PDE approach to valuation and hedging of credit derivatives

PDE approach to valuation and hedging of credit derivatives

TOMASZ R. BIELECKIy, MONIQUE JEANBLANC*z and MAREK RUTKOWSKI§

yDepartment of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA zDe´partement de Mathe´matiques, Universite´ d’E´vry Val d’Essonne, 91025 E´vry Cedex, France

§School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Faculty of Mathematics and Information Science, Warsaw University of Technology,

00-661 Warszawa, Poland

(Received 27 December 2004; in final form 11 April 2005)

This paper presents a PDE approach in a Markovian setting to hedge defaultable derivatives.

The arbitrage price and the hedging strategy for an attainable contingent claim are described in terms of solutions of a pair of coupled PDEs. For some standard examples of defaultable claims, we provide explicit formulae for prices and hedging strategies.

Keywords: Credit derivatives; Hedging; Valuation; PDE approach

1. Introduction

Our aim is to examine the PDE approach to the valuation and hedging of a defaultable claim in various settings; this allows us to emphasize the importance of the choice of the traded assets. We start with a general model for the dynamics of the traded primary assets. Subsequently, we specify particular models and we deal with particular defaultable claims such as, for instance, survival claims.

For the sake of notational simplicity, we deal throughout with a model with only three primary traded assets.

The generalization to the case of k primary assets is rather straightforward, although notationally more cumbersome.

The paper is organized as follows. In section 2, we examine the no-arbitrage property of a model in terms of a martingale measure. The next section is devoted to the study of the PDE approach to the valuation of defaul- table claims and we give the hedging strategies of a contingent claim under the assumption that prices of primary assets are strictly positive. Section 4 shows how to adapt the valuation PDE and replicating strategies if one of the primary assets is a defaultable security with zero recovery, so that its price vanishes after default.

In section 5, we modify the original market model by replacing the fixed horizon date for trading activities by a random time horizon determined by the default time.

Finally, in section 6, we examine briefly the possible extensions to the case of several default times. We refer to the companion works of Bielecki et al. (2004a–d) for notation and related results.

2. Martingale measures

In this section, we start, as is standard, with the historical dynamics of the traded assets. For ease of computation, we restrict our attention to the case of three traded assets and two sources of noise: a Brownian motion and a random default time. Our goal is to derive the PDE satisfied by the price of a defaultable claim in a general model under market completeness. Subsequently, we examine some examples corresponding to specific choices of the underlying assets.

2.1. Default time

Let be a strictly positive random variable on a prob- ability spaceð,G,QÞ, referred to as adefault time. Note thatQis the real-life (or statistical) probability measure.

*Corresponding author. Email: monique.jeanblanc@maths.

univ-evry.fr

Quantitative Finance

ISSN 1469–7688 print/ISSN 1469–7696 online#2005 Taylor & Francis http://www.tandf.co.uk/journals

DOI: 10.1080/14697680500149297

In order to exclude trivial cases, we assume that Qf >0g ¼1 and QfTg>0. Let us introduce the jump process H_t¼I

ftg and denote by H the filtration generated by this process.

Assume that we are given, in addition, a reference filtrationF such that F_t G for everyt2 ½0,T and the -field F₀ is trivial. We set G¼F_H so that G_t¼ F_t_ H_t ¼ðF_t,H_tÞfor everyt2R_þ. The filtration G is referred to as the full filtration; it includes the observations of default events. We assume that any F-martingale is also aG-martingale. Such an assumption is sometimes called Hypothesis H. For more details on this assumption, we refer to Bieleckiet al. (2004a).

We write F_t¼Qftj F_tg, so that G_t¼1F_t¼ Qf >tj F_tgis thesurvival processwith respect toF. It is easily seen that F is a bounded, non-negative, F-submartingale. It is well known that, under Hypothesis H, we haveF_t¼Qftj F₁g, and thus the processF is increasing. Assume, in addition, thatF_t<1 for every t2R_þ. The F-hazard process of a random time with respect to a filtration F is defined through the equality 1F_t ¼e^t, that is_t¼ lnG_t.

It is well known that if theF-hazard processof is a continuous, increasing process, then the process M_t¼ H_tt^, t2R_þ, is a G-martingale. The process M is referred to as the compensated martingale of the default processH. If the hazard process is absolutely continuous with respect to the Lebesgue measure, so that_t¼Rt

0_udu for someF-progressively measurable process, then is called the F-intensity of . In the following, we work mainly with a constant or deterministic intensity , in order to give the main ideas, which, somewhat surpris- ingly, do not seem to be commonly known.

We assume throughout that the F-hazard process is absolutely continuous. Hence, the process

M_t¼H_t Zt^

0

_udu¼H_t

Zt 0

_uð1H_uÞdu¼H_t Z t

0

_udu, ð1Þ where_t¼_tI

ft<g, is aG-martingale underQ. Also, let us recall that if the representation theorem holds for the filtrationFand a finite familyZⁱ,in, ofF-martingales then, under Hypothesis H, it holds also for the filtration Gand with respect to theG-martingalesZⁱ,in, andM.

2.2. Primary traded assets

We assume that we are given a familyY¹,Y²,Y³of semi- martingales defined on the filtered probability space ð,G,G,QÞ. We interpret Y_tⁱ as the cash price at time t of theith primary traded asset, and we examine a market model M ¼ ðY¹,Y²,Y³;Þ, where is the class of all self-financing trading strategies. In order to get more explicit valuation formulae, we postulate that the process Yⁱis governed by the SDE

dY_tⁱ¼Y_tⁱ

_idtþ_idW_tþ_idM_t

, i¼1, 2, 3, ð2Þ

with the initial conditionY₀ⁱ>0. Here W is a standard one-dimensional Brownian motion and the process M, given by (1), is the compensated martingale of the default process H. Note that, in view of Hypothesis H, the process W follows a Brownian motion not only with respect to its natural filtrationF^W, but also with respect to the enlarged filtrationG. In Bielecki et al. (2005), we extend this model to more general dynamics, involving also a Poisson process.

In the present paper, we deliberately restrict our attention to the case where the coefficientsi,i,iand the default intensity > 0 are constant (or at least deterministic). We assume that_i 1 for i¼1, 2, 3 in order to ensure that the price processesYⁱ,i¼1, 2, 3, are non-negative. Note that the equality_i¼0 corresponds to the case where the ith asset is default-free, while the inequality _i6¼0 means that the ith asset is formally classified as a defaultable security. In particular, the equality_i¼ 1 corresponds to the case where the price Yⁱ vanishes after default, i.e. the ith asset is defaultable and exhibits zero recovery (or total default). It should be stressed that most (but not all) of our results can be extended to the case of coefficients with a Markovian-type dependence on the underlying stochastic processes, as made precise by assumption A below. The crucial observation is that, under assumption A, the process ðY¹,Y²,Y³,HÞ, taking values in R³ f0, 1g, possesses the Markov property under the statistical probabilityQ. It is also worth noting that the triple ðY¹,Y²,Y³Þ does not follow a Markov process underQ, in general.

Assumption A: The coefficientsi,i,iin dynamics (2) and the intensity are given by some functions on R_þR³ f0, 1g, so that _i¼_iðt,Y_t¹ ,Y_t² ,Y_t³ ,H_tÞ, _i¼_iðt,Y_t¹,Y_t² ,Y_t³ ,H_tÞ, _i¼_iðt,Y_t¹ ,Y_t² ,Y_t³Þ and ¼ðt,Y_t¹ ,Y_t² ,Y_t³ Þ. Moreover, the coefficients are regular so that the SDE(2)admits a unique strong solution for i¼1, 2, 3.

2.2.1. Recovery schemes. The case where the ith asset pays a pre-determined recovery at default is covered by the present setup. For instance, the case of a constant recovery payoff _i0 at default time corresponds to the coefficient _iðt,Y_t¹ ,Y_t² ,Y_t³ Þ ¼_iðY_tⁱ Þ¹1 and the following the dynamics of theith asset:

dY_tⁱ¼Y_tⁱ ð_idtþ_idW_tþ ð_iðY_tⁱ Þ¹1ÞdM_tÞ:

If the recovery is proportional to the pre-default value and is paid at default time (i.e. under the fractional recovery of market value), we deal with the constant coefficient_i¼_i1, and thus the dynamics ofYⁱbecome

dY_tⁱ¼Y_tⁱ ð_idtþ_idW_tþ ð_i1ÞdM_tÞ:

If theith asset is no longer traded after the default time, we may assume that the price process is stopped at time and thus the coefficients in the dynamics of theith asset vanish after time.

2.3. Change of numeraire

We assume throughout thatYⁱ, i¼1, 2, 3, are governed by (2) and that₁>1 so thatY_t¹>0 for everyt2R_þ. This assumption allows us to take the first asset as a numeraire. Let us recall that the constant coefficient ₁>1 in dynamics (2) corresponds to a fractional recovery of market value for the first asset.

In general, we do not refer to the theory involving the risk-neutral probability associated with the choice of a risk-free asset (a savings account) as a numeraire. In fact, we do not make the assumption that a risk-free security exists. We shall instead use an equivalent martingale measure Q¹, such that under Q¹ the asset prices expressed in units of the numeraireY¹are martin- gales. In other words, the martingale measure Q¹ is characterized by the property that the relative prices YⁱðY¹Þ¹,i¼1, 2, 3, areQ¹-martingales.

We first derive the dynamics of the process Y^i,1¼YⁱðY¹Þ¹ for i¼1, 2, 3. From Itoˆ’s formula, we obtain the dynamics of the processðY¹Þ¹:

d 1 Y_t¹

¼ 1

Y_t¹ ₁þ₁²þ_t 1

1þ₁1þ₁

dt

₁dW_{t 1} 1þ₁ dM_t

: ð3Þ

Consequently, the integration by parts formula yields the following dynamics for the processesY^i,1:

dY_t^i,1¼Y_t^i,1 _i11ð_i1Þ _tð_i1Þ ₁ 1þ₁

dtþ ð_i1ÞdW_tþ_i1 1þ₁ dM_t

:

As a partial check, we can verify that the jump of Y_t^i,1 equals

Y_t^i,1¼Y_t^i,1Y_t^i,1

¼Y_t^i,1 1þ_i 1þ₁1

H_t ¼Y_t^i,1 _i1 1þ₁ H_t: Remarks: Giesecke and Goldberg (2003) examine in detail the case of a particular example of a structural model with incomplete information in which the default time does not admit intensity, but the hazard process is still continuous. They postulate that the risk-free bond is traded, and the interest rate is constant. Finally, they assume the fractional recovery of market value for all defaultable claims.

2.4. Equivalent martingale measure

We now search for a probability measureQ¹, equivalent to the real-life probability Q on ð,G_TÞ, and such that the processesY^i,1, i¼2, 3, follow martingales under Q¹. From Kusuoka (1999), we know that any probability equivalent to Q on ð,G_TÞ is defined by means of its Radon–Nikody´m density processsatisfying the SDE

d_t ¼_tð _tdW_tþ_tdM_tÞ, ₀¼1, ð4Þ

where and are G-predictable processes satisfying mild technical conditions (in particular, _t>1 for everyt2 ½0,T). Since the martingaleMis stopped at, we may, and do, assume in the following that the process is stopped at. Moreover, the processesWWbandMMbgiven by, fort2 ½0,T,

Wb

W_t¼W_t Z t

0

udu,

Mb

M_t¼M_t Z t

0

_uudu¼H_t Zt

0

_uð1þ_uÞdu

¼H_t Zt

0

^_udu,

where ^_u¼_uð1þ_uÞ, are G-martingales under Q¹. The relative prices Y^i,1, i¼2, 3, follow Q¹-martingales if and only if the drift term in their dynamics expressed in terms of WWb and MMb vanishes. This in turn means that the following equality holds, for i¼2, 3 and every t2 ½0,T:

Y_t^i,1 _1iþ ð_1iÞð _t1Þ þ_tð_1iÞ_t1 1þ₁

¼0:

ð5Þ Equivalently, we have, on the setY_t^i,1 6¼0,

_1iþ ð_1iÞð _t1Þ þ_tð_1iÞ_t1

1þ₁ ¼0, i¼2, 3: ð6Þ Remarks: In the case ₁¼₁¼0 and ₁¼r, the dynamics of Y¹ are dY_t¹¼rY_t¹dt, where r is the short- term interest rate. Of course, in this case the process Y¹ represents the savings account and the martingale measureQ¹ is the usual risk-neutral probability.

2.4.1. Case of strictly positive primary assets. We work under the standing assumption that ₁>1 so that Y_t¹ >0 for every t. We assume, in addition, that _i>1 for i¼2, 3, so that the price processes Y² and Y³ are strictly positive as well. From the general theory of arbitrage pricing, it follows that the market modelM is complete and arbitrage-free provided that there exists a unique solution ð ,Þof (5) such that the process is strictly greater than 1. Since Y^i,1>0, we seek a pair of processesð ,Þfor which we have

tð_1iÞ þ_tt1i

1þ₁ ¼_i1þ₁ð_1iÞ þ_tð_1iÞ ₁

1þ₁, i¼2, 3:

ð7Þ Recall that _t¼I

ftg, so that we deal here with four linear equations, specifically,

tð₁₂Þ þ_t12

1þ₁ ¼₂₁þ₁ð₁₂Þ þð₁₂Þ ₁

1þ₁, fort, ð8Þ

tð₁₃Þ þ_t13

1þ₁ ¼₃₁þ₁ð₁₃Þ þð₁₃Þ ₁

1þ₁, fort, ð9Þ

tð₁₂Þ ¼₂₁þ₁ð₁₂Þ, fort> , ð10Þ

tð₁₃Þ ¼₃₁þ₁ð₁₃Þ, fort> : ð11Þ Equations (8) and (9) (equations (10) and (11), respect- ively) are referred to as the pre-default (post-default, respectively) no-arbitrage restrictions. To solve these equations explicitly, we find it convenient to write a¼detA, b¼detB, and c¼detC, where A, B and C are the following matrices:

A¼ _{1212 1313}

, B¼ _{1212 1313}

, C¼ ₁₂₁₂

₁₃₁₃

:

The following lemma follows from (7) by simple algebra.

Lemma 2.1: The pair ( ,) satisfies the following equations:

ta¼₁aþc,

_tta¼_1ta ð1þ₁Þb:

To ensure the validity of the second equation in lemma 2.1 not only prior to, but also after, the default time (i.e. on the set {t¼0}), we need to impose an additional condition,b¼0, or more explicitly,

ð₁₂Þð₁₃Þ ð₁₃Þð₁₂Þ ¼0: ð12Þ If (12) holds, we arrive at the following equations:

ta¼₁aþc,

_tta¼_1ta:

We are thus in the position to formulate an auxiliary result.

Proposition 2.2: Assume that the processes Y¹,Y²,Y³ satisfy withi>1for i¼1, 2, 3.

(i) If a6¼0and b¼0then the unique martingale measure Q¹ has the Radon–Nikody´m density of the form

dQ¹

dQ ¼"_Tð WÞ"_TðMÞ, ð13Þ where the constants and are given by

¼₁þc

a, ¼₁>1, ð14Þ and where we write,for t2 ½0,T,

"_tð WÞ ¼exp W_t1 2

2t

ð15Þ and

"_tðMÞ ¼ 1þI

ftgÞexp

ðt^Þ

: ð16Þ

The modelM ¼ ðY¹,Y²,Y³;Þis arbitrage-free and complete. Moreover, the process ðY¹,Y²,Y³,HÞ has the Markov property underQ¹.

(ii) If a¼0and b¼0then a solutionð ,Þexists provided that c¼0and the uniqueness of a martingale measure Q¹ fails to hold. In this case, the model M ¼ ðY¹,Y²,Y³;Þ is arbitrage-free, but it is not complete.

(iii) If b6¼0then a martingale measure does not exist and the modelM ¼ ðY¹,Y²,Y³;Þis not arbitrage-free.

Proof: All statements in part (i) are rather obvious, except for the last one. The Markov property of the processðY¹,Y²,Y³,HÞ under Q¹ can be easily deduced by observing that the dynamics of Y^i,1, i¼2, 3, under Q¹ are

dY_t^i,1¼Y_t^i,1 ð_i1ÞdWWb_tþ_i1 1þ₁ dMMb_t

, ð17Þ and by combining this observation with the fact that the default intensityb under Q¹ is deterministic, specifically b¼ð1þÞ ¼ð1þ₁Þ. It is interesting to note that the default intensity under Q¹ coincides with the default intensity under the real-life probabilityQ if and only if the process Y¹is continuous. Parts (ii) and (iii) are also easy to check. Let us only observe that, under the assump- tions of part (ii), the logarithmic returns on relative prices Y^{2, 1}andY^{3, 1}are proportional. œ From now on, we work under the assumptions of part (i) of the proposition. Recall that the processes

"ð WÞ and "ðMÞ given by (15) and (16) are unique solutions to the SDEs

d"_tð WÞ ¼ "_tð WÞdW_t, d"_tðMÞ ¼"_tðMÞdM_t,

with the initial condition "₀ð WÞ ¼"₀ðMÞ ¼1. Hence, the product ¼"ð WÞ"ðMÞ satisfies, as expected, the SDE (4) with constant processes and, specifically

d_t¼_{t 1}þc a

dW_tþ₁dM_t

n o

:

Example 2.3: Assume that the assetY¹is risk-free, the assetY²6¼Y¹is default-free, andY³is a defaultable asset with non-zero recovery, so that

dY_t¹¼rY_t¹dt,

dY_t²¼Y_t²ð₂dtþ₂dW_tÞ,

dY_t³¼Y_t³ ð₃dtþ₃dW_tþ₃dM_tÞ:

ð18Þ

We thus have 1¼1¼0, 1¼r, 26¼0, 2¼0, and 36¼0,3>1. Therefore,

a¼₂₃6¼0, c¼₃ðr₂Þ,

and the equality b¼0 holds if and only if 2(r3)¼ 3(r2). It is easy to check that

¼r₂

₂ , ¼0, ð19Þ

and thus under the martingale measure Q¹ we have (irrespective of whether3> 0 or3¼0)

dY_t¹¼rY_t¹dt,

dY_t²¼Y_t²ðrdtþ₂dWWb_tÞ,

dY_t³¼Y_t³ ðrdtþ₃dWWb_tþ₃dM_tÞ:

Note that the risk-neutral default intensity b coincides here with the real-life intensity.

2.4.2. Case of a defaultable asset with zero recovery. In this section, we postulate that _i>1 for i¼1, 2 and ₃¼ 1. This implies that the price of a defaultable asset Y³vanishes after, and thus the findings of the previous section are no longer valid. Indeed, since the processY³ jumps to zero after, the first equality in (6), that is

₂₁þ ð₂₁Þð _t1Þ þ_tð₂₁Þ_t1 1þ₁ ¼0, should still be satisfied for everyt2 ½0,T, but the second equality in (6), namely

₃₁þ ð₃₁Þð _t1Þ þ_tð₃₁Þ_t1 1þ₁ ¼0, is required to hold on the set f >tg only (i.e. when _t¼). Thus, the unknown processes and satisfy the following equations:

₂₁þ ð₂₁Þð _t1Þ ¼0, fort> , ð20Þ ₂₁þ ð₂₁Þð _t1Þ

þð₂₁Þ_t1

1þ₁ ¼0, fort, ð21Þ ₃₁þ ð₃₁Þð _t1Þ

þð1₁Þ_t1

1þ₁ ¼0, fort: ð22Þ This leads to the following lemma.

Lemma 2.4: The pair ð ,Þ satisfies the following equations,for t,

ta¼₁aþc,

_ta¼₁a ð1þ₁Þb:

Moreover,for t> ,

₂₁þ ð₂₁Þð _t1Þ ¼0:

Assume that a6¼0,₁ 6¼₂ and >b=a.Then the unique solutionð ,Þis

t¼I

ftg ₁þc a

þI

ft>g _{112 12}

,

_t¼₁ð1þ₁Þb

a >1, ð23Þ and the unique martingale measure Q¹ is given by the formula

dQ¹

dQ ¼"_Tð WÞ"_TðMÞ:

The model M ¼ ðY¹,Y²,Y³;Þ is arbitrage-free, com- plete,and has the Markov property underQ¹.

Example 2.5: Assume that the asset Y¹is risk-free, the asset Y²6¼Y¹is default-free, and Y³is a defaultable asset with zero recovery (see (18)). This corresponds to the following conditions: 1¼1¼0, 1¼r, 26¼0, 2¼0, and3¼ 1. Hencea¼ 26¼0 and assuming, in addition, that

>b

a¼r₃₃

₂ðr₂Þ, we obtain

¼r₂

₂ , ¼ b a¼1

₃r₃

₂ð₂rÞ

>1:

ð24Þ Consequently, we have, under the martingale measureQ¹,

dY_t¹¼rY_t¹dt,

dY_t²¼Y_t²ðrdtþ₂dWWb_tÞ,

dY_t³¼Y_t³ ðrdtþ₃dWWb_tdMMb_tÞ:

We do not assume here that the equality b¼0 holds;

when it does then ¼0 , as in example 2.3. In general, the risk-neutral default intensityb and the real-life inten- sity are different.

Remarks: If we assume that₂¼₃¼ 1 then the pair ð ,Þ satisfies (21) and (22), so that we have, for t (see (26)),

t¼₁þc

a, _t¼₁ð1þ₁Þb

a >1, ð25Þ provided thata6¼0 and >b=a. The solution of (21) and (22) is not uniquely determined fort> .

2.4.3. Case of a stopped trading. In some circumstances, the recovery payoff at the time of default is exogenously specified in terms of some economic factors related to the prices of traded assets (e.g. credit spreads). In such a case, the valuation problem for a defaultable claim is reduced to finding its pre-default value, and it is natural to seek a replicating strategy up to the default time (default time included), but not after this random time. Hence, it suffices to focus on features of the stopped model, that is, a model in which asset prices and all trading activities are assumed to be frozen at time (see section 5). In this case, we search for a pairð ,Þof real numbers satisfying (8) and (9) (or (21) and (22)). Equivalently,

a¼₁aþc,

a¼₁a ð1þ₁Þb:

We no longer postulate that condition (12) is satisfied.

It is clear that if a6¼0 then the unique solutionð ,Þ to the above pair of equations is

¼₁þc

a, ¼₁ð1þ₁Þb

a >1, ð26Þ

where the last inequality holds provided that >b=a.

As expected, in a stopped model, we obtain the same representation (26) of the unique martingale measure Q¹ for any choice of2and3. Let us repeat that condi- tion (12) (or equality (20)) is needed only if we wish to use the same model (2) to value claims on defaultable and non-defaultable assets. Indeed, according to (2), after time the processes Y¹, Y²and Y³ represent the prices of three assets in a model driven by a single source of randomness (a Brownian motion W), and thus condition (12) (or equality (20)) is necessary to exclude arbitrage opportunities when trading is continued up to time T. These conditions are spurious when trading is stopped at default, so that the effective horizon date becomes^T.

3. PDE approach for strictly positive traded assets We shall first examine the PDE approach in a model in which the prices of all three primary assets are non- vanishing. In this case, it is natural to focus on the case when the market modelM ¼ ðY¹,Y²,Y³;Þis complete and arbitrage-free. To this end, we shall work under the assumptions of part (i) of proposition 2.2.

3.1. Valuation PDE

We are interested in the valuation and hedging of a generic contingent claim with maturityTand the terminal payoff Y ¼GðY_T¹,Y_T²,Y_T³,H_TÞ. As we shall see in the following, the technique derived for this case can easily be applied to a defaultable claim that is subject to a fairly general recovery scheme (including, of course, the zero recovery scheme).

We assume that a6¼0 and b¼0, and we work under the unique martingale measureQ¹ corresponding to the choice ofY¹as a numeraire. Recall that we have

dQ¹

dQ ¼"_Tð WÞ"_TðMÞ,

where the pairð ,Þis given by (14). If the random vari- ableYðY_T¹Þ¹isQ¹-integrable then the arbitrage price of a claim Y can be represented as follows, for every t2 ½0,T,

_tðYÞ ¼Y_t¹E

Q¹ððY_T¹Þ¹Yj G_tÞ ¼Y_t¹

E_Q1ððY_T¹Þ¹GðY_T¹,Y_T²,Y_T³,H_TÞ jY_t¹,Y_t²,Y_t³,H_tÞ, where the second equality is a consequence of the Markov property of ðY¹,Y²,Y³,HÞ under Q¹. Let C:½0,T R³_þ f0, 1g !R be a function such that _tðYÞ ¼ Cðt,Y_t¹,Y_t²,Y_t³,H_tÞfor everyt2 ½0,T. It is clear that we have, forh¼0 andh¼1,

CðT,y₁,y₂,y₃,hÞ ¼Gðy₁,y₂,y₃,hÞ, 8 ðy₁,y₂,y₃Þ 2R³: Moreover, the processCC~_t,t2 ½0,T, given by the formula

C~

C_t¼ ðY_t¹Þ¹Cðt,Y_t¹,Y_t²,Y_t³,H_tÞ, 8t2 ½0,T, is a G-martingale under Q¹. As expected, our next goal is to use this property in order to derive the equation

satisfied by the valuation function C. To this end, we shall apply Itoˆ’s formula to the processCC. For brevity,~ we write @_iC¼@_yiC, @_ijC¼@_yi@_yjC. Also, we denote (it is easy to check that if b¼0 then the right-hand side of the formula below does not depend oni)

¼_iþ_ic a:

Proposition 3.1: Let the price processes Yⁱ, i¼1, 2, 3, satisfy

dY_tⁱ¼Y_tⁱ ð_idtþ_idW_tþ_idM_tÞ,

with_i>1for i¼1, 2, 3.Assume that a6¼0 and b¼0.

Then the arbitrage price of a contingent claim Y with the terminal payoff GðY_T¹,Y_T²,Y_T³,H_TÞequals

_tðYÞ ¼Cðt,Y_t¹,Y_t²,Y_t³,H_tÞ

¼I

ft<gCðt,Y_t¹,Y_t²,Y_t³, 0Þ þI

ftgCðt,Y_t¹,Y_t²,Y_t³, 1Þ for some function C:½0,T R³_þ f0, 1g !R. Assume that for h¼0 and h¼1 the auxiliary function Cð,hÞ:½0,T R³

þ!R belongs to the class C^{1, 2}ð½0,T R³

þ,RÞ.Then the functions Cð, 0Þand Cð, 1Þ solve the following PDEs:

@_tCð, 0Þ þX³

i¼1

ð_iÞy_i@_iCð, 0Þ

þ1 2

X³

i,j¼1

_ijy_iy_j@_ijCð, 0Þ Cð, 0Þ

þ

Cðt,y₁ð1þ₁Þ,y₂ð1þ₂Þ,y₃ð1þ₃Þ, 1Þ Cðt,y₁,y₂,y₃, 0Þ

¼0, and

@_tCð, 1Þ þX³

i¼1

y_i@_iCð, 1Þ

þ1 2

X³

i,j¼1

_ijy_iy_j@_ijCð, 1Þ Cð, 1Þ ¼0,

subject to the terminal conditions

CðT,y₁,y₂,y₃, 0Þ ¼Gðy₁,y₂,y₃, 0Þ, CðT,y₁,y₂,y₃, 1Þ ¼Gðy₁,y₂,y₃, 1Þ:

Proof: The first statement is an immediate consequence of the Markov property of the process ðY¹,Y²,Y³,HÞ underQ¹. Let us denote

Cðt,Y_t¹ ,Y_t² ,Y_t³ Þ

¼Cðt,Y_t¹ ð1þ₁Þ,Y_t² ð1þ₂Þ,Y_t³ ð1þ₃Þ, 1Þ Cðt,Y_t¹ ,Y_t² ,Y_t³ , 0Þ:

We writeC_t¼Cðt,Y_t¹,Y_t²,Y_t³,H_tÞ, and we typically omit the variables ðt,Y_t¹ , Y_t² , Y_t³ , H_tÞ in expressions @_tC,

@_iC,C, etc. An application of Itoˆ’s formula yields

dC_t ¼@_tCdtþX³

i¼1

@_iCdY_tⁱþ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijCdt

þ CX³

i¼1

_iY_tⁱ @_iC

! dH_t

¼@_tCdtþX³

i¼1

@_iCdY_tⁱþ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijCdt

þ CX³

i¼1

_iY_tⁱ @_iC

!

dM_tþ_tdt

¼@_tCdtþX³

i¼1

Y_tⁱ @_iCð_idtþ_idW_tÞ

þ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijCdtþ ðCÞdM_t

þ CX³

i¼1

_iY_tⁱ @_iC

! _tdt

¼ @_tCþX³

i¼1

_iY_tⁱ @_iCþ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijC (

þ CX³

i¼1

_iY_tⁱ @_iC

! _t

) dt

þ X³

i¼1

_iY_tⁱ @_iC

!

dW_tþCdM_t:

We now use the integration by parts formula together with (3) to obtain an SDE for CC. Since d½M~ _t ¼ dH_t ¼dM_tþ_tdt, we obtain

dCC~_t¼CC~_t (

₁þ₁²þ_t 1

1þ₁1þ₁

dt

₁dW_{t 1} 1þ₁dM_t

)

þ ðY_t¹ Þ¹ (

@_tCþX³

i¼1

_iY_tⁱ @_iC

þ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijCþ CX³

i¼1

_iY_tⁱ @_iC

! _t

) dt

þ ðY_t¹ Þ¹X³

i¼1

_iY_tⁱ @_iCdW_t

þ ðY_t¹ Þ¹CdM_t ðY_t¹ Þ¹₁X³

i¼1

_iY_tⁱ @_iCdt ðY_t¹ Þ¹ ₁

1þ₁CðdM_tþ_tdtÞ

¼CC~_{t 1}þ₁²þ_t 1

1þ₁1þ₁

dt þCC~_{t 1}dWWb_t1 dt ₁

1þ₁dMMb_tt1 1þ₁dt

þ ðY_t¹ Þ¹ (

@_tCþX³

i¼1

_iY_tⁱ @_iCþ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijC

þ CX³

i¼1

_iY_tⁱ @_iC

! _t

) dt

þ ðY_t¹ Þ¹X³

i¼1

_iY_tⁱ @_iCdWWb_t

þ ðY_t¹ Þ¹X³

i¼1

_iY_tⁱ @_iCdt

þ ðY_t¹ Þ¹CdMMb_tþ ðY_t¹Þ¹_tCdt ðY_t¹ Þ¹₁X³

i¼1

ⁱY_tⁱ @_iCdt ðY_t¹ Þ¹ ₁ 1þ₁ CðdMMb_tþ_tð1þÞdtÞ

¼C_{t 1}þ²₁þ_t 1

1þ₁1þ₁

dt þCC~_{t 1 t1}

1þ₁

dt þ ðY_t¹ Þ¹

(

@_tCþX³

i¼1

_iY_tⁱ @_iCþ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijC

þ CX³

i¼1

_iY_tⁱ @_iC

! _t

) dt

þ ðY_t¹ Þ¹X³

i¼1

_iY_tⁱ @_iCdtþ ðY_t¹ Þ¹_tCdt

ðY_t¹ Þ¹₁X³

i¼1

_iY_tⁱ @_iCdt ðY_t¹ Þ¹ ₁

1þ_{1 t}ð1þÞCdt þ a martingale underQ¹:

Since the process CC~ follows a martingale under Q¹, the finite variation part in its canonical decomposition necessarily vanishes, that is,

0¼C_tðY_t¹ Þ¹ (

₁þ²₁þ_t 1

1þ₁1þ₁

_{1 t1} 1þ₁

)

þ ðY_t¹ Þ¹ (

@_tCþX³

i¼1

_iY_tⁱ @_iCþ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijC

þ CX³

i¼1

_iY_tⁱ @_iC

! _t

)

þ ðY_t¹ Þ¹X³

i¼1

_iY_tⁱ @_iCþ ðY_t¹ Þ¹_tC

ðY_t¹ Þ¹₁X³

i¼1

_iY_tⁱ @_iC ðY_t¹ Þ¹ ₁

1þ_1tð1þÞC:

Consequently,

0¼C_{t 1}þ²₁₁ þ_t1tð1þÞ ₁ 1þ₁

þ@_tCþX³

i¼1

_iY_tⁱ @_iCþ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijC

þ CX³

i¼1

_iY_tⁱ @_iC

! _t

þX³

i¼1

_iY_tⁱ @_iCþ_tC

₁X³

i¼1

_iY_tⁱ @_iC_tð1þÞC ₁ 1þ₁: Since, in view of (14), we have

₁þ₁²₁ þ_t1tð1þÞ ₁

1þ₁¼ , _iþ_ið ₁Þ _it¼_it, we finally obtain

@_tCþX³

i¼1

_it

ð ÞY_tⁱ @_iC

þ1 2

X³

i,j¼1

_ijY_tⁱ Y_t^j @_ijCC_tþ_tC¼0:

Recall that _t ¼I_ft<g. We conclude that the price of a contingent claim Y with the terminal payoff GðY_T¹,Y_T²,Y_T³,H_TÞ can be represented as Cðt,Y_t¹,Y_t², Y_t³,H_tÞwhere, forh¼0 andh¼1,

CðT,y₁,y₂,y₃,hÞ ¼Gðy₁,y₂,y₃,hÞ, 8 ðy₁,y₂,y₃Þ 2R³_þ, and the auxiliary functions Cð, 0Þ and Cð, 1Þsatisfy the PDEs given in the statement of the proposition. œ Remarks: Note that the valuation problem splits in a natural way into two pricing PDEs that can be solved recursively. In the first step, we solve the PDE satisfied by the post-default pricing function Cð, 1Þ. Next, we substitute this function into the first PDE, and we solve it for thepre-default pricing function Cð, 0Þ. The assump- tion that we deal with only three primary assets and the coefficients are constant can be easily relaxed, but a general result is too heavy to be stated here. It is also interesting to observe that the real-life default intensity , rather than the intensity b under the martingale measureQ¹, enters the valuation PDE. This shows once again that the martingale measureQ¹ is merely a techni- cal tool, and the properties of the default time under the real-life probability are essential for valuation and hedging of a defaultable claim through the PDE approach in a complete market model.

Example 3.2: (Black and Scholes PDE). Let us place ourselves within the setup of example 2.3 (with a6¼0 and b¼0). Assume that a contingent claim Y ¼GðY_T²Þ for some functionG:R!Rsuch thatYðY_T¹Þ¹is integr- able underQ¹. Since, by definition, the valuation function C depends on t, we may, and do, assume, without loss

of generality, that it does not depend explicitly on the variable y1. In fact, it is possible to show that it does not depend ony3either, so that _tðYÞ ¼Cðt,Y_t²Þ. Since now₁¼r and₁¼₂¼0, it is easy to check that the two valuation PDEs of proposition 3.1 reduce here to a single PDE:

@_tCþ ð₂₂ Þy₂@₂Cþ1

2₂²y²₂@₂₂C ð₂₂ ÞC¼0, with ¼ ð₂rÞ=₂. After simplifications, we obtain the following equation:

@_tCþry₂@₂Cþ1

2₂²y²₂@₂₂CrC¼0,

so that we arrive, as expected, at the classic Black and Scholes PDE.

3.2. Replicating strategies

Our next goal is to derive a universal representation for a replicating strategy of a generic claim. Recall that ¼ ð¹,²,³Þis aself-financing strategyif the processes ¹,²,³ areG-predictable and the wealth process

V_tðÞ ¼¹_tY_t¹þ²_tY_t²þ³_tY_t³ satisfies

dV_tðÞ ¼¹_tdY_t¹þ²_t dY_t²þ³_tdY_t³:

We say that replicates a contingent claim Y if V_TðÞ ¼Y. If is a replicating strategy for a claim Y then we have, for everyt2 ½0,T,

_tðYÞ ¼¹_tY_t¹þ²_tY_t²þ³_tY_t³:

The next result shows that, in order to find a replicating strategy, it suffices, as in the classical case, to make use of sensitivities of the valuation function C with respect to prices of primary assets, and to take into account the jumpCassociated with the default event. Recall that

C¼Cðt,Y_t¹ ,Y_t² ,Y_t³ Þ

¼Cðt,Y_t¹ ð1þ₁Þ,Y_t² ð1þ₂Þ,Y_t³ ð1þ₃Þ, 1Þ Cðt,Y_t¹ ,Y_t² ,Y_t³ , 0Þ:

As before, for the sake of easier readability, the variables inC,@_iCandCare suppressed. Note, however, that we deal here with the two functions Cð, 0Þ and Cð, 1Þ depending on whether a replicating portfolio is examined prior to or after default.

Proposition 3.3: Under the assumptions of proposition 3.1, the replicating strategy for a claim GðY_T¹,Y_T²,Y_T³,H_TÞ is ¼ ð¹,²,³Þ, where the compo- nents ⁱ, i¼2, 3, are given in terms of the valuation