Pricing and Hedging CoCos∗

Patrick Cheridito†

ORFE and Bendheim Center for FinancePrinceton University

Princeton, NJ 08544, USA

Zhikai XuORFE

Princeton UniversityPrinceton, NJ 08544, USA

First version: February 1, 2013Current version: June 14, 2013

Abstract

Contingent convertible bonds are typical hybrid products in that they are exposed to differentsources of risk: interest rate risk, equity risk and conversion risk. We develop a general framework fortheir pricing and hedging that can be specified in different ways. We focus on structural and reducedform models driven by a finite-dimensional Markov process. The two approaches are qualitativelydifferent. But both allow to price contingent convertibles and calculate dynamic hedging strategieswith holdings in related instruments such as fixed income products, the issuing company’s stock andcredit default swaps. As case studies we consider contingent convertibles issued by Lloyds BankingGroup in December of 2009 and Rabobank in March of 2010. Both modeling approaches suggest thaton Oct 14, 2011, the contingent convertibles of Lloyds Banking Group traded at a price that was lowrelative to market quotes of interest rate swaps, the firm’s equity and credit default swaps, while theyproduced prices for the Rabobank contingent convertibles that were close to their market value.

Keywords: Contingent convertible bonds, credit default swaps, pricing, calibration, hedging, struc-tural model, reduced form model.

1 Introduction

A typical contingent convertible bond (CoCo) is a corporate bond that converts into equity of the issuingfirm if a prespecified trigger event occurs. However, there exist different variants. Some convert into a cashpayment (write-down), and others just become worthless (write-off). Motivations for issuing CoCos vary.But since the financial crisis of 2007–2009 they have been offered by a number of financial institutionsto protect their capital buffers in times of crisis. For instance, they have been issued by Lloyds BankingGroup, Rabobank, Credit Suisse, Bank of Cyprus, Australia and New Zealand Banking Group, UBS,Zurcher Kantonalbank and Macquarie. The purpose of this paper is to develop a theoretical frameworkfor the pricing, calibration and hedging of CoCos from the point of view of a market participant. A CoCois specified by the following three characteristics:

• Maturity, principal and coupons. Like a standard corporate bond, a CoCo promises to make couponpayments and redeem the principal at maturity. The coupon rate can be fixed or floating.

∗We thank Frederic Abergel, Markus Brunnermeier, Agostino Capponi, Frederic Samama, Taha Vural and AlexanderWugalter for helpful comments.†Partially supported by NSF Grant DMS-0642361

1

• A trigger event causing the CoCo to convert. Different trigger mechanisms are possible. Accountingtriggers are based on accounting measures of capital adequacy. Market triggers are set off by marketevents such as declines in stock prices or indexes. Regulatory triggers allow regulators to imposeconversion on firms in financial distress. Decision triggers leave it to the firm’s management todecide when to convert the CoCo.

• A conversion mechanism describing what happens when the trigger event occurs. A typical CoCoconverts into a prespecified number of equity shares. Write-down CoCos lose part of their principal,and write-off CoCos become worthless. Normally, a CoCo also pays accrued interest at conversion.

Most existing CoCos have an accounting trigger. Some have an additional regulatory trigger. Othersare callable by the issuer, and some include an option for the holder to convert the CoCo early. As casestudies we focus on issuances by Lloyds Banking Group in December of 2009 and Rabobank in March of2010. Lloyds Banking Group issued different CoCos at the end of the year 2009. We consider one thatwas issued on the 1st of December with a maturity of 10 years. It makes fixed coupon payments andconverts into a prespecified number of common equity shares if the Core Tier 1 Ratio of Lloyds BankingGroup falls below 5%. The Rabo CoCo was issued on March 19, 2010 with a maturity of 10 years andfixed coupon payments. Rabobank is a cooperative society without publicly traded stock for the CoCoto turn into. Instead, it converts into an immediate cash payment of 25% of the principal amount ifRabobank’s equity capital ratio falls below 7%. For our theoretical analysis we consider the followingprototype:

• The maturity is T , the principal amount is F , and there is a stream of fixed coupon payments1 ciat times 0 < t1 < · · · < tn = T .

• The trigger event occurs at a random time τ .

• If triggered, the CoCo converts into G shares of equity.

This covers the Lloyds CoCo. The Rabo CoCo is simpler. Instead of G shares of equity it convertsinto a payment of G units of currency. All our formulas can easily be adjusted to this case. A CoCo is atypical hybrid product in that it depends on different sources of risk:

• Interest rate risk. Before conversion, a CoCo is a fixed income product and therefore sensitive tomovements of the risk-free yield curve. This exposure can be hedged with government bonds orinterest rate swaps.

• Conversion risk. If the Coco has a market trigger based on one or more liquidly traded securities,they can be used to hedge conversion risk. Otherwise, we propose to hedge it with CDS’s. Conver-sion risk is related to default risk, and CDS’s are issued with long maturities. Many of them areliquidly traded.

• Equity risk. A CoCo that potentially converts into equity is exposed to equity risk. It can behedged with equity shares, futures or options. Write-down and write-off CoCos have no exposureto equity risk.

1For simplicity we only consider fixed coupon payments. Floating coupon rates can be covered with minor modifications.

2

A general CoCo model should include all relevant sources of risk. Moreover, for calibration andhedging purposes it should lend itself to the efficient valuation of related instruments. For a CoCo that istriggered by the market value of the issuing firm’s equity, it is enough to model the firm’s stock price andrisk-free interest rates. Other trigger mechanisms require additional model components. If conversionis not triggered by liquidly traded instruments, we propose to hedge it with CDS’s. To price them, onemust describe the firm’s default time. Two kind of models have been studied extensively in the creditrisk literature: structural models, which go back to Merton (1974) and describe bankruptcy as an eventin which the value of a firm’s total assets falls below that of its liabilities, and reduced form models,originated by Jarrow and Turnbull (1995), in which credit events happen when a point process jumps.We investigate both approaches to modeling the trigger event and describe default in the same way. Moreprecisely, in a structural model we assume conversion and default to occur at hitting times of a stochasticprocess and in a reduced form model at the first two jump times of a time-changed Poisson process. Inthe structural approach the trigger process could be a market price, an accounting ratio or a quantityobserved by regulators. We assume it to be continuous and observable. Then the prices of the issuingfirm’s stock and the CoCo are not expected to jump at conversion since even if the firm’s equity is dilutedat conversion, investors continuously take this into account before it happens. A reduced form model candescribe any trigger mechanism not directly based on liquidly traded securities. It models conversion anddefault as jumps of a time-changed Poisson process. Therefore, they come as a surprise, and prices ofthe firm’s stock, the CoCo and CDS’s are expected to jump. In both cases it is theoretically possible toperfectly hedge the CoCo by dynamically investing in enough non-redundant securities. In practice it isimportant that they be liquidly traded and offer exposure to the same types of risk as the CoCo. As casestudies we price the CoCos issued by Lloyds Banking Group on Dec 1, 2009 and Rabobank on March19, 2010 by calibrating a structural and reduced form model to market quotes of equity shares, risk-freeyields and CDS spreads. Both approaches need the recovery rate used in the calibration to CDS spreadsas an input. It can be estimated from historical default data or inferred from time series of CDS spreadsas, for instance, in Pan and Singleton (2008). Alternatively, it can be chosen so that the model pricesthe CoCo at market value, yielding a CoCo-implied recovery rate. Both our model specifications pricedthe Lloyds CoCo at market value with an implied recovery rate of 60% or more, significantly higher thanthe 40% often assumed by practitioners. This suggests that on the pricing date, Oct 14, 2011, either themarket price of the Lloyds CoCo was low compared to interest rate swaps, the firm’s stock price andCDS’s, or investors were expecting the recovery rate to be considerably higher than the standard 40%.The Rabo CoCo was priced at market value by both models with an implied recovery rate very close to40%. So under standard assumptions, they priced the Rabo CoCo practically at their market value.

Most of the existing quantitative CoCo studies take a structural approach. Raviv (2004) as wellas Hilscher and Raviv (2012) use the barrier approach of Black and Cox (1976) to price CoCos. DeSpiegeleer and Schoutens (2012) and its generalization Corcuera et al. (2012) focus on modeling theissuing firm’s equity value and approximate accounting triggers by an event where the stock price fallsbelow some level. Papers that use a structural model for studying CoCo designs include Pennachi (2010),Albul et al. (2010), McDonald (2010), Glasserman and Nouri (2010), Koziol and Lawrenz (2011), Boltonand Samama (2012), Buergi (2012), Berg and Kaserer (2012), Brigo et al. (2013), Metzler and Reesor(2013), Pennachi et al. (2013). For a critical assessment of some of the existing CoCo pricing models werefer to Wilkens and Bethke (2012).

The contribution of this paper consists in a general CoCo framework that can be specified in differentways. The structure of the paper is as follows. In Section 2 we provide formulas for the pricing andcalibration of CoCos in the case where conversion happens at a general stopping time. In Section 3 westudy structural models and in Section 4 reduced form models. In both cases we assume the underlying

3

uncertainty to be generated by a finite-dimensional Markov process. Then a CoCo can be hedged witha dynamic trading strategy if there exist enough liquidly traded securities with exposure to the samesources of risk. In Section 5 we use a structural and reduced form model to value CoCos issued by LloydsBanking Group and Rabobank.

2 General formulas for pricing, calibration and hedging

We consider a financial institution with outstanding CoCos and assume that its equity shares pay div-idends at a constant rate q ≥ 02. We model the market price of an equity share and the instantaneousrisk-free interest rate with stochastic processes (St)t≥0 and (rt)t≥0. The filtration generated by all observ-able events is denoted by (Ft)t≥0, and discounted prices of future cash-flows are assumed to be martingales

under a risk neutral probability measure Q. In particular, St := e−∫ t0 rsdseqtSt is a Q-martingale, and the

time-t price of a risk-free zero-coupon bond with maturity s is given by P (t, s) := EQt

[e−

∫ st rvdv

], where

EQt denotes the conditional expectation with respect to Ft. We denote the conversion time by τ and the

default time by θ. We assume both of them to be stopping times with respect to (Ft) and τ ≤ θ, thatis, the firm cannot go into bankruptcy before conversion has been triggered. In Section 3, τ and θ areboth modeled as hitting times of a stochastic process and in Section 4 as the first two jump times of atime-changed Poisson process.

A standard CoCo can be viewed as the sum of a defaultable bond and an option that delivers a fixednumber of equity shares if the trigger event occurs. If the CoCo has not converted by time t < T andcan be hedged with liquid securities, its unique arbitrage-free price is

Ct =∑ti>t

ciEQt

[e−

∫ tit rsds1{τ>ti}

]+∑ti>t

ciEQt

[e−

∫ τt rsds

τ − ti−1

ti − ti−11{ti−1<τ≤ti}

]+ FEQ

t

[e−

∫ Tt rsds1{τ>T}

]+GEQ

t

[e−

∫ τt rsdsSτ1{τ≤T}

].

(2.1)

The first two terms represent the value of future coupon payments together with accrued interest payedupon conversion. The third term is the value of the principal and the last term the value of a possibleconversion into equity. If the CoCo converts into a cash payment instead of equity shares, formula (2.1)is easily adjusted by replacing the last term with

GEQt

[e−

∫ τt rsds1{τ≤T}

]. (2.2)

The following result gives the price in a more convenient form (Qt,Qit,Q∗t denote conditional proba-

bilities with respect to Ft).

Theorem 2.1. If by time t < T the CoCo has not converted yet, its price Ct can be written as∑ti>t

ciP (t, ti)Qit[τ > ti] +

∑ti>t

citi − ti−1

EQt

[e−

∫ τt rsds(τ − ti−1)1{ti−1<τ≤ti}

]+ FP (t, T )Qn

t [τ > T ] +GStEQ∗t

[e−q(τ−t)1{τ≤T}

],

(2.3)

2The model could be extended to include stochastic dividends. But this would only have a minor influence on the priceof a CoCo.

4

where the measures Qi and Q∗ are defined by

dQi

dQ=

e−∫ ti0 rsds

EQ

[e−

∫ ti0 rsds

] anddQ∗

dQ=Sτ∧TS0

. (2.4)

If the CoCo converts into a cash payment instead of equity, the last term of formula (2.3) has to bereplaced with

GEQt

[e−

∫ τt rsds1{τ≤T}

]. (2.5)

In the special case where τ is independent of (rs)t≤s≤T with respect to Qt, the first three terms of formula(2.3) simplify to∑

ti>t

ciP (t, ti)Qt[τ > ti],∑ti>t

citi − ti−1

EQt

[P (t, τ)(τ − ti−1)1{ti−1<τ≤ti}

], FP (t, T )Qt[τ > T ] (2.6)

and the expression (2.5) toGEQ

t

[P (t, τ)1{τ≤T}

].

Remark 2.2. It is possible that if a CoCo converts, there is a structural break in the dynamics of thestock price St. First, the capital structure of the firm changes, and moreover, the management mightimplement a new strategy, or the company is restructured. However, for the valuation of CoCos it isenough to specify the stock price St up to time τ ∧ T and make sure that St∧τ∧T is a Q-martingale. Ifone extends the process St beyond τ ∧ T in such a way that it stays a Q-martingale until time T , onecan define Q∗ by dQ∗/dQ = ST /S0. Then formula (2.3) still holds since the term under the conditionalexpectation EQ∗

t is Fτ∧T -measurable. If the CoCo converts into a cash payment, one does not have tomodel the stock price St at all, except in the case where the conversion trigger depends on it, or it isneeded to model CDS’s.

A meaningful model should also be able to price liquidly traded instruments that are related to CoCossuch as the issuing firms’s stock, fixed income products and CDS’s. The stock price St is a building block

of our model, and risk-free zero-coupon bonds are given by P (t, s) = EQt

[e−

∫ st rvdv

]. Moreover, we assume

that at the pricing date t < T , there exist K liquidly traded CDS contracts with maturities T1 < · · · < TKlying on an equally spaced grid t < s1 < s2 < . . . such that TK ≥ T . If the CoCo has not converted untiltime t, default has not occurred. Therefore, the time-t value of a protection buyer position in a CDSwith coupon times si and maturity Tk is PLt − CLt, where

PLt = (1−R)EQt

[e−

∫ θt rsds1{θ≤Tk}

](2.7)

is the value of the protection leg and

CLt = δ∑

t<si≤Tk

∆sEQt

[e−

∫ sit rsds1{θ>si}

]+ EQ

t

[e−

∫ θt rsds(θ − si−1)1{si−1<θ≤si}

](2.8)

the value of the coupon leg. R is the recovery rate. In reality it is random. But for simplicity, practitionersusually assume it to be constant3 (often around 40%). It could be estimated from past defaults or with

3More generally, one can assume the recovery rate to be a random variable independent of other components of themodel. This adds the flexibility that its expectation can be different under the statistical and pricing measure. Instead ofa fraction of the principal, one could also model the recovery amount as a fraction of the discounted principal or marketvalue; we refer to Duffie and Singleton (1999) for more details.

5

the method of Pan and Singleton (2008) from time series of CDS spreads and then use it for CoCopricing. Alternatively, it can be chosen so that the model prices CoCos at market value, resulting ina CoCo-implied recovery rate. ∆s denotes the time between coupon payments in years. It normally is1/4 or 1/2. δ is the spread and specified in the contract. Formula (2.8) gives the time-t value of futurecoupons payments including accrued interest. Before CDS’s were standardized, the spread was usuallyset so that no initial cash-flow was necessary. Now this is in general no longer possible. But the price ofa CDS with maturity Tk is still quoted in terms of the spread that would make its current price equal tozero:

δ(t, Tk) =(1−R)EQ

t

[e−

∫ θt rsds1{θ≤Tk}

]∑

t<si≤Tk ∆sEQt

[e−

∫ sit rsds1{θ>si}

]+ EQ

t

[e−

∫ θt rsds(θ − si−1)1{si−1<θ≤si}

] . (2.9)

In the models of Sections 3 and 4 below we assume τ and θ to be independent of the short rate. Then itfollows as in the proof of Theorem 2.1 that δ(t, Tk) can be written as

δ(t, Tk) =(1−R)EQ

t

[P (t, θ)1{θ≤T}

]∑t<si≤Tk ∆sP (t, si)Qt[θ > si] + EQ

t

[P (t, θ)(θ − si−1)1{si−1<θ≤si}

] . (2.10)

So to price all relevant instruments in the models of Sections 3 and 4, it will be enough to compute thefollowing quantities:

Qt[τ > ti], EQ∗t

[e−qτ1{τ≤T}

], EQ

t

[P (t, τ)1{τ≤ti}

], EQ

t

[P (t, τ)τ1{τ≤ti}

]Qt[θ > si], EQ

t

[P (t, θ)1{θ≤si}

], EQ

t

[P (t, θ)θ1{θ≤si}

].

(2.11)

Since we are only interested in the price of a CoCo as long as it has not converted, we just have tocalculate the expressions (2.11) in the case t < τ . Moreover, for pricing and calibration we take thewhole curve P (t, .) as given. For finitely many tenors the prices of zero-coupon bonds can be deducedfrom market data of government bonds or interest rate swaps. From there the curve can be completed byinterpolation. So for pricing and calibration, one does not need an explicit model for the short rate (rt).To hedge interest rate risk one can either specify the dynamics of (rt) or just immunize against the mostcommon movements of the risk-free yield curve. The hedging aspect will be discussed in more detail inthe frameworks of the following two sections.

3 Structural models

In this section we assume that the randomness is generated by a d-dimensional diffusion process that canbe realized as the unique strong solution of an SDE of the form

dXt = a(Xt)dt+ b(Xt)dWt, (3.1)

where (Wt) is a d-dimensional Brownian motion and a : Rd → Rd, b : Rd → Rd×d are deterministicfunctions. We suppose that the instantaneous risk-free interest rate is of the form

rt = r(Xt) for a continuous function r : Rd → R. (3.2)

The conversion time τ and default time θ are assumed to be hitting times:

τ = inf {t ≥ 0 : Ht ≤ h∗} , θ = inf {t ≥ 0 : Ht ≤ h∗} , (3.3)

6

where h∗ ≥ h∗ are constants and Ht is a process of the form Ht = h(Xt) for a continuous functionh : Rd → R. We think of Ht as a capital ratio since most existing CoCos have an accounting trigger.Alternatively, it could be the stock price4, an index or a quantity that is observed by the regulator. Tokeep the model tractable, we assume that (Ht) is continuous, observable and independent of (rt). Thenτ and θ are predictable and independent of (rt). This means that conversion does not come as a surprise,and therefore, even though the number of equity shares increases at time τ , it should not induce a jumpin the stock price since investors continuously take the possibility of conversion into account before thetrigger event happens.

We assume that for t ≤ τ , the stock price St equals X1t , and the first component of the SDE (3.1)

reads asdX1

t = X1t (rt − q)dt+X1

t σ(Xt)TdWt

for a volatility function σ : Rd → Rd such that

St = exp

(∫ t

0σ(Xs)

TdWs −1

2

∫ t

0σ(Xs)

Tσ(Xs)ds

)is a martingale under Q. Theoretically, one could add jumps to the process (Ht) or assume it to beonly partially observable. In this case, there would be an element of surprise if the trigger event occurs,and one would expect the stock price to jump. However, it would make the model much more complex.In Section 4 below we discuss the other extreme where conversion happens at the first jump time of atime-changed Poisson process and therefore is not predictable.

3.1 CoCo and CDS pricing with PDEs

The distributions of hitting times are known in closed form only in special cases. But the quantities(2.11) can always be obtained by solving simple parabolic PDEs with Dirichlet boundary conditions.

Let D :={x ∈ Rd : h(x) > h∗

}and denote for all (t, x) ∈ [0, T ] × D by Qt,x the probability Q

conditioned on Xt = x and τ > t. For the corresponding conditional expectation we write EQt,x.

The following four propositions give PDEs for Qt,x[τ > ti], EQ∗t,x[e−qτ1{τ≤T}], EQ

t,x

[P (t, τ)1{τ≤ti}

]and

EQt,x

[P (t, τ)τ1{τ≤ti}

]. The PDEs for Qt,x[θ > si], EQ

t,x

[P (t, θ)1{θ≤si}

]and EQ

t,x

[P (t, θ)θ1{θ≤si}

]are the

same, except that the conversion level h∗ has to be replaced with the default level h∗.

Proposition 3.1. Fix i = 1, . . . , n, and assume there exists a bounded function u : [0, ti] × D → Rsatisfying the PDE

ut(t, x) +Au(t, x) = 0, u(t, x) = 0 for x ∈ ∂D, u(ti, x) = 1D(x), (3.4)

where

Au(t, x) :=∑j

aj(x)∂u

∂xj(t, x) +

1

2

∑j,k

αjk(x)∂2u

∂xj∂xk(t, x), α(x) := b(x)b(x)T .

Thenu(t, x) = Qt,x[τ > ti] for all (t, x) ∈ [0, ti]×D.

4The case where conversion is triggered if the stock price breaches a lower barrier is discussed in the appendix.

7

Proposition 3.2. Assume u : [0, T ]× D → R is a bounded solution of the PDE

ut(t, x) +A∗u(t, x) = 0, u(t, x) = e−qt for x ∈ ∂D, u(T, x) = e−qT 1∂D(x), (3.5)

where

A∗u :=∑j

a∗j (x)∂u

∂xj(t, x) +

1

2

∑j,k

αjk(x)∂2u

∂xj∂xk(t, x), a∗(x) := a(x) + b(x)σ(x).

Thenu(t, x) = EQ∗

t,x[e−qτ1{τ≤T}] for all (t, x) ∈ [0, T ]×D.

Proposition 3.3. Fix i = 1, . . . , n, t < ti and x ∈ D. If u : [t, ti]× D → R is a bounded solution of thePDE

ut(s, y) +Au(s, y) = 0, u(s, y) = P (t, s) for y ∈ ∂D, u(ti, y) = P (t, ti)1∂D(y), (3.6)

thenu(t, x) = EQ

t,x

[P (t, τ)1{τ≤T}

].

Proposition 3.4. Fix i = 1, . . . , n, t < ti and x ∈ D. If u : [t, ti]× D → R is a bounded solution of thePDE

ut(s, y) +Au(s, y) = 0, u(s, y) = P (t, s)s for y ∈ ∂D, u(ti, y) = P (t, ti)ti1∂D(y), (3.7)

thenu(t, x) = EQ

t,x

[P (t, τ)τ1{τ≤T}

].

3.2 Hedging in structural models

We now address the hedging of a CoCo in a structural model. The goal is to replicate the CoCo byinvesting in liquidly traded securities. This hedges a short CoCo position. A long position is hedged withthe opposite strategy. Let us assume that there exists a money market account with instantaneous returnrt and an additional d non-redundant liquid securities. A unit of currency invested in the money marketgrows like π0

t = exp(∫ t

0 rsds), and since Xt is a Markov process, prices of the CoCo and all other securitiesat time t < τ are given by π(t,Xt) and πj(t,Xt) for deterministic functions π, πj : [0, T ]× Rd → R.

3.2.1 Perfect hedge

If the functions π, πj are C1,2 on [0, T )×Rd, it follows from Ito’s lemma that the CoCo can perfectly behedged with a dynamic trading strategy ϑjt , j = 0, . . . , d, satisfying

∂π

∂xi(t,Xt) =

d∑j=1

ϑjt∂πj

∂xi(t,Xt), i = 1, . . . , d, and π(t,Xt) = ϑ0

tπ0t +

d∑j=1

ϑjtπj(t,Xt) (3.8)

at all times t < τ . In theory, since the d-dimensional Brownian motion W has the predictable representa-tion property, any d non-redundant securities can be used as hedging instruments. However, in practiceone would naturally try to use securities with exposure to the same risk sources as the CoCo.

8

3.2.2 Perfect hedge for a CoCo that is not triggered by the stock price

In case conversion is not caused by the stock price, the CoCo can be hedged with stock shares, CDS’sand interest rate swaps. To compute a realistic hedging strategy we make the following assumptions:

a) π1 is the stock S

b) π2, . . . , πm+1 are m CDS’s, Ht only depends on X2t , · · · , Xm+1

t , and the coefficients of the SDEs forX2t , · · · , Xm+1

t do not depend on X1t

c) πm+2, . . . , πd are interest rate swaps, rt only depends on Xm+2t , . . . , Xd

t , and the coefficients of theSDEs for Xm+2

t , · · · , Xdt do not depend on X1

t , . . . , Xm+1t .

Then for t < τ the system of equations (3.8) can be written as

GEQ∗t

[e−q(τ−t)1{τ≤T}

]= ϑ1

t

∂π

∂xi(t,Xt) =

m+1∑j=2

ϑjt∂πj

∂xi(t,Xt) for i = 2, · · · ,m+ 1

∂π

∂xi(t,Xt) =

d∑j=2

ϑjt∂πj

∂xi(t,Xt) for i = m+ 2, . . . , d

π(t,Xt) = ϑ0tπ

0t +

d∑j=1

ϑjtπj(t,Xt).

For fixed t < τ , this is a system of d+1 linear equations in ϑ0t , . . . , ϑ

dt , which, if it is regular, has a unique

solution. Under assumptions a)–c), which are reasonable if the CoCo is not directly triggered by S, stockshares are only used to hedge equity risk. As a consequence, the hedging position in the stock ϑ1

t neverexceeds the number of equity shares G the CoCo converts into if triggered. If conversion is triggeredby the stock price, one invests in equity shares to hedge equity and conversion risk. So investments inthe stock might exceed G; see De Spiegeleer and Schoutens (2012). If the CoCo converts into a cashpayment, ϑ1

t is zero.

3.2.3 Hedging without specifying the interest rate

Instead of specifying the full model and trying to implement a complete hedge, one can invest in thestock and different CDS’s to hedge the equity and conversion risk and then immunize against movementsof the risk-free yield curve. This has the advantage that the short rate (rt) does not have to be modeledexplicitly. For simplicity we here only immunize against parallel shifts of the risk-free yield curve. Butone can also take other movements, like twists and changes in curvature, into account. As before, weconsider a CoCo that is not triggered by the issuing firm’s stock price and assume a)–b) from above. Butwe do not make any assumptions on the interest rate (rt), except that it is a Markov process independentof (Ht).

For given t < τ , one holds ϑ1t = GEQ∗

t

[e−q(τ−t)1{τ≤T}

]stock shares and invests in the m CDS’s such

that

∂π

∂xi(t,Xt) =

m+1∑j=2

ϑjt∂πj

∂xi(t,Xt) for i = 2, · · · ,m+ 1.

9

Then one invests in fixed income products to immunize against parallel shifts of the yield curve. Thesensitivity of the CoCo with respect to a parallel shift of the yield curve is the negative of the absoluteFisher–Weil duration (AFW), which for a standard CoCo, is

Yt =∑ti>t

ci(t− ti)P (t, ti)Qt,s[τ > ti] + F (T − t)P (t, T )Qt,x[τ > T ]

+∑ti>t

citi − ti−1

EQt,x

[(τ − t)P (t, τ)(τ − ti−1)1{ti−1<τ≤ti}

],

(3.9)

and for a CoCo converting into cash,

Yt =∑ti>t

ci(t− ti)P (t, ti)Qt,x[τ > ti] + F (T − t)P (t, T )Qt,x[τ > T ]

+∑ti>t

citi − ti−1

EQt,x

[(τ − t)P (t, τ)(τ − ti−1)1{ti−1<τ≤ti}

]+GEQ

t,x

[P (t, τ)1{τ≤T}

].

(3.10)

Investments in the stock are not sensitive to movements of the risk-free yield curve, and the AFW of aprotection buyer position in a CDS with maturity Tk is

Y kt =(1−R)EQ

t,x

[(θ − t)P (t, θ)1{θ≤T}

]−

∑t<si≤Tk

∆s(si − t)P (t, si)Qt,x[θ > si] + EQt,x

[(θ − t)P (t, θ)(θ − si−1)1{si−1<θ≤si}

]. (3.11)

So parallel shifts of the yield curve can be neutralized by investing in a fixed income portfolio with AFWYt −

∑m+1j=2 ϑjtY

jt .

The durations (3.9)–(3.11) are readily computed if one knows the quantities (2.11) together with

EQt,x

[P (t, τ)τ21{τ≤ti}

]and EQ

t,x

[P (t, θ)θ21{θ≤si}

].

The latter two can be calculated by solving the same PDE as in Proposition 3.4 with adjusted boundaryconditions.

4 Reduced form models

In this section we assume that the underlying noise is generated by (Xt) and (Nt), where (Xt) is ad-dimensional diffusion process of the form (3.1) and (Nt) a one-dimensional standard Poisson process.Since a Brownian motion and a Poisson process defined on the same probability space are automaticallyindependent, it follows that (Xt) is independent of (Nt). As in Section 3, we assume the instantaneousrisk-free interest rate rt to be given by r(Xt) for a continuous function r : Rd → R. In addition weintroduce a continuous function λ : Rd → R+ and set λt := λ(Xt). The conversion time τ and default timeθ are modeled as the first two jump times of the time-changed Poisson process NΛt , where Λt :=

∫ t0 λsds.

It is natural to have negative correlation between the increments of (λt) and (St). When the stock pricedecreases, market participants are anticipating a higher likelihood of conversion and default. However,we assume (λt) and (rt) to be independent. Then τ and θ are independent of (rt). Moreover, we supposethat the first line of equation (3.1) can be written as

dX1t

X1t

= (rt − q − γλt)dt+ σ(Xt)TdWt,

10

and for t ≤ τ , the stock price moves like

dStSt−

= (rt − q − γλt)dt+ σ(Xt)TdWt + γdNΛt , (4.1)

for a function σ : Rd → Rd and a constant γ ≥ −1 such that

St∧τ =

{exp

(∫ t0 σ(Xs)

TdWs − 12

∫ t0 σ(Xs)

Tσ(Xs)ds− γΛt

), t < τ

(1 + γ)Sτ−, t ≥ τ

is a martingale under Q. This means that before conversion, (St) is continuous, and at time τ it jumps byγSτ−. It is difficult to predict whether and by how much the stock price will jump in reality. It will dependon the specifics of the CoCo contract and the situation the company will be in. Furthermore, the stateof the financial sector and the broader economy might play a role. If, for instance, conversion happensduring a systemic crisis, the stock price might react differently than in normal times. For simplicity weassume γ to be a constant. Then one can consider the conclusions of the model for different values ofγ, or one can estimate γ by modeling the balance sheet. For instance, if one assumes that at conversionof the CoCo, the changes in market value of all items on the balance sheet cancel out, one obtains thefollowing equation for the fraction γt by which the stock price would jump if conversion were to happenat time t:

nEγtSt + nC(G(1 + γt)St − Ct) + JLt = 0. (4.2)

Here nE denotes the number of equity shares before conversion, nC the number of outstanding CoCosand JLt the jump in market value of all remaining liabilities if conversion were to happen immediately.So γt really is a stochastic process, and since Ct is a function of future values of γt, solving equation (4.2)for γt would amount to a fixed point problem. However, to price and hedge a CoCo on an interval [t, T ],one can approximate the jump fraction as follows:

1. Calculate Ct assuming the stock does not jump at conversion.

2. Choose γ ∈ R such that equation (4.2) holds at time t.

3. Use γ to price and hedge the CoCo during the interval [t, T ].

4.1 CoCo and CDS pricing with intensities

In the following proposition we provide formulas for the quantities (2.11) in terms of the jump intensity(λt).

11

Proposition 4.1. Assume τ has not occurred until time t < T . Then

Qt,x[τ > ti] = EQt,x

[e−(Λti−Λt)

]EQ∗t,x[e−qτ1{τ≤T}] =

∫ T

te−qsEQ∗

t,x

[(1 + γ)λse

−(1+γ)(Λs−Λt)]ds

EQt,x[P (t, τ)1{τ≤ti}] =

∫ ti

tP (t, s)EQ

t,x

[λse−(Λs−Λt)

]ds

EQt,x[P (t, τ)τ1{τ≤ti}] =

∫ ti

tP (t, s)sEQ

t,x

[λse−(Λs−Λt)

]ds

Qt,x[θ > si] = EQt,x

[(1 + Λsi − Λt)e

−(Λsi−Λt)]

EQt,x

[P (t, θ)1{θ≤si}

]=

∫ si

tP (t, s)EQ

t,x

[λs(Λs − Λt)e

−(Λs−Λt)]ds

EQt,x

[P (t, θ)θ1{θ≤si}

]=

∫ si

tP (t, s)sEQ

t,x

[λs(Λs − Λt)e

−(Λs−Λt)]ds.

The next result is useful for calculating the expectations on the right side of the equations in Propo-sition 4.1.

Proposition 4.2. Denote

φt,x(s, z) := EQt,x

[e−z(Λs−Λt)

], s ≥ t, z ≥ 0,

and assume that EQt,x

[supt≤u≤s λu

]<∞ for all s ≥ t. Then

−∂φt,x∂s

(s, 1) = EQt,x

[λse−(Λs−Λt)

]−∂φt,x

∂z(s, 1) = EQ

t,x

[(Λs − Λt)e

−(Λs−Λt)]

∂2φt,x∂s∂z

(s, 1)− ∂φt,x∂s

(s, 1) = EQt,x

[λs(Λs − Λt)e

−(Λs−Λt)].

Remark 4.3. Analogously to Proposition 4.2 one can introduce

φ∗t,x(s, z) := EQ∗t,x

[e−z(1+γ)(Λs−Λt)

], s ≥ t, z ≥ 0,

and show that if EQ∗t,x

[supt≤u≤s λu

]<∞, then

−∂φ∗t,x∂s

(s, 1) = EQ∗t,x

[(1 + γ)λse

−(1+γ)(Λs−Λt)].

4.2 Hedging in reduced form models

As in Section 3 we assume there exists a money market account growing like π0t = exp(

∫ t0 rsds). To

completely hedge a CoCo in a reduced form model one needs one more liquid instrument than in astructural model due to the additional uncertainty coming from the Poisson process (Nt). (Xt, Nt) is stilla Markov process. So the prices of the CoCo and any d + 1 hedging instruments before conversion aregiven by π(t,Xt) and πj(t,Xt) for deterministic functions π, πj : [0, T ]× Rd → R.

12

4.2.1 Perfect hedge

If the functions π and πj are C1,2 on [0, T ) × Rd, one obtains from Ito’s lemma that a standard CoCo

can be hedged perfectly with a dynamic strategy ϑjt , j = 0, . . . , d+ 1, satisfying for all t < τ ,

∂π

∂xi(t,Xt) =

d+1∑j=1

ϑjt∂πj

∂xi(t,Xt), i = 1, . . . , d

G(1 + γ)X1t − π(t,Xt) =

d+1∑j=1

ϑjt∆πj(t,Xt)

π(t,Xt) = ϑ0tπ

0t +

d+1∑j=1

ϑjtπj(t,Xt),

where ∆πj(t,Xt) denotes the jump-to-conversion, that is, the price jump of the j-th security if the CoCowere to convert at time t. Not all security prices jump if the CoCo converts. But in this model CDS’sdo so since the default intensity jumps up.

4.2.2 Perfect hedge for a CoCo that is not triggered by the stock price

If the CoCo is not triggered by the issuing firm’s stock price, one can hedge it by investing in the stock,CDS’s and interest rate swaps. Let us assume the following:

a) π1 is the stock S

b) π2, . . . , πm+2 are m+ 1 CDS’s, λt only depends on X2t , . . . , X

m+1t , and the coefficients of the SDEs

for X2t , . . . , X

m+1t do not depend on X1

t

c) πm+3, . . . , πd+1 are interest rate swaps, rt only depends on Xm+2t , . . . , Xd

t , and and the coefficientsof the SDEs for Xm+2

t , . . . , Xdt do not depend on X1

t , . . . , Xm+1t .

Then to construct the hedging portfolio ϑt, one has to solve the following system of equations for allt < τ :

GEQ∗t

[e−q(τ−t)1{τ≤T}

]= ϑ1

t

∂π

∂xi(t,Xt) =

m+2∑j=2

ϑjt∂πj∂xi

(t,Xt) for i = 2, . . . ,m+ 1

∂π

∂xi(t,Xt) =

d+1∑j=2

ϑjt∂πj

∂xi(t,Xt) for i = m+ 2, . . . , d+ 1

G(1 + γ)X1t − π(t,Xt) = ϑ1

tγX1t +

m+2∑j=2

ϑjt∆πj(t,Xt)

π(t,Xt) = ϑ0tπ

0t +

d+1∑j=1

ϑjtπj(t,Xt).

13

4.2.3 Hedging without specifying the interest rate

If the interest rate (rt) is not specified one can invest in St and different CDS’s to hedge against eq-uity and conversion risk and then immunize against movements of the risk-free yield curve. As in thestructural model, we only neutralize parallel shifts. Let us assume that the CoCo is not triggered bythe stock price, a)–b) from above hold and (λt) is independent of (rt). Then for t < τ , one buysϑ1t = GEQ∗

t

[e−q(τ−t)1{τ≤T}

]stock shares and invests in the m+ 1 CDS’s such that

∂π

∂xi(t,Xt) =

m+2∑j=2

ϑjt∂πj

∂xi(t,Xt) for i = 2, · · · ,m+ 1,

and

G(1 + γ)X1t − π(t,Xt) = ϑ1

tγX1t +

m+2∑j=2

ϑjt∆πj(t,Xt).

After that one immunizes against parallel shifts of the yield curve by purchasing a fixed income portfoliowith AFW Yt −

∑m+1j=2 ϑjtY

jt . It can be shown as in Proposition 4.1 that

EQt,x

[P (t, τ)τ21{τ≤ti}

]=

∫ ti

tP (t, s)s2EQ

t,x

[λse−(Λs−Λt)

]ds

and

EQt,x

[P (t, θ)θ21{θ≤si}

]=

∫ si

tP (t, s)s2EQ

t,x

[λs(Λs − Λt)e

−(Λs−Λt)]ds.

Together with Proposition 4.2 this allows to calculate the AFWs of the CoCo and the CDS’s.

5 Examples

In this section we apply a structural and reduced form model to price CoCos issued by Llyods BankingGroup and Rabobank. In both models, we use a small number of stochastic factors. This already seemsto yield realistic results. But the models can easily be extended by adding more factors. The short ratedoes not have to be specified since it enters the pricing formulas only through the zero-coupon bondprices P (t, s), which for finitely many s-values can be deduced from government bonds or interest rateswaps. From there the curve can be interpolated.

Like Corcuera et al. (2012) we choose Oct 14, 2011 as the pricing date and use the same data onrisk-free yields and CDS spreads for calibration. The specifics of the two CoCos we consider are as follows:

• The Lloyds Banking Group’s Enhanced Capital Notes (ISIN XS0459089255) were issued on Dec1, 2009 with a maturity of 10 years. They pay semi-annual coupons at an annual rate of 15%. IfLloyds Banking Group’s Core Tier 1 Ratio falls below 5%, each ECN converts into F/K ordinaryequity shares, where F is the principal amount of the note and K equals 59 pence.

In 2011, Lloyds Banking Group reported a Core Tier 1 Ratio of 9.6%, while Basel III regulationrequires it to be at least 4.5%. On the pricing date the ECNs had a time to maturity of 8.19 yearsand traded at 109.9% of their principal amount. The market price of Lloyds Banking Group’s stockwas 33.25 pence.

14

• The Rabobank’s Senior Contingent Notes (ISIN XS0496281618) were issued on March 19, 2010with a maturity of 10 years. They pay yearly coupons at a rate of 6.875%. They are triggered ifRabobank’s Equity Capital Ratio (equity capital/ risk-weighted assets) falls below 7%. If triggered,they are written down by 75%, and each note converts into an immediate cash payment of 25% ofthe principal amount F .

Rabobank reported an Equity Capital Ratio of 14.7% in 2011, and Basel III requires a minimumof 4.5%. On the pricing date, the time to maturity of the SCNs was 8.44 years, and they traded at88.84% of their principal amount.

The following table shows risk-free yields and CDS spreads prevailing on Oct 14, 2011. Lloyds BankingGroup’s stock and the ECNs are denominated in GBP. The risk-free yields were extracted from GBPinterest rate swap data. The SCNs are denominated in EUR. The risk-free yields correspond to EURinterest rate swaps.

GBP risk-free yields and Lloyds Banking Group CDS spreadsTenor in years 1 2 3 4 5 7 10

Risk-free yield in % 1.73 1.38 1.53 1.73 1.94 2.35 2.81CDS spread in bps 253.2 286.4 299.3 315.8 325.9 330.2 337.5

EUR risk-free yields and Rabobank CDS spreadsTenor in years 1 2 3 4 5 7 10

Risk-free yield in % 2.12 1.62 1.75 1.93 2.13 2.47 2.77CDS spread in bps 50.7 73.7 97.6 109.2 116.3 122.4 127.1

5.1 Structural model with an exponential OU accounting ratio

The easiest model for the accounting ratio would be a (exponential) Brownian motion. But we obtaineda better fit to market quotes of CDS spreads with an exponential Ornstein–Uhlenbeck process. So weassume that the stock price follows a geometric Brownian motion

dSt = (rt − q)Stdt+ σSt(√

1− ρ2dW 1t + ρdW 2

t )

and the logarithm of the accounting ratio an Ornstein–Uhlenbeck process

dHt = κ(m−Ht)dt+ ηdW 2t .

Then the measure Q∗ is given by

dQ∗

dQ=ST

S0

= exp

(σ(√

1− ρ2W 1T + ρW 2

T

)− 1

2σ2T

),

and it follows from Girsanov’s theorem that the dynamics of (Ht)t≥0 can be written as:

dHt = κ(m∗ −Ht)dt+ ηdW ∗t ,

where m∗ = m+ησρ/κ and W ∗t = W 2t −σρt is a Brownian Motion under Q∗. The problem of pricing and

calibrating CoCos thus reduces to solving linear PDEs with Dirichlet boundary conditions as described in

15

Section 3.1. We solved the PDEs numerically with the Crank–Nicholson method.5 In the case of LloydsBanking Group, Ht is meant to model the logarithm of the Core Tier 1 Capital Ratio and in Rabobank’scase the logarithm of the equity capital ratio. exp(h∗) is the contractual trigger level (5% for Lloydsand 7% for Rabobank) and exp(h∗) the minimum capital ratio level required by Basel III (4.5% in bothcases).

For a given value of the recovery rate R we calibrated the model by choosing the parameters κ, m, ηand the starting point h of the process (Ht) so that the CDS spreads produced by the model were closeto the market quotes. However, the error function constructed only with CDS spreads is almost flat insome regions, causing problems for gradient based minimization algorithms. Therefore, we also inferredthe Q-survival probabilities from the CDS spread data using the bootstrapping method of O’Kane andTurnbull (2003) and added them to the objective function. More precisely, we chose the parameterswhich minimized the function∑

i=2,3,4,5,7,10

(δmodeli − δmarketi

δmarketi

)2

+

(qmodeli − qOKTi

qOKTi

)2

, (5.1)

for the CDS spreads δmodeli , δmarketi implied by the model and market data and the Q-survival probabilitiesqmodeli , qOKTi produced by our model and the one of O’Kane and Turnbull (2003). It is well-known thatstructural models do not fit short term CDS spreads well; see for instance, Lando (2004). Therefore, wedropped the one year CDS spread from the calibration. The following table lists the optimal parameterscorresponding to the CoCo-implied recovery rates, that is, the recovery rates R which made the modelprices of the CoCos equal to their market prices (76.6% for Lloyds and 42% for Rabobank). RMSEdenotes the root mean squared error of the calibration.

Parameters Lloyds Banking Group Rabobank

κ 1.0488 0.2034exp(m) 9.07% 15.10%

η 0.5853 0.3960exp(h) 10.92% 13.14%R 76.60% 42.00%

RMSE 6.85% 4.74%

The top two panels of Figure 1 compare the Q-survival probabilities of Lloyds Banking Group andRabobank implied by our model to the ones constructed with the O’Kane and Turnbull (2003) approach.It can be seen that they are almost indistinguishable. The panels in the middle of Figure 1 plot the CDSspreads generated by our model against the market quotes. The fit is not perfect. Especially for LloydsBanking Group, the 7- and 10-year CDS spreads predicted by the model are too low. This is in line with,for instance, Eom et al. (2004) or Huang and Zhou (2012), which argue that standard structural modelsdo not fit CDS spread term structures well. In the case of Rabobank the fit is better, which is reflectedin a smaller RSME. The two bottom panels of Figure 1 show CoCo prices produced by the model fordifferent values of the recovery rate R chosen in the CDS calibration. The jaggedness of the curves iscaused by numerical instabilities of the calibration.

In the case of Lloyds Banking Group, the CoCo price also depends on σρ, which cannot be inferredfrom CDS data. We set it equal to 30% to model a strong positive correlation between the increments of

5Going-Jaeschke and Yor (2002) and Alili et al. (2005) derived formulas for hitting time distributions of Ornstein–Uhlenbeck processes. But the expressions given in these papers are so complicated that it was easier for us to solve thePDEs numerically.

16

0 2 3 4 5 7 100.2

0.4

0.6

0.8

1Lloyds: survival probabilities (recovery rate = 76.60%)

Tenor in years

MarketOU model

2 3 4 5 7 10200

250

300

350

400

Tenor in years

Spr

ead

in b

ps

Lloyds: CDS spreads

MarketOU model

20 40 60 8080

100

120

140

160Lloyds: CoCo price as function of CDS recovery rate

Recovery rate in %

Pric

e in

% o

f the

prin

cipa

l

0 2 3 4 5 7 100.2

0.4

0.6

0.8

1RaboBank: survival probabilities (recovery rate = 42.00%)

Tenor in years

MarketOU model

2 3 4 5 7 1050

75

100

125

150

Tenor in years

Spr

ead

in b

ps

RaboBank: CDS spreads

MarketOU model

20 40 60 8050

60

70

80

90

100

110RaboBank: CoCo price as function of CDS recovery rate

Recovery rate in %

Pric

e in

% o

f the

prin

cipa

l

MarketOU model

MarketOU model

Figure 1: Q-survival probabilities, CDS spreads and CoCo prices with exponential OU accounting ratio

the stock price and the accounting ratio. However, it can be seen from the value decomposition in Figure2 below that a possible conversion into equity accounts for only about 20% of the value of the ECNs.So the choice of σρ has only a minor influence on the model price of the ECNs. The calibration yieldeda long term mean for the accounting ratio of 9.07% and an implied accounting ratio at the pricing dateof 10.92%, slightly higher than the reported 9.6%. The CoCo-implied recovery rate R is 76.60%, muchhigher than the usually assumed 40%. This suggests that on Oct 14, 2011, either the ECNs traded lowrelative to the stock of Lloyds, interest rate swaps and CDS’s, or market participants were expectinga higher recovery rate than usual. In any case, it turned out that the price of the ECNs increased toaround 130% of the principal amount during the first half of 2012.

17

For Rabobank, the estimated long term mean of the accounting ratio came out as 15.10%, and theimplied capital ratio at the pricing date as 13.14%, slightly below the reported 14.7%. The CoCo-impliedrecovery rate for Rabobank was 42.00%. So under standard assumptions, the model would have pricedthe SCNs very close to their market value on Oct 14, 2011.

Figure 2 shows the decompositions of the values of the ECNs and the SCNs into the parts stemmingfrom the coupon payments, the redemption of the principal and a possible conversion. The recovery ratechosen in the calibration ranges from 10% to 90%. It can be seen that for both CoCos future couponpayments and a possible redemption of the principal account for most of the total value. For increasingvalues of the recovery rate, the model-implied Q-survival probabilities decrease and the conversion valuesbecome larger.

5.2 Reduced form model with a CIR jump intensity

We now consider a reduced form model with a mean-reverting jump intensity. To ensure that it does notbecome negative, we model it with a Cox–Ingersoll–Ross process. It is then possible to deduce closedform expressions for the functions φt,x and φ∗t,x introduced in Proposition 4.2 and Remark 4.3.

More precisely, we assume that the stock price follows the dynamics

dStSt−

= (rt − q − γλt) dt+ σ1dW1t + σ2

√λtdW

2t + γdNΛt

and the jump intensity is a CIR process:

dλt = κ(m− λt)dt+ η√λtdW

2t .

Then Q∗ is given by

dQ∗

dQ=

{exp

(σ1W

1T + σ2

∫ T0

√λtdW

2t − 1

2

(σ2

1T + σ22

∫ T0 λtdt

)− γΛT

)if τ > T

(1 + γ) exp(σ1W

1τ + σ2

∫ τ0

√λtdW

2t − 1

2

(σ2

1τ + σ22

∫ τ0 λtdt

)− γΛτ

)if τ ≤ T.

It follows from Girsanov’s theorem that the dynamics of (λt) can be written as

dλt = κm+ (ησ2 − κ)λtdt+ η√λtdW

∗t ,

where W ∗t = W 2t −σ2

∫ t0

√λsds is a Brownian Motion under Q∗. To calculate the expressions of Proposi-

tion 4.1 with the methods of Proposition 4.2 and Remark 4.3 we have to compute the Laplace transforms

φt,λ(s, z) := EQt,λ

[e−z(Λs−Λt)

]and φ∗t,λ(s, z) := EQ∗

t,λ

[e−z(1+γ)(Λs−Λt)

].

Due to the affine structure of the CIR process one has

φt,λ(s, z) = eA(s−t,z)λ+B(s−t,z)

for

A(u, z) =−2z(1− e−α(z)u)

2α(z)− (α(z)− κ)(1− e−α(z)u)

and

B(u, z) = −κmη2

(2 ln

(1− α(z)− κ

2α(z)(1− e−α(z)u)

)+ (α(z)− κ)u

),

18

Lloyds: CoCo value decomposition

Recovery rate in %

Pric

e in

% o

f the

prin

cipa

l

10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

160CouponsPrincipalEquity

Rabobank: CoCo value decomposition

Recovery rate in %

Pric

e in

% o

f the

prin

cipa

l

10 20 30 40 50 60 70 80 900

20

40

60

80

100

120CouponsPrincipalCash Payment

Figure 2: CoCo value decompositions in the structural model

where α(z) =√κ2 + 2η2z. It follows that

∂φt,λ∂s

(s, z) = φt,λ(s, z)

(∂A

∂u(s− t, z)λ+

∂B

∂u(s− t, z)

)∂φt,λ∂z

(s, z) = φt,λ(s, z)

(∂A

∂z(s− t, z)λ+

∂B

∂z(s− t, z)

)∂2φt,λ∂z∂s

(s, z) = φt,λ(s, z)

(∂2A

∂z∂u(s− t, z)λ+

∂2B

∂z∂u(s− t, z)

)+

(∂A

∂z(s− t, z)λ+

∂B

∂z(s− t, z)

)∂φt,λ∂s

(s, z).

19

Analogous expressions can be found for φ∗t,λ(s, z) and ∂φ∗t,λ(s, z)/∂s. For a given zero-coupon curveP (t, .), one can evaluate the integrals of Proposition 4.2 numerically to obtain approximations to all thequantities of (2.11).

For a given recovery rate R, we chose the parameters κ,m, η and the initial value λ of the jumpintensity (λt) so as to produce CDS spreads consistent with market quotes. As in the structural model,we added the Q-survival probabilities extracted from CDS spreads with the method of O’Kane andTurnbull (2003) to the objective function and minimized the squared error function (5.1). The followingtable gives he resulting parameters corresponding to the CoCo-implied recovery rates (60.6% for Lloydsif γ = 0 and 39% for Rabobank). It can be seen that the root mean squared errors are considerablysmaller than for the structural model of Subsection 5.1.

Parameters Lloyds Banking Group Rabobank

κ 0.9519 0.2530m 0.16 0.04η 0.0014 0.0001λ 0.53 0.14R 60.60% 39.00%

RMSE 1.98% 2.81%

The two panels at the top of Figure 3 show the Q-survival probabilities produced by our reducedform model and the ones extracted with the method of O’Kane and Turnbull (2003). As in the structuralapproach, they are practically equal. The panels in the middle of Figure 3 show CDS spreads generatedby the model compared to the market quotes. The fit is better than in the structural model of Subsection5.1 above. The last two panels of Figure 3 show CoCo model prices as a function of the recovery ratechosen in the CDS calibration.

The conversion value of the ECNs also depends on the product ησ2 and the jump fraction γ, whichcannot be deduced from CDS spreads. To obtain a negative correlation between the increments of thestock price and the conversion intensity, we chose σ2 equal to -30%. However, since in this example theempirical value of η is very small, the choice of σ2 has practically no influence on the model price of theECNs. For gamma we picked values of -20%, 0% and 20%. The resulting differences in ECN prices areminor. In all three cases the CoCo-implied recovery rate is around 60%, above the standard 40%. So,as in Subsection 5.1, the model suggests that on Oct 14, 2011 the market value of the ECNs was lowcompared to related products, or investors expected a higher recovery rate than usual. The market priceof the SCNs was consistent with market quotes of interest rate swaps and CDS’s for a recovery rate veryclose to 40%.

Figure 4 shows the value decompositions of the ECNs and SCNs into the components correspondingto coupon payments, principal redemption and conversion value. The value of the recovery rate R rangesfrom 10% to 90%, and the jump fraction γ of the stock price of Lloyds Banking Group was set equalto 0. Like in the structural model, the CoCo prices are decreasing in the recovery rate, and the maincontribution to their values comes from coupon payments and a possible redemption of the principal.

6 Conclusion

This paper develops a framework for the pricing and hedging of CoCos. It introduces a general modelthat can price CoCos together with related products such as fixed income instruments, equity sharesand CDS’s. We concentrated on structural and reduced form specifications based on a finite-dimensional

20

0 1 2 3 4 5 7 100.4

0.5

0.6

0.7

0.8

0.9

1Lloyds: survival probabilities (recovery rate = 60.60%)

Tenor in years

MarketCIR model

1 2 3 4 5 7 10200

250

300

350

400

Tenor in years

Spr

ead

in b

ps

Lloyds: CDS spreads

20 40 60 80

80

100

120

140

Lloyds: CoCo price as function of CDS recovery rate

Recovery rate in %

Pric

e in

% o

f the

prin

cipa

l

Marketγ = 0%γ = +20%γ = −20%

0 1 2 3 4 5 7 100.4

0.5

0.6

0.7

0.8

0.9

1RaboBank: survival probabilities (recovery rate = 39.00%)

Tenor in years

MarketCIR model

1 2 3 4 5 7 1020

40

60

80

100

120

140

160

Tenor in years

Spr

ead

in b

ps

RaboBank: CDS spreads

MarketCIR model

20 40 60 8050

60

70

80

90

100Rabobank: CoCo price as function of CDS recovery rate

Recovery rate in %

Pric

e in

% o

f the

prin

cipa

l

MarketCIR model

MarketCIR model

Figure 3: Q-survival probabilities, CDS spreads and CoCo prices with CIR jump intensity

Markov process. The two approaches are qualitatively different. In a structural model the conversion timeis predictable, and consequently, the prices of CoCos, the issuing firm’s stock and CDS’s are continuousat conversion. In a reduced form model, conversion comes as a surprise, and prices jump. But bothapproaches can be taken to calculate CoCo prices and dynamic hedging strategies. As case studies, wecalibrated a structural and a reduced form model to market quotes of equity, interest rate swaps andCDS’s to price CoCos issued by Lloyds Banking Group on Dec 1, 2009 and Rabobank on March 19,2010. On Oct 14, 2011 both models would have priced the Lloyds CoCo at market value with an impliedrecovery rate of more than 60%. This suggests that on the pricing date, the Lloyds CoCos either tradedat price that was low relative to market quotes of related instruments, or investors were expecting a

21

Lloyds: CoCo value decomposition

Recovery rate in %

Pric

e in

% o

f the

prin

cipa

l

10 20 30 40 50 60 70 80 900

50

100

150CouponsPrincipalEquity

Rabobank: CoCo value decomposition

Recovery rate in %

Pric

e in

% o

f the

prin

cipa

l

10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

100CouponsPrincipalCash Payment

Figure 4: CoCo value decompositions in the reduced form model

considerably higher recovery rate than the standard 40% in case of a default of Lloyds Banking Group.Both models priced the Rabobank CoCos at market value with an implied recovery rate very close to40%. Consistently with the credit risk literature, we found that it was easier to reproduce the termstructure of CDS spreads with a reduced form model, and the calibration of a structural model posedsome numerical challenges.

22

A Proofs

Proof of Theorem 2.1It is clear that the first and third term of (2.1) can be written as∑

ti>t

ciEQt

[e−

∫ tit rsds1{τ>ti}

]=∑ti>t

ciP (t, ti)Qit[τ > ti] (A.1)

andFEQ

t

[e−

∫ Tt rsds1{τ>T}

]= FP (t, T )Qn

t [τ > T ]. (A.2)

To transform the last term, one uses that (St) is a Q-martingale. Therefore conditioned on τ > t, onehas

GEQt

[e−

∫ τt rsdsSτ1{τ≤T}

]= GStEQ

t

[Sτ

Ste−q(τ−t)1{τ≤T}

]= GStEQ∗

t

[e−q(τ−t)1{τ≤T}

].

If τ is independent of (rs)t≤s≤T with respect to Qt, the measures Qit in (A.1)–(A.2) can be replaced with

Qt, and by first conditioning on τ , one obtains∑ti>t

citi − ti−1

EQt

[e−

∫ τt rsds(τ − ti−1)1{ti−1<τ≤ti}

]=∑ti>t

citi − ti−1

EQt

[P (t, τ)(τ − ti−1)1{ti−1<τ≤ti}

],

GEQt

[e−

∫ τt rsds1{τ≤T}

]= GEQ

t

[P (t, τ)1{τ≤T}

].

Proof of Propositions 3.1–3.4Propositions 3.1–3.4 are Feynman–Kac type results. If u is a bounded solution of the PDE (3.4), thenthe process Mt = u(t∧ τ,Xt∧τ ) is a bounded martingale. So on the set {Xt = x, τ > t}, one has u(t, x) =Mt = EQ

t,x [Mti ] = Qt,x[τ > ti], which shows Proposition 3.1.

By Girsanov’s theorem, W ∗t = Wt −∫ t∧τ

0 σ(Xs)ds is a Brownian motion under Q∗, and one canwrite dXt = a∗(Xt)dt + b(Xt)dW

∗t , t ≤ τ . So if u is a bounded solution of the PDE (3.5), the process

Mt = u(t ∧ τ,Xt∧τ ) is a bounded Q∗-martingale. On the set {Xt = x, τ > t}, this gives u(t, x) = Mt =EQ∗t,x[MT ] = EQ∗

t,x[e−qτ1{τ≤T}], which proves Proposition 3.2.If u is bounded solution of the PDE (3.6), then Ms = u(s ∧ τ,Xs∧τ ) is a bounded martingale. So on

the set {Xt = x, τ > t} one has

u(t, x) = Mt = EQt,x[MT ] = EQ

t,x

[P (t, τ)1{τ≤T}

].

This shows Proposition 3.3. The proof of Proposition 3.4 is exactly the same.

Proof of Proposition 4.1Under Q, conditioned on t < τ and (λs), the density of τ is given by λse

−(Λs−Λt) and the one of θby λs(Λs − Λt)e

−(Λs−Λt). So all equalities except the second one follow by first conditioning on (λs).Moreover, it follows from Girsanov’s theorem, that under Q∗, conditioned on t < τ and (λs), the densityof τ is given by (1 + γ)λse

−(1+γ)(Λs−Λt). This yields the second equality.

Proof of Proposition 4.2One has

φt,x(s, 1) = EQt,x

[e−(Λs−Λt)

]= 1−

∫ s

tEQt,x

[λue−(Λu−Λt)

]du,

23

and since EQt,x

[supt≤u≤s λu

]< ∞, one deduces from Lebesgue’s dominated convergence theorem that

EQt,x

[λue−(Λu−Λt)

]is continuous in u. So φt,x(s, 1) is continuously differentiable in s with derivative

−EQt,x

[λse−(Λs−Λt)

]. Moreover, for z > 0, one has

φt,x(s, z)− φt,x(s, 1) = −∫ z

1EQt,x

[(Λs − Λt)e

−u(Λs−Λt)]du,

and (Λs−Λt)e−u(Λs−Λt) is uniformly bounded in u. It follows that EQ

t,x

[(Λs − Λt)e

−u(Λs−Λt)]

is continuous

in u. Therefore, φt,x(s, z) is continuously differentiable in z with derivative −EQt,x

[(Λs − Λt)e

−z(Λs−Λt)].

Analogously, it follows that −EQt,x

[(Λs − Λt)e

−z(Λs−Λt)], is continuously differentiable in s with derivative

EQt,x

[zλs(Λs − Λt)e

−z(Λs−Λt) − λse−z(Λs−Λt)]. This completes the proof.

B CoCos with stock price triggers

Market triggers are transparent but have the drawback that they are prone to manipulation. Here weshortly discuss how they fit into the general framework developed in this paper. Let us consider a CoCowhich converts into equity if the stock price St reaches a lower level S∗ < S0. That is, the conversiontime is of the form

τ = inf {t ≥ 0 : St = S∗} .

In the benchmark case where the interest rate is equal to a constant r and the stock price evolves like

St = S0 exp

{(r − q − 1

2σ2

)t+ σWt

}(B.1)

for a constant volatility σ and a Q-Brownian motion W , expressions for the quantities (2.11) can bederived from well-known results on hitting times of Brownian motion. Conditioned on τ > t, one has

τ = inf {s ≥ t : Ws = −αt + βs}

for

αt =logSt − logS∗

σand β =

q + σ2/2− rσ

.

So conditioned on τ > t, the distribution of τ under Qt is given by Qt[τ ≤ s] =∫ st ft(u)du for

ft(u) =αt√2πu3

exp

(− 1

2u(αt − βu)2

);

see for instance, Steele (2001). This allows to compute the quantities

Qt[τ > ti], EQt

[P (t, τ)1{τ≤ti}

]and EQ

t

[P (t, τ)τ1{τ≤ti}

].

Under the distorted measure Q∗ = ST /S0 = exp(σWT − σ2T/2), W ∗t = Wt − σt is a Brownian motion.Therefore, conditioned on τ > t, one has Q∗t [τ ≤ s] =

∫ st f∗t (u)du for

f∗t (u) =αt√2πu3

exp

(− 1

2u(αt − β∗u)2

), where β∗ = β − σ.

This can be used to evaluate the expectation EQ∗t

[e−qτ1{τ≤T}

].

24

If rt and St follow more general diffusion dynamics, CoCos with stock price triggers can be priced bysolving PDEs like in Section 3. Calibration and hedging can also be done according to Section 3. Onlynow equity shares and options are more closely related to the trigger event than CDS’s. However, equityoptions often only exist with short maturities and strikes around the money, while CDS contracts aretraded with long maturities; compare to the discussion in Corcuera et al. (2012).

References

[1] Albul B., Jaffey D.M., Tchistyi A. (2010). Contingent convertible bonds and capital structure deci-sions. Working Paper, University of California at Berkeley.

[2] Alili L., Patie P., Pedersen J.L. (2005). Representations of the first hitting time density of anOrnstein–Uhlenbeck process, Stochastic Models, 21(4), 967–980.

[3] Berg T., Kaserer C. (2012). Does contingent capital induce excessive risk-taking and prevent anefficient recapitalization of banks? Preprint.

[4] Black F., Cox J. (1976). Valuing corporate securities: some effects of bond indenture provisions.Journal of Finance 31, 351–367.

[5] Bolton P., Samama F. (2012) Capital access bonds: contingent capital with an option to convert.Economic Policy 27(70), 275–317.

[6] Brigo D., Garcia J., Pede N. (2013). CoCo bonds valuation with equity- and credit-calibrated firstpassage structural models. Preprint.

[7] Buergi M. (2012). A tough nut to crack: On the pricing of capital ratio triggered contingent con-vertibles. Preprint.

[8] Corcuera J.M., De Spiegeleer J., Ferreiro-Castilla A., Kyprianou A.E., Madan, D.B., Schoutens, W.(2012). Pricing of contingent convertibles under smile conform models. Preprint.

[9] De Spiegeleer J., Schoutens W. (2012). Pricing contingent convertibles: a derivatives approach.Journal of Derivatives 20(2), 27–36.

[10] Duffie D., Singleton K. (1999). Modeling term structures of defaultable bonds. Review of FinancialStudies 12, 687–720.

[11] Eom Y., Helwege J., Huang J. (2004). Structural models of corporate bond pricing: an empiricalanalysis. Review of Financial Studies 17, 499–544.

[12] Glasserman P. and Nouri B. (2010). Contingent capital with a capital-ratio trigger. SSRN Preprint.

[13] Going-Jaeschke A., Yor M. (2003). A survey and some generalizations of Bessel processes. Bernoulli9(2), 313–349.

[14] Hilscher J., Raviv A. (2012). Bank stability and market discipline: the effect of contingent capitalon risk taking and default probability. SSRN Preprint.

[15] Huang J., Zhou H. (2008). Specification analysis of structural credit risk models. SSRN Preprint.

25

[16] Jarrow R., Turnbull S. (1995). Pricing derivatives on financial securities subject to credit risk. Journalof Finance 50, 53–85.

[17] Koziol C., Lawrenz J. (2011). Contingent convertibles: solving or seeding the next banking crisis?Journal of Banking & Finance 36(1), 90–104.

[18] Lando D. (2004). Credit Risk Modeling: Theory and Applications. Princeton University Press.

[19] McDonald R.L. (2010). Contingent capital with a dual price trigger. Preprint.

[20] Merton R.C. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journalof Finance 29, 449–470.

[21] Metzler A., Reesor R. (2013). Valuation and analysis of contingent capital bonds in the structuralframework. Preprint.

[22] O’Kane D., Turnbull S. (2003). Valuation of credit default swaps. Lehman Brothers, Fixed IncomeQuantitative Credit Research.

[23] Pan J., Singleton K. (2008). Default and recovery implicit in the term structure of sovereign CDSspreads. Journal of Finance 63, 2345–2384.

[24] Pennacchi G. (2010). A structural model of contingent bank capital. FRB of Cleveland WorkingPaper No. 10-04.

[25] Pennachi G., Vermaelen T., Wolff C. (2013). Contingent capital: the case for COERCs. Preprint.

[26] Raviv A. (2004). Bank stability and market discipline: debt-for-equity swap versus subordinatednotes. Preprint.

[27] Steele, J.M. (2001). Stochastic Calculus and Financial Applications. Springer-Verlag, New York.

[28] Wilkens S., Bethke N. (2012). Contingent convertible bonds: a first empirical assessment of selectedpricing models. Preprint.

26