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OVCV Model Description

David Frank

Quantitative Finance Development

Bloomberg L.P.

May 15, 2014

Abstract

This document details the Jump-Diffusion and Black-Scholes models used for Convertible bondsin the function OVCV

Keywords. Convertible Bond, Model Description.

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Contents

1 Introduction 3

2 The Stock Process Under the Jump-Diffusion Model 3

3 Derivation of the Convertible Bond PDE Under the Jump-Diffusion Model 4

4 Convertible Bonds Under the Black-Scholes Model 6

5 Convertible Bond Features 7

5.1 Dividend Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.2 Soft Calls With N-of-M Triggers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.3 Contingent Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 Cross-Currency Convertibles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.5 Mandatory Convertibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.6 Exchangeables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.7 Reset Convertibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.8 Make-Whole Calls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.9 Dividend-Forfeit Clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.10 CoPay Clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.11 Mandatory Convertibles Averaging Period . . . . . . . . . . . . . . . . . . . . . . 14

6 Calibration of the Models 15

6.1 Calibration of the Black-Scholes Model. . . . . . . . . . . . . . . . . . . . . . . . . 15

6.2 Calibration of the Jump-Diffusion Model. . . . . . . . . . . . . . . . . . . . . . . . 15

6.3 Calibration of the Hazard Rate to the CDS Spread . . . . . . . . . . . . . . . . . . 16

7 Calibration With Stochastic Credit: the Equity-to-Credit Link 16

8 Delta and Gamma Calculations 19

9 Borrow Cost 20

10 Computation of Expected Life 20

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1 Introduction

In this document, we describe the Jump-Diffusion and Black-Scholes models used for pricing con-

vertible bonds in the function OVCV. We first describe the process followed by the stock priceunder the Jump-Diffusion model. We then derive the partial differential equation (PDE) whosesolution gives the convertible bond price as a function of stock price and time under this model. Wethen consider the Black-Scholes model, a subcase of the Jump-Diffusion model. Next, we discussthe modeling of additional features of convertible bonds, such as dividend protection and soft calls.Next we discuss model calibration, that is, how the model parameters are chosen to match inputvolatilities and CDS spreads (in the Jump-Diffusion case). Next, we describe delta and gammacalculation, and how borrow cost impacts the calculations. Lastly, we explain how we computeexpected life for convertibles.

2 The Stock Process Under the Jump-Diffusion Model

In this section, we describe the stock dynamics followed under the Jump-Diffusion model.

The convertible bond is priced using a one factor model. We assume the stock price follows theusual Black-Scholes, lognormal stock process with time-dependent rates and volatilities, with theaddition of an independent Poisson process to model defaults. The following list describes notationused throughout this document:

Bt

Bond price, including accrued interest (dirty price)F Face value (par value) of the bondSt Stock price at timetr Time-dependent instantaneous forward interest rateq Time-dependent instantaneous forward continuous dividend rate Time-dependent instantaneous forward volatilityR The recovery rateK Time-dependent conversion ratioD The value of the convertible bond after default, including recoveryh Time-dependent hazard rate Fractional loss in the stock price on defaultWt A standard Brownian motion

Ut A Poisson process, independent ofWt Time of (first) default

The lognormal stock process can be described as:

dSt = [r(t) q(t)] Stdt + (t)StdWt (1)

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The addition to these dynamics of a jump in the stock price on default leads to the followingdynamics:

dSt= [r(t) q(t) + h(t)] St dt + (t)St dWt St dUt (2)

where U is an independent Poisson process used to model defaults, with dUt= 1 with probabilityh(t)dt, and 0 otherwise. The notation St is used to denote the stock price immediately beforeany jump at time t. The parameter h(t) is known as the hazard rate. The hazard rate function iscalibrated to Credit Default Swap data if such data is available for the bond issuer. On default,the stock price is assumed to jump downward by exactly the fraction of its pre-default value.

As with the Black-Scholes model for stock options, the model described above leads via the usualarbitrage arguments to a PDE for the convert price. The actual solution method used is to solve

that PDE over a grid in the two dimensions of stock price and time, with boundary conditionsappropriate to the convertible bonds conversion, call, and put provisions. We derive the PDE inthe next section.

3 Derivation of the Convertible Bond PDE Under the Jump-

Diffusion Model

In this section, we derive the partial differential equation which holds for the price of a convertiblebond with default risk under our model. Henceforth we employ subscripts on the variable B todenote partial derivatives of the convertible bond price.

First, consider the case of the convertible bond without default, that is, where h(t) is zero inequation (2). If we form a portfolio consisting of one convertible bond and shares of thestock, then by Itos Formula we arrive at the following PDE for changes in the value of this portfolio:

d =

Bt+

1

2(t)2S2BSS

dt + BS dS(dS+ q(t)Sdt)

Using the standard Black-Scholes argument, we can eliminate risk from the portfolio by choosing

= BS, in which case the portfolio must grow at the risk free rate. We can thus derive a PDE forthe bond price under the no-default assumption.

With the addition of the risk of default, we arrive at the Jump-Diffusion model. We assume thatthe probability of default in the interval [t, t + dt] is h(t)dt, and that after default, the bond valuefalls to some value D, a function ofR and other factors (we provide the exact form ofD later).Then, assuming default risk is fully diversifiable, there is no excess expected return above the risk

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free rate earned for holding credit risk. We can then form a portfolio as above but now includingone risky convertible bond and shares of the stock. The change in value of this portfolio isgiven by

d = [1 dUt ]

Bt+ 12

(t)2S2BSS

dt + BS dS

[1 dUt ]

dS+ q(t)S dt

+ dUt [(D B) + S]

where the first line contains terms that represent the change in the value of the bond if there is nodefault during the period dt, the second line has terms that represent the change in value of theshort stock position if there is no default, and the third line is the change in value of the bond andshort stock when there is a default.

If we now eliminate the stock risk from the portfolio by again choosing =BS, and take expecta-tions with respect to the risk neutral measure we find

[d] = [1 h(t) dt ]

Bt+1

2(t)2S2BSS

dt + BS dS

[1 h(t) dt ]

BSdS+ BSq(t)S dt

+ h(t) dt [(D

B) + SBS]

Now by eliminating terms of order higher than dt and by dropping the canceling dS terms, theequation reduces to

[d] =

Bt+

1

2(t)2S2BSS

dt BSq(t)S dt + h(t)

D B+ SBS

dt (3)

The assumed diversifiability of credit risk implies that the expected return on the portfolio is againthe risk free rate:

r(t) dt= E[d], where =B BSS

This last equation combined with (3) gives us the PDE which we solve to price the convertiblebond under Jump-Diffusion. Direct substitution into (3) gives:

r(t) (B BSS) dt =

Bt+1

2(t)2S2BSS

dt BSq(t)S dt + h(t)

D B+ SBS

dt

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By dividing out dtand simplifying, we find

[r(t) + h(t)] B = Bt+1

2(t)2S2BSS+ [r(t)

q(t) + h(t)] BSS+ h(t)D

Assuming the value on default D is the maximum of the recovery value on the bond, and theremaining post-default conversion value, this leads to the final PDE:

[r(t) + h(t)]B = Bt +1

2(t)2S2BSS+ [r(t)q(t) + h(t)]BSS+ h(t)max[RF, K(t)S(1) ] (4)

We solve the above equation with further modifications to handle discrete dividends and coupon

payments. Further, the convertible bond may have time-varying put, call, and conversion features.These are modeled as constraints which are enforced when the various features are in effect. Some-times, these constraints are simple, and they are applied in an obvious, straightforward manner,e.g. capping the value of the bond at the hard call price at points on the grid where the hardcall is in effect. In other cases, the constraint is more complex, and the techniques used to enforcethe constraint are more complicated, as discussed below in the sections on the various convertiblebond features.

4 Convertible Bonds Under the Black-Scholes Model

Under the Black-Scholes model, the stock price follows the lognormal process given in equation(1). The Black-Scholes model can be viewed as a subcase of the Jump-Diffusion model, where thehazard rate h(t) is zero for all times t. Equation (2) reduces to equation (1) when h(t) is zerobecause the term dUt is zero with probability one, and so the terms h(t) and S

t dUt both dropout of equation (2). Since there are no jumps under Black-Scholes, the stock process is continuousalmost surely, and thus St and St are the same. When Pricing a bond under the Black-Scholesmodel, we solve the same PDE as under Jump-Diffusion, but with the hazard rate h(t) set to zero.

The Black-Scholes model accepts as input an OAS level. In solving the PDE under Black-Scholes,we add the OAS to the time-dependent instantaneous forward interest rate r(t). Thus, the OASis applied as a parallel shift to the instantaneous forward curve.

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5 Convertible Bond Features

In this section, we describe how we model various popular convertible bond features currentlysupported by OVCV.

5.1 Dividend Protection

Dividend Protection typically provides an upward adjustment to the conversion ratio triggered bypayment of a dividend in excess of a specified threshold. The threshold may be specified as anabsolute amount or as a percentage of the stock price, while dividends may be similarly specifiedas either absolute or proportional dividends. To describe our approach to modeling dividendprotection, we first provide formulae for the two forms of conversion ratio adjustment supportedby the model, then describe the numerical approach to modeling with this feature, and finally, weprovide some comparative results showing the effect of dividend protection on bond prices.

We will use the following notation in the discussion of dividend protection:

S The stock price prior to the ex-dividend dateCR0 Conversion ratio before the ex-dividend date, i.e. pre-adjustmentCR1 Conversion ratio after the ex-dividend date, i.e. post-adjustmentD(t) Discrete dividend(s) paid over the relevant period up to timetDR(t) D(t)/S, i.e. the dividend yield over the relevant period up to time tT(t) Threshold level above which dividend protection is applied.K(t) Trigger level which must must be reached for dividend protection to take effect

Note that we must have T K, and in the vast majority of cases, T =K.

Form 1 - Absolute Dividend Protection: This form of dividend protection adjusts the conver-sion ratio for absolute (that is, actual) dividends in excess of a given cash amount. In the simplestcase,

CR1 = C R0S

S C

whereSis the stock price prior to the ex-dividend date, andCis the cash dividend paid in excess of

a threshold amountT. That is, C= max(D(t) T, 0), whereD(t) is the actual dividend amount.

An additional level of complexity is added if the threshold amounts Tare a function of time, i.e.replace T byT(t), giving

CR1 = C R0S

Smax(D(t) T(t), 0)

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Finally, we can add a higher trigger level K, where we allow for dividend protection when thedividend is in excess ofK, but providing protection down to the lower threshold level T, leadingto the first modeled form of dividend protection:

CR1= C R0S

S if(D(t)> K(t), D(t) T(t), 0) (5)

Form 2 - Relative Dividend Protection: This form of dividend protection adjusts the conver-sion ratio for dividends in excess of a given percentage of the then-current stock price. The dividendprotection is similar to Form 1, except the protection is based on the dividend yield rather thanthe absolute dividend amount.

In the simplest form,C R1= C R0[1+ max(DRT, 0)], whereTis again the threshold above which

the conversion ratio is adjusted (now as a percentage, e.g. .02 for 2 percent), andDR, the dividendyield, is calculated by dividing the total dividends in the relevant period by the last observed shareprice.

An added level of complexity is introduced if dividends above trigger Kpercent are protected andthen you get protection for the part above the lower threshold T, leading to

CR1 = C R0 [1 + if(DR > K, DR T, 0)]

Generalizing this to time dependent thresholds and triggers gives

CR1=C R0 [1 + if(DR(t)> K(t), DR(t) T(t), 0)] (6)

Modeling the Conversion Ratio Adjustment: Under both forms of dividend protection,the modeled future dividends at a given time may depend on the stock price (for proportionaldividends), and the threshold may also depend on the stock price (for a proportional threshold).This dependency is handled in the model by adjusting the dividends and the threshold at eachnode of the PDE grid as required.

Modeling dividend protection thus requires keeping track of the evolution through time of theconversion ratio as a function of stock price and time. The evolution of the conversion ratiothrough time will in general depend on the path taken by the stock price through time (ratherthan depending only on the realized stock price at some future horizon date; that is, how we getto the end point matters). We model dividend protection by adding an additional state variablefor the conversion ratio to the PDE grid. The PDE is solved in the usual manner for each of manyconversion ratio levels, that is, as if the different conversion ratio level grids were independent PDEgrids, except that at the designated dividend protection observation times values flow from one

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level to the next as determined by the dividend protection adjustment to the conversion ratio ateach grid point. The modeling approach of adding a second state variable for the conversion ratiofully captures the dependency of the conversion ratio on the stock price path.

Effect of Dividend Protection on Bond Pricing: Next we present some results showingthe effect of dividend protection on convertible bond theoretical prices. We consider a 5 yearconvertible bond with current stock price 9, convertible any time at a strike of 10, r=3%, d=2.5%,vol=25, with a 5 year default probability of .133, equivalent to a flat CDS spread of approximately170bps. The following plot compares this bond with absolute dividend protection to an otherwiseidentical bond with relative dividend protection, starting from comparable thresholds. We use aproportional dividend assumption, at a 2.5% percent annual rate, paid quarterly. (We remarkthat assuming known absolute dividends over the life of a convertible bond having more than say,one year to maturity, seems to conflict with the typical stock volatility that would allow the stockprice to range over a factor of 5, 10, or larger; that is, assuming dividend payments remain thesame regardless of the path the stock price takes seems contradictory to both expectations and

experience).

The x-axis indicates the level above which dividend protection adjustments are done. In the relativeprotection case, the x-axis simply indicates the identical percentage threshold and trigger levels(T(t) and K(t)), while in the absolute protection case, the threshold and trigger levels are set tothe absolute dividend level that is the x-axis percentage of the initial stock price.

We see that in the case of full protection (level zero on the x-axis), the two forms are nearlyidentical, but for partial protection, absolute protection produces a higher theoretical bond price.Since the dividends are set at 2.5%, with relative protection at or above that level the bondholder

receives no benefit from dividend protection. But with an absolute threshold set at 2.5% of theinitial stock price (that is, at 0.25), if the stock runs up the absolute amount of the dividendswill be 2.5% of the larger stock price and will exceed the threshold, thus causing a conversion ratioadjustment and still providing some dividend protection value to the holder. But for a sufficientlylarge threshold (in this case, around 9.5-10.0%), the increase in the level of stock price required forthe dividend to exceed the threshold has such a small probability that the effect becomes negligible,and the convert price under the two forms of protection converge to a common value.

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0 2 4 6 8 10109

110

111

112

113

114

115

116Absolute and Relative Dividend Protection Vs. Protection Threshold

Dividend Protection Threshold Percentage

Bond

Price

Per100

Absolute Dividend Protection

Relative Dividend Protection

5.2 Soft Calls With N-of-M Triggers

N-of-M triggers are seen in soft calls, and also in contingent conversion provisions. Both of theseprovisions allow some action at time t (for soft calls, the issuer may call the bond, for contingentconversion, the holder may convert) provided the stock price S() has exceeded some given barrierB(t) for N (possibly consecutive) days of the last M consecutive days, where N M; typicalvalues for N-of-M provisions are 20-of-30. Exact modeling of such contract provisions is extremelytime consuming, requiring a prohibitive 2M PDE grid levels for an exact solution. We model theseprovisions by calculating an equivalent barrier B(t) such that the probability of S() exceedingthe barrier B(t) for the required N of the previous M days is equal to the probability of theS(t), the stock at time t, exceedingB(t). (As N is required to be consecutive days or not effectsthe probability calculation, and we take this difference into account). This reduces the N-of-Mbarrier to an equivalent 1-of-1 barrier. We then apply the relevant constraint at time t based onthe equivalent barrier B(t). This approximation captures most (but not all) of the value of softcall and contingent conversion provisions, and the technique is widely used in the market. Theequivalent barrier B (t) for the least applicable time t (that is, the earliest time at which there isan N-of-M trigger) is shown in OVCV as the Effective Trigger.

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5.3 Contingent Conversion

Contingent conversion provisions add several subtleties to the modeling. A straightforward appli-cation of contingent conversion is possible when the stock price is observed at time t to determine

convertibility at timet. However, the more common form of contingent conversion requires periodic(typically, quarterly) observation to determine whether conversion is allowed over the subsequentperiod. We model this provision by use of a discrete state variable to track whether or not thelast observation allowed conversion, and thus solve the PDE in two distinct and independent grids,one where contingent conversion is always allowed, one where contingent conversion is never al-lowed. As we move backward in time to solve the PDE numerically, when we cross the periodicobservation times we get values from one plane or the other depending on whether the conditionfor contingent conversion is satisfied at a particular node. An additional complexity which mustbe modeled arises from the standard provision waiving the contingent conversion requirement inthe event of a call. In such case, the issuer will not call even if the bond price is above the callprice, if such a call benefits the holder by then allowing an advantageous early conversion. We use

appropriate tests in the PDE grid to avoid issuers making such disadvantageous call decisions.

5.4 Cross-Currency Convertibles

When the bond is issued in a different currency from the one in which the underlying stock isdenominated, we assume currency exchange rates are fixed and price the bond entirely in the bondcurrency. That is, we model the stock process as if it traded in the bond currency rather than thestock currency, and convert all stock-currency values to bond currency at the appropriate forwardrate.

We treat the stock price in the bond currency as the new underlying variable. The FX volatility andcorrelation between the stock price and exchange rate impact the stock process in the bond currency.We use a term structure FX(t), which is calibrated from the FX at-the-money volatilities. Thevolatility of the new underlying (stock in bond currency) is then effectively

Equity(t)2 + FX(t)2 + 2 Equity(t)FX(t)

where is the correlation between (1) percentage move of stock price in its own currency and (2)percentage move of foreign exchange rate, in units of stock currency per bond currency.

5.5 Mandatory Convertibles

Distinctive handling for mandatory convertibles includes the use of a final boundary conditionreflecting the final payoff, and use of two input volatilities, to capture the fact that the payoffresembles positions in options with two distinct strikes. The two input volatilities are used tocreate a local volatility surface, and the bond is priced using this local volatility surface.

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5.6 Exchangeables

Exchangeable bonds are convertible issues where the conversion is into a stock other than that ofthe issuer. Thus, of the two credits one might consider (that of the issuer, or that of the company

whose stock is received on conversion), the credit having the largest impact on such a bond is thatof the issuer. The issuers credit is generally of greater import since an issuer default will impair theability to collect coupons and principal, and possibly, the stock as well, depending on the presenceor absence of an escrow or trust holding the stock for safe conveyance even in the event of issuerdefault. For this reason we use the issuers credit when pricing Exchangeables.

Since we are not using the credit of the stock, the stock is assumed to follow the process in theBlack-Scholes model, that is, geometric brownian motion, without jumps, and we currently assumeno correlation between the stock price and issuer hazard rates. However, though there are no stockjumps, when using the Jump-Diffusion model, we still model issuer defaults. This occurs due tothe presence of the recovery term in the final PDE solved to price the bond, Equation ( 4). In the

case of exchangeables, the values ofh(t) multiplying B and BSare set to zero, since those hazardrates represent the hazard rate associated with the stock. But the hazard rate h(t) multiplyingthe recovery term is the issuer hazard rate, and this term is still present in the PDE. Note alsothat the recovery term for exchangeables has a different form from that given in Equation (4). Inthe event of default, the payoff is based on the recovery rate and whether the stock is pledged bythe issuer or not. For pledged stock, the payoff to the holder on default is the greater of recoveryvalue or the full conversion value into stock. For unpledged stock, the payoff is recovery valuebased on a bankruptcy claim on the larger of the face value and the full conversion value, that is,the bankruptcy payoff is the assumed bond recovery times the larger of face value and stock pricetimes conversion ratio.

5.7 Reset Convertibles

Reset Convertibles are convertibles where the conversion ratio is reset based on the stock pricebehavior, typically by resetting the conversion ratio upward if the stock price falls more than somethreshold level. Resets can be categorized as either static or dynamic resets. Dynamic resets arecharacterized by the fact that the conversion ratio can reset at any time, whereas for a static resetthe conversion ratio is reset at fixed observation times. We handle static resets, but not dynamic.

Static resets are defined as follows. There is a date schedule consisting of a set of pre-determineddates

{tk, k= 1

n}

which we call reset dates. At each reset date tk, the reference price (usuallydefined as the Volume Weighted Average Price (VWAP) of the underlying stock over a certaintime period immediately prior to the reset date) is calculated. If the reference price is lower thanthe trigger level Pk defined in the bonds term sheet, the conversion price is reset to a level whichis the product of the reference price and a gearing factor gk. For the majority of convertibles whichhave reset features, the conversion price can be reset downward only to compensate investors for adecrease in the bonds parity value. In most cases, we also find the reset of the conversion price isbounded from below by a reset price floor, which puts a limit on this kind of protection.

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We have found there are two ways to define the trigger level: either it is defined as an absolute stockprice level (or equivalently, the percentage pk of the initial conversion price), or as the percentagepk of the prevailing conversion price immediately prior to the reset date. Similarly, the reset price

floor is defined either as the percentage fk of the initial conversion price, or the precentage fk ofthe prevailing one.

Modeling of static resets uses the same approach as modeling of dividend protection. We use theconversion ratio as an additional state variable. Assume that at time tk ,the kth element of thereset schedule, the associated trigger level is pk (in percentage), the reset price floor is fk (alsoin percentage), the gearing factor is gk, the face or par value of the bond is F, and the initialconversion price is K. For the node (Si, CRj) on the 2-dimensional PDE grid, at time tk, weupdateC Rj to CRj using the following formula:

CRj = CRj ifSi>=Pk

max(CRj , min(Ck, FSigk )) otherwise

where the real trigger level Pk is defined as:

Pk =

Kpk ifpk is defined as a percentage of the initial conversion priceF

CRjpk ifpk is defined as a percentage of the prevailing conversion price

and the cap Ck is defined as:

Ck = FKfk iffk is defined as a percentage of the initial conversion price

CRjfk

iffk is defined as a percentage of the prevailing conversion price

The reset of the conversion price can also additionally be floored by an absolute number P. In thiscase the updating formula is changed to:

CRj =

CRj ifSi>=Pkmax(CRj , min(Ck,

F

P, FSigk

)) otherwise

Except for conversion ratio updates as described here at reset dates, we strictly follow the methoddefined in the section on modeling the conversion ratio adjustment on page 8 to solve the two-

dimensional PDE.

5.8 Make-Whole Calls

A Make-Whole call provision is a conditional payment usually contingent on a soft call, but possiblyon a hard call. During the soft call period, when the convertible is called and (or) investors areforced into conversion following a call, investors are entitled to a lump sum payment which is the

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present value of future coupons they wont get because of the early call. The PV calculation isbased on a constant yield or a benchmark yield curve (usually the Treasury curve) plus some spread.Since the Make-Whole provision raises the call value and the conversion value simultaneously, itsignificantly decreases the chance of the bond being called early.

5.9 Dividend-Forfeit Clauses

The Dividend-Forfeit clause is quite typical for French convertibles. For bonds with this feature, onconversion, issuers can choose between delivering existing shares, which are called treasury shares,and issuing new shares and delivering them to the bond holder. The newly issued shares are notentitled to the dividend(s) being paid in the current fiscal year. Since it is therefore advantageousfor the issuer to deliver new shares, we assume they will.

In OVCV, we adjust a bonds conversion value if it has a Dividend-Forfeit clause. At the valuation

times, we find the cut-off time t of the fiscal year in which s belongs to, and calculate the dividendpayment Abetween time s and time tbased on the following formula:

A=ni=1

er(tis)[(1 eq(titi1))Sti + Di] + er(ts)(1 eq(ttn))St

where t0 = s < t1

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6 Calibration of the Models

Calibration of a model is the process of finding model parameters such that the calibration in-struments are priced by the model to match the market prices for those instruments. In this case,

the calibration instruments are vanilla european calls on the convertibles underlying stock, and,for the Jump-Diffusion model, CDS prices appropriate to the credit characteristics of convertiblebond.

Black-Scholes volatilities are by definition the volatilities used in the Black-Scholes SDE for thestock, that is, equation (1), when(t) is constant. Black-Scholes volatilities are what is commonlyquoted in the market; therefore, Black-Scholes volatilities are used in OVCV as user input toindicate the market prices of the vanilla options. As distinguished from Black-Scholes volatilities,we will refer to the (t) seen in equation (2) as jump-diffusion volatilities, since these volatilitiesdrive the part of the jump-diffusion stock process without the jumps. Given a set of Black-Scholesvolatilities, we must calibrate the model to determine the jump-diffusion volatilities which correctlymatch the option prices.

6.1 Calibration of the Black-Scholes Model

In the case of the Black-Scholes, there is nothing to calibrate. The input volatilities (t) (whethera single value or time-dependent) are used directly in the PDE. The only additional computationis that in the case of time-dependent volatilities, the term structure of volatilities is convertedto instantaneous volatilities assuming piece-wise constant instantaneous volatilities between thesupplied term points.

6.2 Calibration of the Jump-Diffusion Model

In the case of the Jump-Diffusion model, our goal is to determine a term structure of jump-diffusionvolatilities which correctly price options at each term in the OVCV Volatility Tab term structure,out to the bond maturity. If a flat volatility is used, we treat the flat volatility as if it werethe quoted volatility at every term on the Volatility Tab term structure; and if E2C is used, wemust jointly calibrate hazard rates and volatility, so we consider the union of the terms on theVolatility Tab and the Credit Tab. (We describe calibration with E2C in more detail in the nextsection). Now we proceed to find piecewise-constant jump-diffusion volatilities by bootstrapping,that is, taking each term in sequence. For the first such term, we determine the constant (t)such that an option priced under the Jump-Diffusion process, with that jump-diffusion volatility,matches the BS price using the Black-Scholes volatility. For each subsequent term, we extend thepiecewise constant function (t) from the previous term so that this terms option priced underthe Jump-Diffusion process, with that volatility function, matches the Black-Scholes price usingthe Black-Scholes volatility.

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6.3 Calibration of the Hazard Rate to the CDS Spread

We convert the CDS spread into piecewise-constant hazard rates using the CDS model describedin DOCS 2057273. This CDS model gives us the time-dependent hazard rates h(t) used in the

PDE we solve, equation (4).

7 Calibration With Stochastic Credit: the Equity-to-Credit Link

Under the model as specified by the PDE of equation (4), the hazard rate at some future time tdoes not vary asSt is higher (or lower) relative to the original stock price S0. This is counter to ourexpectation that as the companys stock rises (falls) we would expect a concomitant improvement(deterioration) in credit quality. That is, we expect the future hazard rate to vary inversely tothe future stock price. Equity-to-Credit (E2C) adds a link between the future stock price and the

future hazard rate which models this behavior. We first calibrate a time-dependent base hazardrate h0(t) and then use this to determine the future hazard rate at time t as a function of both tand St:

h(St, t) =h0(t)

S0St

p

wherep is a positive parameter which can be interpreted as the ratio of the stocks jump-diffusionvolatility to the spread volatility (as shown for the case where = 1 in the paper Calibration andImplementation of Convertible Bond Models by Andersen and Buffum).

The calibrated valuesh0(t) must preserve the expected hazard rates (expectation with respect tothe stock price) implied by the CDS spreads. Thus, the calibration problem at a timetis as follows:ifg (t) is the expected hazard rate at time t as calculated from CDS data, then g(t) is conditionedon survival until time t, and so we must have (letting represent the default time)

g(t) = [h(t, St)| t] =

h0(t)

S0St

p t =h0(t)Sp0

1

Spt

t

This last equation is solved to determine h0(St, t). We also point out that the volatilities used inthe jump-diffusion must be calibrated such that options are priced consistently with market prices

under the stock process including the E2C link. Since a change in the diffusion volatilities affectsthe expectation on the right-hand side of the last equation where we determine h0(t) and a changein h0(t) affects the pricing of options, we jointly calibrate the jump-diffusion volatilities and h0(t).

We next show some calibration results. Here, for various values ofp, we show the effect of convert-ing Black-Scholes volatilities to Jump-Diffusion volatilities, and we show the function h0(t). Wecalibrate to a flat volatility of 40% for calls struck at the ATM-forward, with an initial stock price

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of 50, r(t) = 4%, d(t) = 2%,, and a constant credit spread of 5% (so the survival probability up toany time T is simply e.05T). The calibration fit was performed in monthly increments. (For thesake of comparison, this example was chosen to be identical to the calibration example shown inFigure 6 of Andersen and Buffums paper).

Here we show the calibrated function h0(t) for various values ofp:

0 2 4 6 8 10 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Hazard Rates: h0

Time

P=0

P=.5

P=1

P=2

Here we show the calibrated function (t) of Equation (4) for various values ofp:

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0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

JumpDiffusion Vols:(t)

Time

P=0

P=.5P=1

P=2

To see the effect of E2C on pricing, we consider a 5 year convertible bond with current stock price9, convertible any time at a strike of 10, r=3%, d=0%, vol=50, with a 5 year default probabilityof .133, equivalent to a flat CDS spread of approximately 170bps. If we look at the model priceof this bond a year in the future without E2C, we see the bond price asymptotically approaches aflat line at the present value of the coupons discounted at the risk-adjusted rates. However, whenwe add E2C, the bond floor disappears: as the stock price goes to zero, the bond price also dropsoff to zero, as the credit quality deteriorates. We see this drop off using two different values (0.5and 1.5) for the exponent parameter p.

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0 5 10 1520

40

60

80

100

120

140

160

180Comparison of Bond Price in 1 Year High, Low and No E2C

Stock Price in 1 Year

Bond

Price

Per100

No E2C: P=0.0

Lo E2C: P=0.5

Hi E2C: P=1.5

8 Delta and Gamma Calculations

Bloomberg now offers a choice of calculation methods for Delta and Gamma: users can requesteither bump-and-reprice values for these sensitivities, or grid values. We explain below how thetwo types are calculated, and how they differ.

Bump-and-reprice values are computed by independently repricing the bond at higher and lowerstock prices. Let S0 represent the current stock price, dSrepresent a shift in the stock price, andB(S) represent the bond price computed from the model with initial stock price S. Then Deltaand Gamma can be estimated as

Delta = B(S0+ dS) B(S0 dS)

2 dS

Gamma = B(S0+ dS) + B(S0 dS) 2B(S0)

(dS)2

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Grid values are computed by using values for B(S0dS) computed on the PDE pricing grid. Theseprices are available from the PDE grid used to compute B(S0). Since the bond prices for shiftedstock prices come from the same PDE grid as the bond price for the original stock price, the gridDelta and Gamma will generally be smoother and more accurate estimates than values computed

from bump-and-reprice. We therefore recommend using grid values.

Without E2C, while the two methods produce different results, they are different estimates ofthe same theoretical values. But with E2C the methods are different. The bump-and-repricevalues assume the initial credit spread does not change as the stock price is bumped up and down.Implicit in the grid values however, the credit spread is assumed to change consistently with theE2C assumption as the stock price is shifted. Thus, the grid Delta includes a credit hedge usingstock, whereas the bump-and-reprice delta does not include a credit hedge.

9 Borrow Cost

Borrow cost is a fee paid to borrow stock for the purpose of shorting. This fee appears to thestockholder as a source of income, just like a dividend. Thus, for modelling purposes, we treat theborrow cost as a continuous dividend, that is, as an addition to the variable q(t), the instantaneousforward continuous dividend rate. However, this increment to the dividend is not included in thedividends used to determine conversion ratio adjustments triggered by dividend protection.

10 Computation of Expected Life

Letu(t, St) represent the expected life of the bond, as seen from time t, stock price St, under therisk neutral measure. Assume the stock follows the jump-diffusion process given in (2).

Consider timest and t + dt. Assume the bond has not been early-terminated up to time t, and thatjumps can only occur at times t andt + dt, with all the jump probability occurring in the intervaldtapplied to the probability of a jump at t + dt. Further, assume that for each feature which mightlead to early exercise (such as puts, calls, conversion, etc.) there is a critical boundarySt whichis the least/greatest price at which early exercise occurs for that feature at time t, and that thiscritical price S is a continuous function oft. We will show that the probability ofSt crossing thecritical boundarySt is o(dt).

Assume without loss of generality that St

< St ; a similar argument applies for a barrier below

St. Fix a time interval dt and let S= min(S

u, u (t, t+ dt). There exists some dt such thatS St > 0. Then for any time increment dt < dt,

P(St+dt> S

t+dt)< P(St+dt>S) (7)

Then the probability of crossing S between times tand t + dt is given by

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N

log(St/S) + (r q+ h 2/2)dt

dt

=N

log(St/S)

dt+

(r q+ h 2/2)dt

This probability iso(dt).1 Combining the previous two equations, we can conclude thatP(St+dt>St+dt) is also o(dt). By next-step analysis it then follows that

u(t, s) =dt + Et[u(t + dt,St+dt)] + o(dt) (8)

By Itos formula for jump-diffusion processes, we have

u(t + dt,St+dt) = u(t, S) + utdt + uS[(r q+ h)Sdt + SdW]+12uSS

2S2dt + [u(t + dt,St+dt) u(t, s)] dUtTaking the risk neutral expectation at time tof this equation yields

Et[u(t + dt,St+dt)] = u(t, S) + utdt + uS(r q+ h)dt + 12uSS2S2dt+Et([u(t + dt,St+dt) u(t, s)]dUt) (9)

We now compute the expectation of the jump term, the last term in the equation above. We notethat dUt is one with probability hdt, and when it is one we have a default, so the expected lifechanges from its pre-default value to zero; and when dUt is zero, the entire term is zero. Thus

Et([u(t + dt,St+dt) u(t, s)]dUt) =u(t, S)h dt

Substituting this back into equation (9) yields

Et[u(t + dt,St+dt)] =u(t, S) + ut dt + uS(r q+ h) dt +12

uSS2S2 dt u(t, S)h dt

Finally, substituting this back into equation (8) yields

u(t, s) =dt + u(t, S) + utdt + uS(r q+ h)dt +12

uSS2S2 dt u(t, S)hdt + o(dt)

Canceling the u(t, S) terms on either side and dividing by dt, and ignoring terms smaller thanorder dt, we get a PDE which must be satisfied by the function u:

0 = 1 + ut+ uS(r q+ h) + 12

uSS2S2 uh

1Note first that St < S, so log(St/S) is negative. Thus, as dt goes to zero, the log term goes to negative infinity.The second term of the argument to the cumulative normal function goes to zero and so can be ignored. Thuswe are looking at the rate of decay of the left tail of the normal distribution, which is the same as the rate ofdecay of the right tail. It is easy to see that this tail of the normal distribution decays exponentially. For large x,

x exp(y2/2)dy

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We solve this PDE with boundary conditions at maturity u(t, S) = 0 for all S, and forcing u(t, S)to zero on conversion, put, call, or other life-terminating event explicitly handled by a boundarycondition in the PDE.