BSDE Modeling of Financial Derivatives · (iii) ForanyP 2M;theprocess deﬁnedby,fort2[0;T] :...

Binomial Tree Set-UpSemimartingale Set-Up

Markovian Set-UpExtensions

BSDE Modeling of Financial Derivatives

S. Crépey

Équipe Analyse et ProbabilitéUniversité d’Évry Val d’Essonne

91025 Évry Cedex, France

Rencontre ITN Marie curie project ‘BSDE, control and SPDE’

December 2010, Marrakech

S. Crépey BSDE Modeling of Financial Derivatives



Martingale Modeling

Crépey, S.: About the Pricing Equations in Finance. Forthcoming inParis-Princeton Lectures in Mathematical Finance, Lecture Notes inMathematics, Springer.

Direct work under a risk-neutral measure ℙ, rather than under thehistorical measure ℙOnly valid in ‘perfect markets’

Complete or at least very liquid marketsNo ‘frictions’ of any kind




1 Binomial Tree Set-Up

2 Semimartingale Set-UpPricing by ArbitrageConnection with Hedging

3 Markovian Set-UpMarkovian BSDE ApproachGeneric Factor ProcessApplicationsMarkovian Reflected BSDEs and Obstacles PDE ProblemsDiscussion of Various Hedging Schemes

4 ExtensionsMore General NumerairesDefaultable DerivativesIntermittent Call Protection




Risk-neutral Black–Scholes model

dSt = St(�dt + �dWt) (1)

W a standard Brownian Motion under the risk-neutralmeasure ℙ,

� a constant volatility parameter,� = r − q for a constant riskless interest rate r and a constant

dividend yield q on S




Cox-Ross-Rubinstein tree

Simplest approximation to the risk-neutral Black–Scholes modelReplace the risk-neutral Black–Scholes continuous dynamics by thefollowing discrete Markov chain, parameterized by two positiveconstants 0 < d < u : Sh

0 = S0 and for i = 0, . . . , n − 1 :

Sh(i+1)h =

{u Sh

ih with probability pd Sh

ihwith probability 1− p (2)

with h = Tn , where

T is the time to maturity of an option with payoff function �(S),n is the number of time steps,

andu = e�

√h, d = e−�

√h, p = e�h−d

u−d (3)




Denote the time in the tree by i rather than ihSh

i Shih

Ei conditional expectation with respect to the �-algebragenerated by (Sh

0 , . . . ,Shi ),

T hi (with also T h

0 = T h) Set of stopping times � taking theirvalues in {i , . . . , n}

Also denote by Sh a generic space level in the tree.In the next result we study hedging in the sense of (super-)replication ofthe option’s payoff, using the underlying stock and the riskless asset ashedging instruments.




Proposition 1 (European Case)

The price process given by, for i = 0, . . . , n :

Πi = e−r(n−i)hEi�(Shn ) = uh

i (Shi ) (4)

is the unique replication price process for the European option withpayoff �(Sn), with an associated replication strategy given as

Δi =uhi+1(uSh

i )−uhi+1(dSh

i )

(u−d)Shi

(5)

where the European pricing function uh is defined by uhn (Sh) = �(Sh),

and for i = n − 1, . . . , 0 :

uhi (Sh) = e−rh

[puh

i+1(uSh) + (1− p)uhi+1(dSh)

](6)




(Cont’d, American Case)

The minimal super-hedging price of the option with payoff process(�(Si ))0≤i≤n is given by, for i = 0, . . . , n :

Πi = max�∈T hiEie−r(�−i)h�(Sh

� ) = vhi (Sh

i ) (7)

with a related super-hedging strategy defined as

Δi =vhi+1(uSh

i )−vhi+1(dSh

i )

(u−d)Shi

(8)

where the American pricing function vh is defined by vhn (Sh) = �(Sh),

and for i = n − 1, . . . , 0 :

vhi (Sh) = max

(�(Sh), e−rh

[pvh

i+1(uSh) + (1− p)vhi+1(dSh)

])(9)




Proof. Case n = 1. Assume r = q = 0 for notational simplicity. Firstconsidering an European option with payoff function �(Sh

1 ), one seeks fora ‘price’Π0 and a ‘strategy’ Δ0 in the stock at time 0, solving thefollowing elementary ‘one time-step BSDE’ in the unknowns Π0,Δ0:

Π0 = �(Sh1 )−Δ0(Sh

1 − S0) , a.s. (10)

So in particular Π0 = E�(Sh1 ). Note that (10) is equivalent to the

following linear system in Π0,Δ0:{Π0 = �(uS0)−Δ0(u − 1)S0Π0 = �(dS0)−Δ0(d − 1)S0

which is well-posed, since d < u (unless �(uS0)�(dS0) = u−1

d−1 , in which case thesystem admits an infinity of solutions). The related ‘delta’ Δ0 writes,

Δ0 =�(uS0)− �(dS0)

(u− d)S0.




If the option is American, let (Π0,Δ0, "0) denote the solution to thefollowing ‘one time-step reflected BSDE’:{

Π0 = �(Sh1 ) + "0 −Δ0(Sh

1 − S0) , a.s.Π0 ≥ �(S0) , "0 ≥ 0 , (Π0 − �(S0))"0 = 0 .

So (this can be proven elementarily by inspection)

(Π0,Δ0, "0) =(max(�(S0),E�(Sh

1 )),�(uS0)− �(dS0)

(u− d)S0, Π0 − E�(Sh

1 )).

By construction the (issuer) ‘hedging strategy’ (Π0,Δ0) ‘superhedges’the American payoff �(Sh), for any holder stopping policy � ∈ T h.Moreover for any superhedging strategy (Y0,Z0) one has thatY0 ≥ �(S0) and Y0 + Z0(Sh

1 − S0) ≥ �(Sh1 ), almost surely (to

super-hedge the payoffs which are due in cases � = {0} and � = {1},respectively). Hence the inequality

Y0 ≥ max(�(S0),E�(Sh1 )) = Π0

follows. Therefore Π0 = max(�(S0),E�(Sh1 )) is the minimal initial wealth

of an issuer superhedging strategy.S. Crépey BSDE Modeling of Financial Derivatives



General Case. Applying step by step backwards in the tree the resultsof part (i) we deduce in the European case that the price process definedby the r.h.s. in (4), along with the hedging strategy (5), for the pricingfunction uh as of (6), replicates the option’s payoff �(Sh

n ) at time T .The probabilistic representation in the l.h.s. of (4) immediately follows.In the American case, the price process defined by the r.h.s. in (7), alongwith the hedging strategy (8), for the pricing function vh as of (9), is aminimal super-hedging strategy for the American option with payofffunction �.




In view of (9), (vi (Shi ))0≤i≤n is a supermartingale dominating

(�(Shi ))0≤i≤n, which implies the ‘≤’ inequality in the r.h.s. identity of

(7). To prove that equality actually holds in this identity, note that onehas, for every � ∈ T h

i ,

Πi = Π� +�∑

j=i+1

"j−1 −�∑

j=i+1

Δj−1(Shj − Sh

j−1)

for non-negative "’s such that "j−1 = 0 unless Πj−1 = �(Shj−1). Letting

� i = inf{j ≥ i , Πj = �(Shj )}, thus

Π� i = �(Sh� i ) , "j−1 = 0 for i + 1 ≤ j ≤ � i ,

and therefore � i achieves the maximum in (7). □




By inspection of the proof, it is also clear that these results holdindependently of the exact definition of u and d, provided 0 < d < u.Setting further u and d as in (3), one can show that theCox-Ross-Rubinstein tree model converges in law (convergence ofEuropean and American option prices) to the related Black–Scholesmodel as h→ 0.




Pricing by ArbitrageConnection with Hedging









Primary Market Model

Riskless discount factor � defined as the inverse of the savingsaccount B

�t = B−1t = exp(−

∫ t

0ru du) . (11)

Primary risky assets with cumulative dividend value process D andprice process PCumulative price P of the primary risky assets

Pt = Pt + �−1t

∫[0,t]

�u dDu. (12)

assumed a locally bounded semimartingale





ArbitragePossibility of locking, by trading in the primary market, a positiveprofit at a future time, starting from a null wealth at time 0, withoutinbetween injection or withdrawal of cash

No Free Lunch with Vanishing Risk (NFLVR) conditionA specific no arbitrage condition involving wealth processes ofadmissible self-financing primary trading strategies





Definition 1

A primary trading strategy (�0, �) in the primary market is anℝ× ℝ1⊗d -valued process, with � predictable and locally bounded, where�0 and the row-vector � respectively represent the number of units heldin the savings account and in the primary risky assets. The related wealthprocess W is given by, for t ∈ [0,T ] :

Wt = �0t Bt + �tPt . (13)

Accounting for dividends, one says that the strategy is self-financing if

dWt = �0t dBt + �t (dPt + dDt) (14)

or, equivalently,d(�tWt) = �t d(�t Pt) . (15)

If, moreover, the discounted wealth process �W is bounded from below,the strategy is said to be admissible.





Further imposing �0 predictable in these definitions would notrestrict the class of the wealth processes W of admissibleself-financing strategies, since by continuity of B, one canequivalently let �0

t or �0t− in (14).

Given the initial wealth w of a self-financing primary tradingstrategy and the strategy � in the primary risky assets, the relatedwealth process is thus given by, for t ∈ [0,T ] :

�tWt = w +∫ t0 �u d(�uPu) . (16)

The process �0 giving the number of units held in the savingsaccount, is then uniquely determined as, for t ∈ [0,T ] :

�0t = �t (Wt − �tPt) .

In the sequel we restrict ourselves to self-financing trading strategies.One thus may and do redefine a (self-financing) primary trading strategyas a pair (w , �), made of an initial wealth w ∈ ℝ, and an ℝ1⊗d -valuedpredictable locally bounded primary strategy in the risky assets �, withrelated wealth process W defined by (16).





Proposition 2 (Fundamental Theorem of Asset Pricing)

The NFLVR condition is equivalent to the existence of a risk-neutralmeasure on the primary market, soℳ ∕= ∅, whereℳ denotes the set ofprobability measures ℙ ∼ ℙ such that �P is a ℙ – local martingale.

NFLVR condition assumed throughout this lectureExistence of a risk-neutral measure ℙ ∈ℳ, soℳ ∕= ∅.

Characterization of uniqueness of a risk-neutral measure in Proposition 2Non arbitrable price processes P for which the setℳ is reduced to asingleton.Equivalent to the market being complete, which means that everyEuropean derivative in the market is replicable, that is, perfectlyhedgeable.





Financial Derivatives

Extending the financial market by introducing a financial derivativeon the primary marketFinancial claim between an investor or holder of a claim, and itscounterparty or issuer, involving, as made precise in Definition 2below, some or all of the following cash flows (or payoffs), as seenfrom the holder’s point of view:

A bounded variation cumulative dividend process D = (Dt)t∈[0,T ];Terminal cash flows, consisting of:

A payment � at maturity T , where � denotes a real-valued, boundedfrom below random variable,And, in the case of American or game products with early exercisefeatures, put and/or call payoff processes L = (Lt)t∈[0,T ] andU = (Ut)t∈[0,T ], given as real-valued, bounded from below, càdlàgprocesses, such that L ≤ U and LT ≤ � ≤ UT .





The put payoff Lt corresponds to a payment made by the issuer tothe holder of the claim, in case the holder of the claim would decideto terminate (‘put’) the contract at time t.The call payment Ut corresponds to a payment made by the issuerto the holder of the claim, in case the issuer of the claim woulddecide to terminate (‘call’) the contract at time t.

Moreover, there may be a call protection, modeled in the form of astopping time � such that issuer calls are not allowed before time �.

Call protections quite typical in the case of real-life callable productslike, for instance, convertible bonds, to the effect of making theproduct cheaper to the investor.The introduction of call protections in our definitions also allows oneto consider an American claim as a game claim with call protection� = T .





Of course, there is also the initial cash flow, namely the purchasingprice of the contract paid at the initiation time by the holder andreceived by the issuer.The terminology ‘derivative’ comes from the fact that all the abovecash flows are typically given as functions of the underlying ‘primary’asset price processes P.In the following definitions, the put time (put or maturity time, tobe precise) �, and the call (or maturity) time �, represent stoppingtimes at the holder’s and at the issuer’s convenience, with � ≥ � incase of a call protection before a stopping time �.





Definition 2

(i) An European claim is a financial claim with dividend process D, andwith payment � at maturity T .(ii) An American claim is a financial claim with dividend process D, andwith payment at the terminal (put or maturity) time � given by,

1{�<T}L� + 1{�=T}� . (17)

(iii) A game claim is a financial claim with dividend process D, and withpayment at the terminal (call, put or maturity) time � = � ∧ � given by,

1{�=�<T}L� + 1{�<�}U� + 1{�=T}� . (18)





In the sequel, the statement ‘Π is an arbitrage price for a derivative’is to be understood as, ‘(P,Π) is an arbitrage price for the extendedmarket consisting of the primary market and the derivative’. Thisimplies that we consider a liquidly traded derivative, so that thederivative and the primary market can be viewed as an extendedperfect market.The notion of arbitrage price process of a financial derivative referredto in the next result, is then nothing but the classical NFLVR notionalready introduced above regarding the ‘European’ primary market,suitably extended to game (including American) claims.

Let Tt and Tt (or simply T and T , in case t = 0) denote the set ofall [t,T ]-valued and [t ∨ �,T ]-valued stopping times.Let also � stand for � ∧ �, for any (�, �) ∈ Tt × Tt .As above,ℳ denotes the set of the risk-neutral probabilitymeasures over the primary market, withℳ ∕= ∅ under our standingnon arbitrage assumption.





Proposition 3

(i) For any ℙ ∈ℳ, the process Π defined by, for t ∈ [0,T ] :

�tΠt = Eℙ

{∫ Tt �u dDu + �T �

∣∣ℱt

}, (19)

is an arbitrage price of the related European claim. Moreover, anyarbitrage price of the claim is of this form providedsupℙ∈ℳ Eℙ

{∫[0,T ]

�u dDu + �T �}<∞

(ii) For any ℙ ∈ℳ, the process Π defined by, for t ∈ [0,T ] :

�tΠt = esssup�∈TtEℙ

{∫ �

t�u dDu + ��

(1{�<T}L� + 1{�=T}�

) ∣∣ℱt

}, (20)

is an arbitrage price of the related American claim, as soon as it is asemimartingale. Moreover, any arbitrage price of the claim is of this formprovidedsupℙ∈ℳ

Eℙsupt∈[0,T ]

{∫[0,t]

�u dDu + �t(1{t<T}Lt + 1{t=T}�

)}<∞ ; (21)





(Cont’d)

(iii) For any ℙ ∈ℳ, the process Π defined by, for t ∈ [0,T ] :

esssup�∈Tt essinf�∈TtEℙ

{∫ �

t�udDu + ��

(1{�=�<T}L� + 1{�<�}U� + 1{�=T}�

)∣∣ℱt

}= �tΠt = essinf�∈Tt esssup�∈TtEℙ

{∫ �

t�udDu + ��

(1{�=�<T}L� + 1{�<�}U� + 1{�=T}�

)∣∣ℱt

} (22)

with � = � ∧ � , is an arbitrage price of the related game claim, as soonas it is a well-defined semimartingale. Moreover, any arbitrage price ofthe claim is of this form assuming (21).

Arbitrage prices Π of the form (19), (20) or (22) will be called ℙ-prices inthe sequel. Given such a ℙ-price Π, let the discounted cumulative price ofthe option be given as, for t ∈ [0,T ] (cf. (12)),

�tΠt = �tΠt +

∫[0,t]

�u dDu . (23)





Under our assumptions, the discounted cumulative price process ofan European option thus defined with Π as of (19) therein, is anℝ ∪ {+∞}-valued martingale under some ℙ ∈ℳ, and not only alocal martingale under some ℙ ∈ℳ.In the American case, the discounted cumulative price defined by(23) with Π as of (20) therein, corresponds to the process known asthe Snell envelope of the payoff process defined by the integral in(21). The Snell envelope of a payoff process is the smallestsupermartingale which dominates it.In the game case, the discounted cumulative price (23) with Π as of(22) there, corresponds to the Dynkin game conditional valueprocess of the lower and upper payoff processes respectively definedby the integral in (21), and this integral with L replaced by U.





In view of Proposition 3(i) and (ii), one can interpret an European claimas an American claim with a fictitious put payment process L defined by�L = −c , where −c is a minorant of

∫ T⋅ �u dDu + �T �.

Since American options are ways more complex mathematically thanEuropean ones, interpreting an European option as an Americanoption may seem a rather convoluted approach.But it is in fact a simple way to establish the admissibility oftentative hedging strategies for an European option.

For the sake of unification, by ‘financial derivative’, or ‘option’, we meanhenceforth by default ‘game claim’, possibly with a call protection �,including American claim (case � = T , in particular European claim withL as specified above) as a special case.





Connection with Hedging

We consider in the following definition an issuer hedge starting at time 0.The adaptation of this definition to an holder hedge, and/or to an hedgestarting at an arbitrary time t ∈ [0,T ], is straightforward.Recall that we denote � = � ∧ �.

Definition 3

(i) An hedge with semimartingale cost process Q for a game option isrepresented by a triplet (w , �, �) such that:∙ (w , �) is a primary trading strategy,∙ the call time � belongs to T ,∙ the wealth process W of the strategy (w , �) satisfies for every put time� in T , almost surely,

��W� +∫ �0 �udQu ≥

∫ �0 �udDu + ��

(1{�=�<T}L� + 1{�<�}U� + 1{�=�=T}�

).

(24)





(Cont’d)

(ii) In the special case of an American (in particular, European) option,so � ≡ T , the set of admissible call times T is reduced to the constanttime T . The previous definition thus reduces to a primary tradingstrategy (w , �) with related wealth process W such that for every puttime � in T , almost surely,

��W� +∫ �0 �udQu ≥

∫ �0 �udDu + ��

(1{�<T}L� + 1{�=T}�

). (25)

(iii) In the special case of an European option, and if moreover equalityholds in (25) for � ≡ T , then, almost surely,

�TWT +∫ T0 �udQu =

∫ T0 �udDu + �T � . (26)

In this case one effectively deals with a replicating strategy with cost Q.





The process Q is to be interpreted as the cumulative financing cost,that is, the amount of cash added to (if dQt ≥ 0) or withdrawn from(if dQt ≤ 0) the hedging portfolio in order to get a perfect, but nolonger self-financing, hedge.

Hedges at no cost (that is, with Q = 0) are thus in effectsuper-hedges.

In relation with admissibility issues, note that the left hand side of(24) (discounted wealth process with financing costs included) isbounded from below, for any hedge (w , �, �) with cost Q.





In the American case, càdlàg properties of the processes involved in(25) implies that the latter being satisfied for every put time � in Tis equivalent to: almost surely, for every t ∈ [0,T ],

�tWt +∫ t0 �udQu ≥

∫ t0 �udDu + �t

(1{t<T}Lt + 1{t=T}�

).

(27)In the special case of European options, requiring (25) for every puttime � in T , or equivalently (27), may seem too strong, since anEuropean option can only be exercised at time � ≡ T .Note however that in the European case the put payoff L is definedin a very specific way, so that the ‘undue’ requirement that (27) alsoholds for t < T , and not only at t = T , is merely a simple way toencompass admissibility of the strategy (w , �).





As Proposition 5 below will show, Definition 3 has the merit to beconsistent with the concept of arbitrage pricing of Proposition 3(iii)for a game option, including American and European options asspecial cases.Of course, this general connection between pricing and hedging inthe weak sense of Definition 3, will definitely need to be completedby the analysis and discussion of various, more explicit hedgingstrategies, which will also be accomplished later in this course.To emphasize the very weak sense of Definition 3 and warn thereader against any possible misunderstanding of these results, notethat as will shall see in Proposition 5 ‘no hedge’, so � ≡ 0, or in factmore generally, any primary strategy (w , �), is a hedge in the senseof this definition, with cost process essentially given by the trackingerror process e = e(�) in the sense of the following definition.





Definition 4

For any primary strategy (w , �), the issuer’s Profit and Loss (or TrackingError) process (et)t∈[0,T ] relative to the price process Π of Proposition 4is given by e0 = 0 and, for t ∈ [0,T ],

�tet =∫ t0

(−d(�uΠu) + �ud(�uPu)

), (28)

where the cumulative price Π was defined in (23).

The tracking error process e = e(�) thus corresponds, during thelifetime of the option (until it is first put, called, or expired), to thenotion of the cumulative profit and loss of a trader hedging oneshort option position by the strategy (w , �) in the primary market.Accordingly with this interpretation and consistently with thenotation, note that process e = e(�) only depends on �, not on w .





One theoretically pleasant property of hedges with semimartingalecost processes in Definition 3 is that they inherit from the class ofsemimartingales the property of invariance under equivalent (orabsolutely continuous more generally) changes of probabilitymeasures.However the class of hedges with semimartingale cost processes isobviously too large for any practical purpose, so we shall now restrictour attention to hedges with a local martingale cost Q under aparticular risk-neutral measure ℙ.





Henceforth in this course, we thus work under a fixed but arbitraryrisk-neutral measure ℙ, with ℙ-expectation denoted by E.All the measure-dependent notions implicitly refer to this probabilitymeasure ℙ.In practical applications, it is convenient to think of ℙ as ‘the pricingmeasure chosen by the market’ to price a contingent claim. Forpricing and hedging purposes this measure is typically estimated bycalibration of a model to market data.





BSDE Modeling

We assume further for the sake of simplicity that the derivative’scumulative dividend process D is time-differentiable, so dDt = Ctdt, forsome progressively measurable time-integrable coupon rate process C .

Reflected BSDE with data �,C , �, L,U, � :

⎧⎨⎩�tΠt = �T � +

∫ Tt �uCudu +

∫ Tt �u(dKu − dMu), t ∈ [0,T ]

Lt ≤ Πt ≤ Ut , t ∈ [0,T ]∫ T0 (Πu − Lu) dK+

u =∫ T0 (Uu − Πu) dK−u = 0 ,

(29)

where, with the convention that 0×±∞ = 0 in the third line,

Ut = 1{t<�}∞+ 1{t≥�}Ut . (30)

Given that � = e−∫ ⋅0 rtdt , note that the first line of (29) is equivalent to

Πt = � +∫ Tt (Cu − ruΠu)du + (KT − Kt)− (MT −Mt), t ∈ [0,T ] .

(31)S. Crépey BSDE Modeling of Financial Derivatives




Definition 5

By a solution to (29), we mean a triplet of processes (Π,M,K ) such thatall conditions in (29) are satisfied, where:∙ the state-process Π is a real valued, càdlàg process,∙ M is a martingale vanishing at time 0,∙ K is a finite variation continuous process null at time 0, and K±

denote the components of the Jordan decomposition of K .

By the Jordan decomposition of K in the last bullet point, we mean theunique decomposition K = K+ − K− of K as difference of twonon-decreasing processes K± null at 0, defining mutually singular randommeasures on [0,T ].

So, loosely speaking, the non-decreasing processes K+ and K−

‘cannot increase simultaneously’.





Since it involves the martingale M, the notion of a solution to (29),is of course measure-dependent, contingent on the prevailing marketpricing measure ℙ which underlies its definition.The first line of (29) can be interpreted as giving the Doob-Meyerdecomposition

∫ t0 �u(dKu − dMu) of the special semimartingale

�tΠt := �tΠt +

∫ t

0�uCudu . (32)

So an equivalent definition of a solution to (29) would be that of aspecial semimartingale Π (rather than a triplet of processes(Π,M,K )) such that all conditions in (29) are satisfied, where Mand K therein are to be understood as the canonical localmartingale and finite variation predictable components of process∫

[0,⋅] �−1t d(�tΠt).

The ‘price’ notation Π in the solution of (29) and the ‘cumulativeprice’ notation Π in (32) (cf. (23)) will be justified shortly.





Under suitable L2-integrability conditions on the data and thesolution, existence and uniqueness of a solution to (29) areessentially equivalent to the so-called Mokobodski condition

Existence of a square integrable quasimartingale Y (specialsemimartingale with additional integrability properties) such thatL ≤ Y ≤ U on [0,T ].

Existence and uniqueness of a solution to (29) thus hold when oneof the barriers is a quasimartingale

In particular, when the lower barrier is given as S ∨ c, where S is asquare-integrable Itô process and c is a constant in ℝ ∪ {−∞}.

This covers, for instance, the put payment process L of an Americanvanilla option, or of a vanilla convertible bond.Moreover one typically has K = 0 in the case of an Europeanderivative.





We thus work henceforth in this part under the following hypothesis.

Assumption 6

Equation (29) admits a solution (Π,M,K ), with K equal to zero in thespecial case of an European derivative.

The following result states that the value component Π of the solution ofthe stochastic pricing equation (29), indeed yields an arbitrage price ofthe related derivative, the price relative to the measure ℙ underlying (29).

Proposition 4

Π is the ℙ-price process of the derivative.

Proof. By a rather standard verification principle, Π satisfies (22), whichin the special case of an American (resp. European) option reduces to(20) (resp. (19)). □





Observe in view of (32) that the tracking error process e = e(�) isnow a special semimartingale.Let the ℙ – local martingale � = �(�) be such that �0 = 0 and∫ ⋅0 �td�t is the local martingale component of the specialsemimartingale �e, so (cf. (28), (31))

d�t = dMt − �t �−1t d(�t Pt) (33)

�tet =∫ t0 �udKu −

∫ t0 �ud�u . (34)

Let also the call time �∗ ∈ Tt be defined by

�∗ = inf{u ∈ [t ∨ �,T ] ; Πu ≥ Uu

}∧ T . (35)

Using the minimality condition (third line) in (29) and the continuityof K±, note that one has,

K− = 0 and K = K+ ≥ 0 on [0, �∗] , Π�∗ = U�∗ on {�∗ < T} .(36)





Proposition 5

(i) For any primary strategy �, (Π0, �, �∗), is an hedge with ℙ – local

martingale cost �(�);(ii) Π0 is the minimal initial wealth of an hedge with ℙ – localmartingale cost;(iii) In the special case of an European derivative with K = 0, then(Π0, �) is a replicating strategy with ℙ – local martingale cost �. Π0 isthus also the minimal initial wealth of a replicating strategy with ℙ –local martingale cost.

Special case � = 0Model completeness

Replicability of European options

Holder hedgesHedging a defaultable game option starting at any date t ∈ [0,T ],rather than at time 0 above.





Proof. (i) One must show that for any � ∈ T , almost surely:

Π0 +

∫ �∗∧�

0�ud(�uPu) +

∫ �∗∧�

0�ud�u ≥ (37)∫ �∗∧�

0�uCudu + ��∗∧�

(1{�∗∧�=�<T}Lt + 1{�∗<�}U�∗ + 1{�∗=�=T}�

)or equivalently, using (33):

Π0 +

∫ �∗∧�

0�udMu ≥ (38)∫ �∗∧�

0�uCudu + ��∗∧�

(1{�∗∧�=�<T}L� + 1{�∗<�}U�∗ + 1{�∗=�=T}�

)where by the first line in (29):

Π0 +

∫ �∗∧�

0�udMu =

∫ �∗∧�

0�uCudu + ��∗∧�Π�∗∧� +

∫ �∗∧�

0�udKu .

Inequality (38) then follows from (36) and from the following relations,which are valid by the terminal and put conditions in (29):

ΠT = � , Π� ≥ L� .





(ii) There exists an hedge with initial wealth Π0 and ℙ – local martingalecost, by (i) applied with, for instance, � = 0. Moreover, for any hedge(w , �, �) with ℙ – local martingale cost Q, one has by (24), for t ∈ [0,T ]:

w +

∫ �∧t

0�ud(�uPu) +

∫ �∧t

0�udQu ≥ (39)∫ �∧t

0�uCudu + ��∧t

(1{�∧t=t<T}Lt + 1{�<t}U� + 1{�=t=T}�

)The left hand side is thus a bounded from below local martingale, henceit is a supermartingale. Moreover, (24) states that (39) should hold moregenerally with any stopping time � ∈ T instead of t therein. So, bytaking expectations in (39) with � instead of t therein:

w ≥ E{∫ �∧�

0�uCudu + ��∧�

(1{�∧=�<T}L� + 1{�<�}U� + 1{�=�=T}�

)}.

Hence w ≥ Π0 follows, by (22).

(iii) In the special case of an European derivative, the stated resultsfollow by setting K = 0 in the previous points of the proof. □




Markovian BSDE ApproachGeneric Factor ProcessApplicationsMarkovian Reflected BSDEs and Obstacles PDE ProblemsDiscussion of Various Hedging Schemes









Markovian BSDE Approach

Definition 7

We say that the BSDE (29) is Markovian if its input data r ,C , �, L and Uare given by Borel-measurable functions of some ℝq-valued(F,ℙ)-Markov factor process X , so

rt = r(t,Xt) , Ct = C (t,Xt) , � = �(XT ) , Lt = L(t,Xt) , Ut = U(t,Xt) ,(40)

and if the call protection time � is the first time of entry, capped at T , ofthe process (t,X ), into a given closed subset of [0,T ]× ℝq.





From the point of view of interpretation, the components of X areobservable factors

Most factors typically given as primary price processes, or furtherfactors required to explain path dependent product payoffs or modeldynamicsSome of the primary price processes may not be needed as factors,but are used for hedging purposes.

Observability of the factor process X in the mathematical sense ofF-adaptedness not sufficient in practice

Calibrability issue





Generic Factor Process

dXt = b(t,Xt) dt + �(t,Xt) dWt + �(t,Xt−)dNt , (41)

W a q-dimensional Brownian motionN a compensated integer-valued random measure with finite

jump intensity measure �(t,Xt , dx), for some deterministicfunction �.�(t,Xt−)dNt A short-hand for∫

x∈ℝq �(t,Xt−, x)N(dt, dx), where x can beunderstood as a ‘mark’ of the jump of X in�(t,Xt−, x),

Further notation:Jt a random variable on ℝq with law �(t,Xt−,dx)

�(t,Xt−,ℝq) conditionalon Xt−,

(tl) the ordered sequence of the times of jumps of N (welldefined in case of a finite jump measure �)





Let for every function u on [0,T ]× ℝq,

�u(t, x , y) = u(t, x + �(t, x , y))− u(t, x) , �u(t, x) =

∫ℝq�u(t, x , y)�(t, x , dy)

�ut = �u(t,Xt−, Jt) , �ut = �u(t,Xt−)

and in particular

�(t, x) := �Idℝq (t, x) =

∫ℝq�(t, x , y)�(t, x , dy)

�t = �(t,Xt−, Jt) , �t = �(t,Xt−)

So, with also b(t, x) = b(t, x)− �(t, x) in the last line

�(t,Xt−)dNt = d(∑

tl≤t

�tl)− �tdt (42)

dXt = b(t,Xt)dt + �(t,Xt) dWt + d(∑

tl≤t

�tl)− �tdt

= b(t,Xt)dt + �(t,Xt) dWt + d(∑

tl≤t

�tl)

(43)





Itô Formula and Model Generator

In view of (43), the following variant of the Itô formula holds, for anyreal-valued function u of class C1,2 on [0,T ]× ℝq :

du(t,Xt) = Gu(t,Xt) dt +∇u(t,Xt)�(t,Xt) dWt + d

⎛⎝∑tl≤t

�utl

⎞⎠ (44)

with

Gu(t, x) = ∂tu(t, x) +∇u(t, x)b(t, x) +12Tr[a(t, x)∇2u(t, x)] (45)

where a(t, x) = �(t, x)�(t, x)T, and where ∇u and ∇2u denote therow-gradient and the Hessian of u with respect to x . So in particular

Tr[a(t, x)∇2u(t, x)] =∑

1≤i,j,k≤q

�i,k(t, x)�j,k(t, x)∂2xi ,xj

u(t, x) .





Using the short-hand �u(t,Xt−)dNt =∫x∈ℝq �u(t,Xt−, x)N(dt, dx), note

that one also has (cf. (42)),

�u(t,Xt−)dNt = d(∑

tl≤t �utl

)− �utdt . (46)

The Itô formula (44) may thus be rewritten as

du(t,Xt) = Gu(t,Xt) dt +∇u(t,Xt)�(t,Xt) dWt + �u(t,Xt−)dNt(47)

where we set

Gu(t, x) = Gu(t, x) + �u(t, x)

= ∂tu(t, x) +∇u(t, x)b(t, x)

+12Tr[a(t, x)∇2u(t, x)] + �u(t, x)−∇u(t, x)�(t, x) .

(48)

The process X is thus a Markov process with generator G.





Applications

Versatile specifications of our ‘abstract’ jump-diffusion X rangingfrom pure diffusions or (resorting to unbounded jump measures)Lévy processes, to continuous-time Markov chainsRich enough for most applications in financial derivatives modeling

Equities

Includes in particular the most common forms of stochastic volatilityand/or jump equity derivatives pricing models, such as the Black-Scholesmodel, local volatility models, the Merton model, the Heston model, theBates model, or Lévy pricing models.





Interest-RatesAs will be explained later in this course, the risk-neutral modelingapproach can be readily extended to a martingale modeling approachrelatively to an arbitrary numeraire, rather than the savings accountin the risk-neutral approach.This allows one to extend the previous models to interest-rates andforeign exchange derivatives

Black model, SABR model





Single-Name Credit-Risk and Equity to Credit

Moreover, as we shall see later in this course, one can easilyaccommodate in the risk-neutral (or in a more general martingale)modeling approach defaultable derivatives with terminal payoffs ofthe form 1T<��(XT ), where � represents the default-time of areference entity.This allows one to deal further with equity-to-credit derivatives, like,for instance, convertible bonds. A model X as of (41) is thentypically used in the mode of a pre-default factor process model.

Portfolio Credit RiskFinally continuous-time Markov chains, or continuous-time Markov chainsmodulated by diffusions, which can all be considered as specific instancesof the general jump-diffusion framework (41), cover most of theMarkovian models used in the field of portfolio credit derivatives





Markovian Reflected BSDEs and Obstacles PDE Problems

Call protection of the form

� = inf{ t > 0 ; Xt /∈ O} ∧ T (49)

for a constant T ∈ [0,T ] and an open subset O ⊆ ℝq satisfying suitableregularity properties.

Proposition 6

(i) (Post-protection price). On [�,T ], the ℙ-price process Π can berepresented as Πt = u(t,Xt), where u is the unique viscosity solution ofthe following double obstacle problem:

min(max

(Gu(t, x) + C (t, x)− r(t, x)u(t, x),

L(t, x)− u(t, x)),U(t, x)− u(t, x)

)= 0, t < T , x ∈ ℝq,

(50)

with terminal condition u(T , x) = �(x).





(Cont’d)

(ii) (Protection price). On [0, �], the ℙ-price process Π can berepresented as Πt = u(t,Xt), where the function u is the unique viscositysolution of the following variational inequality (lower obstacle problem):

max(Gu(t, x) + C(t, x)− r(t, x)u(t, x), L(t, x)− u(t, x)

)= 0, t < T , x ∈ O, (51)

with boundary condition u = u on([0,T ]× ℝq

)∖([0,T )×O

).

(iii) In case the pricing functions u and u are sufficiently regular for anItô formula to be applicable, one has further, for t ∈ [0,T ],

dMt = ∇��(t,Xt)dWt + ��(t,Xt−)dNt , (52)

where the random function � therein is to be understood as u for t > �and u for t ≤ �.

Jumps in X → Needs to deal with the ‘thick’ parabolic boundary([0,T ]× ℝq

)∖([0,T )×O

).





Discussion of Various Hedging Schemes

In view of Proposition 6, the first line of (29) takes the followingform (cf. (31)):

− d�(t,Xt) = (C − r�)(t,Xt)dt + dKt −∇��(t,Xt)dWt − ��(t,Xt−)dNt (53)

where the function � therein is defined as u for t > � and u fort ≤ �.Let us assume an analogous structure (except for the barriers) onthe primary market price process P, so Pt = v(t,Xt) for adeterministic function v(t, x), and

− dv(t,Xt) = (C − rv)(t,Xt)dt −∇v�(t,Xt)dWt − �v(t,Xt−)dNt , (54)

where C(t,Xt)dt represents a primary dividend process.Note that v is an ℝd -valued function, so in particular ∇v lives inℝd⊗q, and identity (54) holds in ℝd .





The cost � associated to the strategy � (and thus the related trackingerror e, cf. (33)-(34)), can then be expressed in terms of the pricingfunctions u and v and the related delta functions, as follows.

Proposition 7

Under the previous conditions in the Markovian jump-diffusion set-up(41), the dynamics (33) of the cost process � = �(�) may be rewritten as(using the notation introduced in (42)):

d�t =(∇��(t,Xt)− �t∇v�(t,Xt)

)dWt

+(��(t,Xt−)− �t�v(t,Xt−)

)dNt

(55)





It is thus possible to perfectly hedge the market risk W by setting,provided ∇v� is left-invertible,

�t = ∇��(∇v�)−1(t,Xt) (56)

In the simplest case where q = d and ∇v and � are invertible, thisformula further reduces to

�t = ∇�(∇v)−1(t,Xt) (57)

Plugging this strategy into (55), one is left with the cost process

�t =∫ ⋅0

(��(t,Xt−)− �t�v(t,Xt−)

)dNt (58)

with � defined by (56) or (57).Perfect strategy from the point of view of hedging the market riskW , but potentially creates some jump risk on the other hand via thedependence on � of the integrand in (58).





At the other extreme, in case the jump measure has finite support(case of continuous-time finite Markov chains for example), it isalternatively possible to perfectly hedge the jump risk N by setting,provided �v(t,Xt−) is left-invertible,

�t = ��(�v)−1(t,Xt−) . (59)

Plugging this strategy into (55), one is left with the cost process

�t =∫ ⋅0

(∇��(t,Xt)− �t∇v�(t,Xt)

)dWt (60)

with � defined by (59).This strategy potentially creates market risk via the dependence in �of the integrand in (60).

Credit derivativeshedging spread risk W vs hedging default risk N





Min-Variance Hedging

Perfect hedge (� = 0) hopeless unless the jump measure of X hasfinite support.→ Best hedgeOptimality criterion required

Risk-neutral min-variance hedging strategy �va given by the followingGaltchouk-Kunita-Watanabe decomposition of

∫ ⋅0 �dM with respect to

�P:

�tdMt = �vat d(�t Pt) + �td�va

t (61)

for some ℝd -valued �P-integrable process �va and a real-valued squareintegrable martingale �td�va

t strongly orthogonal to �P.





Letting in vector-matrix form

< A,B >= (< Ai ,B j >)ji , < A >=< A,A > ,

one thus has by (61), assuming the P-conditional covariance matrix to beinvertible:

�vat = d<Π,P>t

dt

( d<P>tdt

)−1= (�, v) (v , v))−1

(t,Xt−

)(62)

where we denote, for any vector-valued functions u and v on [0,T ]× ℝq

such that the matrix-product uvT makes sense:

(u, v)(t, x) = (∇u)a(∇v)T(t, x) +

∫y∈ℝq

�u(�v)T(t, x , y)�(t, x , dy) . (63)

In the vector-valued case ∇u and ∇v are defined component bycomponent, and can thus be identified to the Jacobian matrices of u andv .




More General NumerairesDefaultable DerivativesIntermittent Call Protection









More General Numeraires

B here a general locally bounded positive semimartingale, notnecessarily of finite variation nor (absolutely before) continuous.Interpretation of B as savings account and of � = B−1 as risklessdiscount factor now replaced by the interpretation of B as anumeraire, referring to the fact that other price processes will beexpressed as relative (rather than discounted) prices �P.Understanding a discounted price as a relative price and arisk-neutral model as a martingale model relatively to the numeraireB, the risk-neutral modeling approach developed in the previoussections holds mutatis mutandis.

The self-financing condition still assumes the form of equation (15),though this is not as obvious as in the special case where B was afinite variation and continuous process.The concept of arbitrage is now to be understood relatively to thenumeraire B (the set of admissible strategies being a numerairedependent notion).





Defaultable Derivatives

Defaultable derivatives with terminal payoff of the form 1T<��(ST ),where � represents the default-time of a reference entity

or payoff 1�<��(S�) upon exercise at a stopping time �, in case ofAmerican or game claims

In the reduced-form approach to credit risk, defaultable claims canbe handled in essentially the same way as default-free claims,provided

1 the default-free discount factor process � is replaced by a credit-riskadjusted discount factor �,

Note the ‘original default-free’ discount factor � = e−∫ ⋅0 rtdt , can

itself be regarded as a default probability, at the killing rate rt in (11).2 a fictitious dividend continuously paid at the default intensity rate

is introduced to account for recovery on the claim upon default.





Intermittent Call Protection

Callable products with more realistic forms of intermittent call protectionCall protection whenever a certain condition is satisfied, rather thanmore specifically call protection before a stopping time earlier in thispart.Effective call payoff process

Ut = Ωct∞+ ΩtUt , (64)

for given càdlàg event-processes Ωt ,Ωct = 1− Ωt .

Call possible whenever Ωt = 1, otherwise call protection is in force.Protection until � corresponds to the special case for whichΩt = 1{t≥�} in (64).





The identification between the arbitrage, or infimal super-hedging,ℙ-price process of the derivative at hand, and the state-process Π ofa solution (Π,M,K ), assumed to exist, to the generalization of theBSDE (29) with U given by (64) therein, can be established by astraightforward adaptation of the arguments of Section 2.In the Markovian jump-diffusion model X defined by (41), andassuming

Ωt = Ω(t,Xt) (65)

one then has the ‘usual representation’

Πt = u(t,Xt) , dMt = ∂u(t,Xt)�(t,Xt)dWt + �u(t,Xt−)dNt

on [0,T ], for a suitable pricing function u.


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