S∂qσ 2_{φ +} ∂H ∂S(r(t) − qt)St +∂H ∂q (θ(ω − qt) − λq+ φ(r(t) − qt)) − r(t)St ∂H ∂S dt, (3.40)

which can be written as dΠ = (A + βB)dt. We have the following equation:

A + βB = r(G + ∆S + βH). (3.41)

Substitution of ∆ and β and rearranging gives

A − rG + rS∂G_∂S ∂G ∂q = B − rG + rS ∂H ∂S ∂H ∂q . (3.42)

The left hand side is a function of G only and the right hand side is a function of H only.

As a result, both sides can be written as a function f (S, q, t) of S, q, and t. f (S, q, t) may be interpreted as the dividend yield risk premium. Under the risk neutral measure Q this premium is 0, leading to the following equation:

∂G ∂t + 1 2 ∂2G ∂S2σ 2_S2₊ 1 2 ∂2G ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2G ∂S∂qσ 2_{φ +} ∂G ∂S(r(t) − qt)St +∂G ∂q(θ(ω − qt) − λq+ φ(r(t) − qt)) − r(t)St ∂G ∂S − rG + rS∂G ∂S = 0 ∂G ∂t + 1 2 ∂2G ∂S2σ 2_S2₊ 1 2 ∂2G ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2G ∂S∂qσ 2_{φ +} ∂G ∂S(r(t) − qt)St +∂G ∂q(θ(ω − qt) − λq+ φ(r(t) − qt)) − r(t)G = 0. (3.43) Solving this equation leads to the price of any contingent claim when there is no credit risk present.

Following the reasoning and making the same assumptions upon default as the previous subsection, the change of value of the hedge portfolio during t to t + dt yields:

dΠ∗_t = (1 − λdt)dΠt− λdt(G + ∆Stn + βH) + λdtmax(ηS(1 − n), RX)

+ βλdtmax(ηS(1 − n), RX) + o(dt)

= dΠt− λdt(G + ∆Stn + βH) + λdtmax(ηS(1 − n), RX)

+ βλdtmax(ηS(1 − n), RX) + o(dt), (3.44)

(20)

portfolio must earn the risk free rate such that dΠ∗ = rΠ∗dt. This leads to

(r(t) + λ)(G + βH)dt = dΠdt − ∆St(r(t) + nλ) + (1 + β)λdtmax(ηS(1 − n), RX) = ∂G ∂t + 1 2 ∂2G ∂S2σ 2_S2₊1 2 ∂2G ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2G ∂S∂qσ 2_{φ +} ∂G ∂S(r(t) − qt)St +∂G ∂q(θ(ω − qt) − λq+ φ(r(t) − qt)) + (nλ)St ∂G ∂S dt + β ∂H ∂t + 1 2 ∂2H ∂S2σ 2_S2₊ 1 2 ∂2H ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2H ∂S∂qσ 2_{φ +} ∂H ∂S(r(t) − qt)St +∂H ∂q(θ(ω − qt) − λq+ φ(r(t) − qt)) + (nλ)St ∂H ∂S dt + (1 + β)λdtmax(ηS(1 − n), RX). (3.45)

This can be written as

(r(t) + λ)(G + βH) = A + Bβ. (3.46)

Substituting for ∆ and β and rearranging gives A − (r + λ)G ∂G ∂q = B − (r + λ)G_∂H ∂q . (3.47)

Following the same derivation without credit risk and using f (S, q, t) = 0 under Q, the PDE becomes ∂G ∂t + 1 2 ∂2G ∂S2σ 2_S2₊1 2 ∂2G ∂q2(ν 2_{+ φ}2_σ2_{) +} ∂2G ∂S∂qσ 2_{φ − (r + λ)G + (r − q + nλ)S}∂G ∂S +∂G ∂q(θ(ω − qt) + φ(rt− qt) − λq) + λmax(ηS(1 − n), RX) = 0. (3.48) The parameter λq is assumed to be constant and defines the risk premium for bearing

dividend yield risk.

The same boundary conditions apply:

G(S, q, T ) =max(ηS, F V + CT), (3.49)

G ≥κS, (3.50)

G ≤max(Bc, ηS). (3.51)

3.4 Valuation using Monte Carlo algorithms

This section follows the least squares Monte Carlo algorithm set out by Lvov et al.

(2004) in order to price convertible bonds. First N paths are simulated of the equity process or the equity and dividend yield process. Given those paths, the algorithm runs recursively, rolling back one step at a time. At each step the conversion, call, or continuation decision is determined. The convertible bond price is then obtained by averaging path wise over the discounted cashflows by the bond.

All cash flows are assumed to only occur at K discrete times forming a set Φ. All coupon dates and exercise dates belong therefore to this set Φ. K is made sufficiently large to approximate the value of the convertible bond. The bond payoff at maturity, conditional on no default prior to the last simulation time step and no conversion prior to maturity, is given by G(tK; tK−1, T ) = ( max(ηS(tK), F + CK) τ > T max(ηnS(tK), RX) τ < tK−1, (3.52)

(21)

where τ represents the default time of the firm, n represents the recovery value of the stock, and R the redemption value of the bond.

The convertible bond holder has the right to convert the bond into shares or continue holding the bond at any each point tk prior to maturity and subject to any restrictions.

This is determined by taking the conditional expectation of the bond’s discounted future payoff. The conditional expectation is calculated by regression using the information contained in a sample of the simulated paths. The optimal stopping time is denoted by τ∗ and describes the point in time at which the bond is converted. After this event the bondholder is not entitled to any coupon payments or any redemption value if the firm defaults.

The investor determines if conversion at time tkis optimal for each path in Φtk given

the convertible bond value. Φtdenotes the subset of paths for which the default time is

greater than time t. The holder converts the bond if the conversion value is larger than the continuation value G(tk, T ):

ηS(tk) > G(tk, T ). (3.53)

The risk neutral continuation value of the convertible bond at time tk, tk∈ Φ, given

no default, no conversion, and no call at or prior to time tk, is given by

G(tk, T ) = EQ   K X j=k+1 e− Rtj tk(r(s)+λ(s))ds_C(t_j_{; t}_k_{, T )|F}_t k  , (3.54)

where C(s; t, T ) denotes the cash flow at time s, t < s ≤ T , r(s) the riskfree rate and λ(s) = γS(ts)α the credit spread.

At time tkthe functional form of G(tk, T ) is unknown. However, it can be represented

as a linear combination of a countable set of FtK−1 measurable basis functions:

G(tk, T ) = ∞

X

j=0

αjLj(ξ(tk)), (3.55)

where Lj(.) is the jth basis function and ξ(t) denotes the value at time t of the vector

of simulated state variables containing the following variables: the underlying stock price and the dividend yield when it is assumed to be stochastic. In practice, only the first N < ∞ basis function are used to approximate the convertible bond price. This approximation is denoted by:

GN(tk, T ) = N

X

j=0

αjLj(ξ(tk)). (3.56)

Once a subset of basis functions has been specified the coefficients αj are estimated as

a solution to the following equation: ˆ

α = min

α (

T_{| = γ − Υα),} _(3.57)

where vector γ contains the discounted values of C(tK; tk, T ) for paths in Φtk, matrix Υ

contains values of N basis functions for paths in Φtk, and α is the vector of coefficients in

the previous equation. The fitted value of this regression GN(tk, T ) converges in mean

square and in probability to G(tk, T ) as the number of paths in Φtk goes to infinity

(Longstaff and Schwartz,2001).

Let Bc(t) denote the call price at time t. The bond is called by the issuer when the

call price is less then the continuation value:

(22)

where Bc(t) equals the face value when the bond is callable and equals ∞ when the bond

cannot be called. When the bond is called the bondholder has the option to convert the bond into shares or take the cash amount. The payoff is

max(ηS, Bc(t)). (3.59)

When the firm defaults, the recovery payoff is assumed to be received at the time of default. In paths where default occurs between tK−2and tK−1, the payoff at time tK−1

is given by

max(ηnS(τ ), RX), (3.60)

together with setting all following cashflows in that path to zero.

Summarizing, paths in ΦtK−1 for which conversion at time tk is optimal, the cash

flow at time tk equals coupon plus conversion value, setting future cashflows in that

path to zero. Paths that default at tk choose between the recovered conversion value

or the recovery value of the bond, with the future cashflows equal to zero in that path. Called bonds have the option between the call price and the conversion value. After the call the holder is not entitled to any payments. For the rest of the paths in Φtk, cash

flow at time tK−1 is equal to the coupon due at that time.

As noted the algorithm runs recursively, rolling back one step at a time, estimating the continuation values and updating path-wise cumulative discounted cash-flows from continuation at each step. The convertible bond price is then obtained by averaging path-wise cumulative discounted cash flows generated by the bond, when the bondholder applies exercise rules produced by the algorithm along each path.

(23)

Analysis

This chapter contains the analysis of incorporating stochastic dividend yield in convert-ible bond pricing models. In the next section the data is given. Sections 2 and 3 provide the numerical estimation techniques for the different models. Mispricing measures are given in Section 4. The last sections contains the results of two numerical examples and multiple pricing results of models calibrated to real convertible bonds.

4.1 Data

An overview and description of the convertible bonds used in this thesis is provided in this section. A total of 176 convertible bonds issued in the United States are identified from a Bloomberg terminal. These bonds have an issuer that has a nonzero dividend yield. All cross currency convertibles are then removed from the dataset. As a liquidity requirement I consider only convertible bonds where the convertible bond and stock market price is available on every day considered. The dataset has 73 points, starts 1 March 2017 and ends 13 June 2017. After filtering the sample with these criteria, a final sample of 13 bonds is obtained. Besides the convertible bond prices, time series of the stock price, dividend yield, and conversion ratio during this period are available.

Several characteristics of the convertible bonds are given in Table 4.1. None of the bonds considered has a put feature. However, 5 bonds have a call feature, allowing the issuer to repurchase the bond. When the issuer decides to call, the investor has to be notified a certain period in advance. The call notice period is 30 days in the United States. In addition to the call notice period, early redemption for these callable convertible bonds is restricted to a certain period in which the issuer is not allowed to call the bond. In order to call the bond out of this period an additional condition must be met. Calling is allowed if the parity ηS exceeds a call trigger for 20 out of the last 30 trading days.

As a result of this last requirement, the call feature is highly path dependent. In order to approximate this, the call price is multiplied by a factor 1 + π. This excess value accommodates for the presence of a call notice period and for the effective call price for which the bond is ultimately called. Upon notice of an early call, the bondholder may choose to convert its bond or receive the call price.

The contingent conversion clause (Coco) restricts the holder from converting. This feature is present in 8 of the bonds. The bond is eligible for conversion when the stock price exceeds a prespecified percentage of the conversion price for 20 out of the last 30 trading days. This feature is neglected in the modeling and is approximated by converting the bond as soon as it is profitable.

The maturity shown in the table is defined as the remaining time in years from March 1, 2017 until the maturity date. The moneyness S/K is defined as the conversion ratio times the stock price divided by the investment value on March 1, 2017. The average coupon 2.63, the average maturity is 9.52, and the average moneyness of the

(24)

Table 4.1: Characteristics of the bonds

Bond Coupon Maturity S/K Coco Softcall Sector Issuer

1 4.25 1.46 0.93 No Yes Real estate/REIT Forest City Realty Trust Inc. 2 6.50 12.33 0.57 No No Basic materials Ashland Global Holdings Inc. 3 2.25 25.25 0.97 Yes Yes Technology Vishay Intertechnology Inc. 4 2.95 18.79 0.97 No Yes Technology Intel Corp.

5 1.50 2.29 0.88 No No Consumer Goods Newell Brands Inc. 6 0.13 2.08 0.70 Yes No Services Expedia Inc. 7 1.88 1.54 1.00 Yes No Consumer Goods Newell Brands Inc.

8 4.00 30.04 1.00 No Yes Healthcare West Pharmaceutical Services 9 1.63 0.37 0.76 Yes No Basic Materials Newmont Mining Corp. 10 2.13 20.79 1.00 Yes Yes Technology Microchip Technology Inc. 11 0.50 3.62 1.00 Yes No Technology Western Digital Corp. 12 2.00 4.58 0.42 Yes No Services TiVo Corp.

13 4.50 0.62 0.94 Yes No Real estate/REIT Starwood Waypoint Homes

bonds is 0.7.

The sector of the issuing firms varies from basic materials to real estate investment trust. The latter is an interesting case as it is legally obligated to pay dividend to its investors to maintain its status as a investment trust.

Several original issuers have merged or were acquired by another firm during the life of the bond. In most cases the holder of the bond receives an extra cash amount at maturity or if the bond is converted. An exception is the convertible bond issued by Ashland Global Holdings Inc., this firm may defer dividend payments up to 20 quarters. This bond was initially issued by Hercules Inc. before the firm was acquired by Ashland in 2008. Bonds 5 and 7 were issued by Jarden Corp. before Newell Brands Inc. merged with the firm in 2016. In the same year the issuer of Bond 11, Sandisk Corp., was acquired by Western Digital Corp. Expedia Inc. acquired HomeAway Inc, issuer of Bond 6, in 2016. The holders of these bonds receive an extra amount of cash when the bond is conversed.

As a proxy for the riskfree rate I use the short rate implied from the US dollar swaprate. Figure 4.1 shows that the yield curve is upward curving. The 10Y US Dollar short rate is increasing during the period.

Figure 4.1: Interest rate graphs

For the purpose of finding if models assuming stochastic dividend yields yield better pricing results, the data is divided into two sub-samples. The historical sub sample contains 50 points and the forecasting sub sample is 23 days long. The parameters of the model are calibrated using the data in the historical sub sample. With these estimated parameters the prices in the forecasting subsample are calculated and compared to the

(25)

true prices.

Not all issuing companies have ordinary bonds outstanding. For this reason the credit spread cannot be estimated directly from these bonds. Moreover, other parameters for the underlying dividend yield process and the default rate are not directly observable. Therefore, I followZabolotnyuk et al.(2010) and use their technique that allows for the calculation of model prices when the values of the parameters are not observable. In this technique all parameters are estimated from the historical convertible bond prices.

4.2 Estimation of the finite difference method

This section provides the framework for both the constant dividend yield PDE approach and the approach assuming stochastic dividend yield. As these PDEs are parabolic, an analytical solution is not know for the given constraints. These are solved numerically by the finite difference method.

The finite difference method uses a grid of the underlying risk drivers to approximate the PDE. This grid is constructed in the following manner. The domain of the stock price S is [0, ∞). This upper limit must be approximated by a sufficiently large number Smax. The domain for the dividend yield q is [0, 1]. However, the upper limit can be

approximated by a more realistic value qmax. The regions for the τ = T − t transformed

PDEs are therefore [Smin, Smax] × [0, T ] and [Smin, Smax] × [qmin, qmax] × [0, T ] when

the dividend yield is assumed to be stochastic. The following table summarizes the regions:

Table 4.2: Boundaries

min max

S 0 10 × S0

q 0 0.35

Letting ∆S = (Smax− Smin)/N , ∆q = (qmax − qmin)/K, and ∆τ = T /M . The

spatial grid points are

Sn= Smin+ n∆S, n = 0, 1, . . . , N (4.1)

qk = qmin+ k∆q, k = 0, 1, . . . , K (4.2)

τm= m∆τ, m = 0, 1, . . . , M (4.3)

In default the stock value drops to the recovery value of (1−n)Stand the convertible

bond pays a fixed recovery payment equal to a fraction R ∈ [0, 1] on the face value X.

Andersen and Buffum (2002) states that this is consistent with the market practice of what happens in bankruptcy proceedings. The effect of the parameter n on the bond value is researched byZabolotnyuk et al. (2010). In this study they find that the value of n does not have a significant effect on the pricing accuracy and consequently I will assume the stock jump parameter n to be 1, letting the stock fall to 0 in default. The same paper finds the bond recovery parameter R to have a significant effect on the convertible bond price and is therefore estimated from the data in this thesis.

4.2.1 Finite difference without stochastic dividend yield

The finite difference solution for the constant dividend PDE is derived in this subsection. Let λ denote the credit spread λ = γSα and let the stock jump parameter be n = 1. The PDE is given by:

∂G ∂t + (r(t) + λ)S ∂G ∂S + σ2S2 2 ∂2G ∂S2 − (r(t) + λ)G + λRX = 0. (4.4)

(26)

First the PDE is transformed using τ = T − t such that, −∂G ∂τ + (r(τ ) + λ)S ∂G ∂S + σ2S2 2 ∂2G ∂S2 − (r(τ ) + λ)G + λRX = 0. (4.5)

The PDE is discretized using an explicit scheme for n = 1, . . . , N −1 and m = 0, . . . , M − 1, using central differences in the spatial dimension. The partial derivatives are given by: ∂G ∂S(Sn, τm+1) ≈ Gm_n+1− Gm n−1 2∆S = GS, (4.6) ∂2G ∂S2(Sn, τm+1) ≈ Gm_n+1− 2Gm n + Gmn−1 ∆S2 = GSS, (4.7) and ∂G ∂τ (Sn, τm+1) ≈ Gm+1_n − Gm n ∆τ . (4.8)

The transformed PDE can be approximated by − G m+1 n − Gmn ∆τ + (r(τ ) + λ)SnGS+ σ2_S2 n 2 GSSn− (r(t) + λ)G m n + λRX = 0. (4.9)

The boundary conditions are given by:

Gm+1₀ = (1 − ∆τ (r(t) + λ))Gm₀ + ∆τ λRX = 0 S = 0 (4.10)

Gm+1_N = ηSN S = SN, (4.11)

for m = 0, . . . , M − 1.

At every node Gm_n the decision is made whether the holder wants to convert:

Gm_n = max(ηSn, Gmn). (4.12)

The issuer decides if the bond will be called:

Gm_n = min(Gm_n, max(Bc, ηSn)). (4.13)

When there is no call feature present or if the bond cannot be called the call price is set to Bc= ∞.

4.2.2 Finite difference with stochastic dividend yield

The PDE for the convertible bond price with stochastic dividend yield, with assuming n = 1 and letting λ = γSα denote the default probability, is given by:

∂G ∂t + 1 2 ∂2G ∂S2σ 2_S2₊ 1 2 ∂2G ∂q2 (ν 2_{+ φ}2_σ2_{) +} ∂2G ∂S∂qσ 2_{φ − (r(t) + λ)G + (r(t) − q + λ)S}∂G ∂S +∂G ∂q(θ(ω − qt) + φ(r(t) − q) − λq) + λRX = 0. (4.14)

Transforming the PDE using τ = T − t, gives −∂G ∂τ + 1 2 ∂2G ∂S2σ 2_S2₊1 2 ∂2G ∂q2(ν 2_{+ φ}2_σ2_{) +} ∂2G ∂S∂qσ 2_{φ − (r(τ ) + λ)G + (r(τ ) − q + λ)S}∂G ∂S +∂G ∂q(θ(ω − q) + φ(r(τ ) − q) − λq) + λRX = 0. (4.15)

(27)

The PDE is approximated by using the following approximations of the partial deriva-tives: ∂G ∂t(Sn, qk, τm+1) ≈ Gm+1_n,k − Gm n,k ∆τ , (4.16) ∂G ∂S(Sn, qk, τm+1) ≈ Gm_n+1,k− Gm n−1,k 2∆S = GS, (4.17) ∂2_G ∂S2(Sn, qk, τm+1) ≈ Gm_n+1,k− 2Gm n,k+ Gmn−1,k ∆S2 = GSS, (4.18) ∂G ∂q(Sn, qk, τm+1) ≈ Gm_n,k+1− Gm n,k−1 2∆q = Gq, (4.19) ∂2G ∂q2 (Sn, qk, τm+1) ≈ Gm_n,k+1− 2Gm n,k+ Gmn,k−1 ∆q2 = Gqq, (4.20) ∂2G ∂S∂q(Sn, qk, τm+1) ≈ Gm_n+1,k+1+ Gm_n−1,k−1− Gm n,k−1− Gmn−1,k 4∆q∆S = Gsq. (4.21)

The PDE can now be approximated by the following equation:

− G m+1 n,k − G m n,k ∆τ + σ2S_n2 2 GSS+ (ν2+ φ2σ2) 2 Gqq+ σ 2_φG sq+ (r(m) + λ − qk)SnGS + Gq(θ(ω − qk) + φ(r(m) − qk) − λq) + λRX − Gmn,k(r(m) + λ). (4.22)

The conversion ratio adjusts with the dividend yield. In order to approximate these changes any thresholds are neglected and the conversion rate is modeled in the following way: ηn,k = η Sn Sn− (qkSn− Dn) = η 1 1 − (qkS_Sn_n − qk) . (4.23)

When this firm is in default, i.e. S = 0, the partial derivatives regarding q are zero as the stock has zero recovery and the sensitivity with respect to the dividend yield is zero. This yields the following boundary condition:

Gm+1_0,k = ∆τ λRX + Gm_n,k(1 − ∆τ (r + λ)). (4.24) The holder converts the bond when the stock reaches its maximum S = SN. The

boundary condition is given by

Gm+1_N,k = ηN,kSN. (4.25)

The maximum value of the dividend yield is q = qK, where qK is chosen to be sufficiently

higher than the given coupon rate, I assume that the bondholder would want to convert at any time

Gm+1_n,K = ηn,KSn. (4.26)

The boundary condition when the dividend yield is zero is approximated using:

Gm+1_n,0 = 2Gm+1_n,1 − Gm+1_n,2 . (4.27) At every node Gm_n the decision is made whether the holder wants to convert:

Gm_n = max(ηn,kSn, Gmn). (4.28)

The issuer decides if the bond will be called:

Gm_n = min(Gm_n, max(Bc, ηn,kSn)). (4.29)

Again, when no call feature present or if the bond cannot be called the call price equals Bc= ∞.

(28)

4.3 Estimation of the Least squares Monte-Carlo method

The least squares Monte Carlo method uses simulated processes to value the bond. A numerical solution is found by discretization of these underlying processes using the Euler-Maruyama method. The time interval [0, T ] is partitioned into M equal subinter-vals of width ∆t = T /M > 0. The stock recovery (1 − n) is assumed to equal 0, identical to the finite difference method.

The algorithm to price convertible bonds with constant dividend yield is given by, for every i ∈ [0, T ],

1. Calculate the first stopping time τ ∼ Exp(_λ 1

i−1∆t), if τ < 1 the firm defaults.

2. Si+1= 1τ >i(Si+ (ri− q + λi)Si∆t + σSi

√

∆tW ) + 1τ >i(1 − n)Si.

3. λi+1= γSi+1α .

4. Given the stock prices and credit spreads, the Least Squares Monte Carlo algo-rithm is solved recursively.

Let W ∼ N (0, 1), because Wtk+1− Wtk ∼ N (0, ∆t). Furthermore, Exp denotes the

exponential distribution and 1 denotes the indicator function.

The algorithm for pricing a convertible bond subject to stochastic dividend yield is as follows:

1. Calculate the first stopping time τ ∼ Exp(_λ 1

i−1∆t), if τ < 1 the firm defaults.

2. Si+1= 1τ >i(Si+ (ri− qi+ λi)Si∆t + σSi √ ∆tW ) + 1τ >n(1 − n)Sn. 3. ∆qi+1= ω(θ − qi)∆t − λq∆t + φSi−S_S_ii−1 + ν √ ∆tZ 4. qi+1= max(qi+ ∆qi, 0), ensuring non-negativity.

5. λi+1= γSi+1α .

6. Given the stock prices and dividend yields, the conversion ratio is given by ηi =

η0_S_i−(qiSSii−Di) = η

1 1−(qiSi_Si−qi)

.

7. The LSMC algorithm is solved recursively given the stock prices, default proba-bilities, dividend yields, and conversion ratios.

With Z ∼ N (0, 1), by the same arguments.

4.4 Measurement of mispricing

The comparison is based on measures showing the ability of the models to generate prices close to the market price. The first score is the mean absolute deviation (MAD):

M AD = 1 N N X i=1 |Pmarket− Pmodel Pmarket |. (4.30)

The MAD takes into account deviations from market prices from both sides. The second score is the mean deviation (MD), which is given by

M D = 1 N N X i=1 Pmarket− Pmodel Pmarket . (4.31)

(29)

The MD reports the average model over- or underpricing of the convertible bond. An-other indicator of model fit is the root mean squared error (RMSE):

RM SE = v u u t 1 N N X i=1 Pmarket− Pmodel Pmarket 2 . (4.32)

The mean absolute deviation and mean deviation scores assign the same weights to all errors. No extra penalty is given when the model price is far from the market price. The RMSE gives larger weights to larger errors.

4.5 Results

This section holds the analysis of the pricing models. The next two subsections contain numerical analyses of the constant dividend yield pricing model and the stochastic dividend yield model. The last subsection yields the pricing results for convertible bonds from the U.S. market.

4.5.1 Numerical example constant dividend yield

An example of the constant dividend yield pricing model is analyzed in this subsection. The date is assumed to be March 1, 2017 together with following characteristics of the bond:

Table 4.3: Example values

parameter value parameter value parameter value

coupon 1.50% σ 0.15 n 1

maturity 02/12/2019 γ 0.25 r 2%

frequency 2 α -1.20 q 3%

par 1000 R 0.40 η 25

The interest rate and conversion ratio are assumed to be constant over time. The sensitivity of the price with respect to the stock price is visualized in Figure 4.2 for a number of scenarios. The blue line represents a bond when the default probability is zero. The three other lines represent the three different examples as given in Table 4.4.

Figure 4.2: The bond price

The value of the bond is increasing for larger stock values as the bond holder converts if the bond is in the money, i.e. when the conversion value ηS exceeds the par value 1000. The bond will not be converted for stock prices below this value and leads to the so called bond floor. This is clearly shown by the riskless convertible bond, where the

(30)

value of the bond is 100 for every stock price under 30. Above this value the probability of conversion increases and so does the price of the bond.

Accounting for the default probability λ = γSα lowers the value of the out of the money bonds, this is because these are more likely to default and consequently trade at a discount. Due to the inverse relation between stock price and default probability, lower values of S lead to higher default probabilities. The default probability of the in the money bonds is low for values of S and therefore are the prices equal for stock values higher than 45. The value of the bond is equal to the recovery value of 40 when the stock price approaches zero.

The bond price rises when the volatility increases for stock values higher than 20. A higher stock volatility increases the probability of higher returns, but because of the bond floor, the investor has its downside risk covered. This combination results in a higher price for the convertible bond. A higher volatility lowers the bond price for stock values under 20. This is because the chance of default increases when the volatility is high and the stock value is low.

The convertible bond prices of the three cases shown in Figure 4.2 are given in Table 4.4 for two stock values, together with the default probabilities (credit spreads) from λ = γSα. The prices obtained by the two numerical methods are compared in this table. The prices obtained from the FD method are in the 95% confidence interval derived from the standard deviation of the LSMC method. In most examples the FD method returns a higher price than the LSMC method.

Table 4.4: Different parameter combinations

Case 1 2 3 S 50 20 50 20 50 20 R 0.4 0.4 0.4 0.4 0.4 0.4 γ 0.25 0.25 0.25 0.25 0.25 0.25 σ 0.15 0.15 0.4 0.4 0.15 0.15 α -1.2 -1.2 -1.2 -1.2 -0.5 -0.5 credit spread 0.23% 0.69% 0.23% 0.69% 3.54% 5.59% FD 125.8867 97.5002 136.0993 98.0606 125.1282 78.3023 LSMC 124.7946 95.4880 136.2544 96.6303 124.4691 75.4331 MC stdev 1.1216 1.6624 6.3207 3.7573 0.6042 2.5998 Lowerbound 122.5963 92.2297 123.8658 89.2660 123.2849 70.3375 Upperbound 126.9929 98.7463 148.6430 103.9946 125.6533 80.5287

The effect of stochastic dividend yield on the pricing of convertible bonds

on the pricing of convertible bonds

Thijs van der Groen

Contents

Introduction

Chapter 2

Literature review

2.1

Convertible bond pricing models

2.2

Conversion strategies

2.3

Empirical results

Chapter 3

Valuation framework

3.1

Contract features

3.2

Dynamics

3.3

Valuation using PDEs

3.4

Valuation using Monte Carlo algorithms

Analysis

4.1

Data

4.2

Estimation of the finite difference method

4.3

Estimation of the Least squares Monte-Carlo method

4.4

Measurement of mispricing

4.5

Results