Filtration Shrinkage and Credit RiskSecond Princeton Credit Risk Conference, May

2008

Philip Protter, Cornell University

May 23, 2008

Credit Risk

Credit risk investigates an entity (corporation, bank, individual)that borrows funds, promises to return them in a specifiedcontractual manner, and who may not do so (default)

Mathematical framework: (Ω,G,P, G) are given,G = (Gt)t≥0, 0 ≤ t ≤ T , usual hypotheses. For sake of this talk,let us consider a firm. The asset value process is

dAt = Atα(t,At)dt + Atσ(t,At)dWt

where α and σ are such that A exists, is well defined, and positive.

Assume the liability structure of the firm is a single zero-couponbond with maturity T and face value 1, and default occurs only attime T , and only if AT ≤ 1.

• The probability of default is P(AT ≤ 1).

• Time zero value of the firm’s debt is

v(0,T ) = EQ((AT ∧ 1 exp(−∫ T

0rsds)).

This is the Black-Scholes-Merton model (from the early1970s) viewed as a European call option on the firm’s assets,maturity T , and strike price equal to the value of the debt.

• The Black-Scholes-Merton model has been extended todefault before time T by considering a barrier L = (Lt)t≤0.Augment the information to include the L information.LetHt = σ(As , Ls ; s ≤ t).

• The probability of default is P(AT ≤ 1).• Time zero value of the firm’s debt is

v(0,T ) = EQ((AT ∧ 1 exp(−∫ T

0rsds)).

This is the Black-Scholes-Merton model (from the early1970s) viewed as a European call option on the firm’s assets,maturity T , and strike price equal to the value of the debt.

• The Black-Scholes-Merton model has been extended todefault before time T by considering a barrier L = (Lt)t≤0.Augment the information to include the L information.LetHt = σ(As , Ls ; s ≤ t).

• The probability of default is P(AT ≤ 1).• Time zero value of the firm’s debt is

v(0,T ) = EQ((AT ∧ 1 exp(−∫ T

0rsds)).

This is the Black-Scholes-Merton model (from the early1970s) viewed as a European call option on the firm’s assets,maturity T , and strike price equal to the value of the debt.

• The Black-Scholes-Merton model has been extended todefault before time T by considering a barrier L = (Lt)t≤0.Augment the information to include the L information.LetHt = σ(As , Ls ; s ≤ t).

• Default time becomes a first passage time relative to thebarrier L:

τ = inf{t > 0 : At ≤ Lt}.

• The value of the firm’s debt is

v(0,T ) = E

[{1{t≤T}Lτ + 1{τ>T}1

}exp(−

∫ T0

rsds)

].

• The previous models are known as the structural approachto credit risk. The default time is predictable.

• Default time becomes a first passage time relative to thebarrier L:

τ = inf{t > 0 : At ≤ Lt}.

• The value of the firm’s debt is

v(0,T ) = E

[{1{t≤T}Lτ + 1{τ>T}1

}exp(−

∫ T0

rsds)

].

• The previous models are known as the structural approachto credit risk. The default time is predictable.

• Default time becomes a first passage time relative to thebarrier L:

τ = inf{t > 0 : At ≤ Lt}.

• The value of the firm’s debt is

v(0,T ) = E

[{1{t≤T}Lτ + 1{τ>T}1

}exp(−

∫ T0

rsds)

].

• The previous models are known as the structural approachto credit risk. The default time is predictable.

• Alternative: the reduced form approach of Jarrow–Turnbull,and Duffie–Singleton, of the 1990s. The observer sees onlythe filtration generated by the default time τ and a vector ofstate variables Xt .

Ft = σ(τ ∧ s,Xs ; s ≤ t) ⊂ Gt

• Assume a trading economy with a risky firm with outstandingdebt as zero coupon bondsAssuming no arbitrage (but not completeness), there is anequivalent local martingale measure Q.

• Let

Mt = 1{t≥τ} −∫ t

0λsds = a Q F−martingale.

Recovery rate given by (δt)0≤t≤T ; So change F:

Ft = σ(τ ∧ s,Xs , δs ; s ≤ t) ⊂ Gt

FXt = σ(Xs ; s ≤ t).

• Alternative: the reduced form approach of Jarrow–Turnbull,and Duffie–Singleton, of the 1990s. The observer sees onlythe filtration generated by the default time τ and a vector ofstate variables Xt .

Ft = σ(τ ∧ s,Xs ; s ≤ t) ⊂ Gt

• Assume a trading economy with a risky firm with outstandingdebt as zero coupon bondsAssuming no arbitrage (but not completeness), there is anequivalent local martingale measure Q.

• Let

Mt = 1{t≥τ} −∫ t

0λsds = a Q F−martingale.

Recovery rate given by (δt)0≤t≤T ; So change F:

Ft = σ(τ ∧ s,Xs , δs ; s ≤ t) ⊂ Gt

FXt = σ(Xs ; s ≤ t).

• Alternative: the reduced form approach of Jarrow–Turnbull,and Duffie–Singleton, of the 1990s. The observer sees onlythe filtration generated by the default time τ and a vector ofstate variables Xt .

Ft = σ(τ ∧ s,Xs ; s ≤ t) ⊂ Gt

• Assume a trading economy with a risky firm with outstandingdebt as zero coupon bondsAssuming no arbitrage (but not completeness), there is anequivalent local martingale measure Q.

• Let

Mt = 1{t≥τ} −∫ t

0λsds = a Q F−martingale.

Recovery rate given by (δt)0≤t≤T ; So change F:

Ft = σ(τ ∧ s,Xs , δs ; s ≤ t) ⊂ Gt

FXt = σ(Xs ; s ≤ t).

Default prior to time T :

Q(τ ≤ T ) = EQ{

EQ(NT = 1|FX )}

= EQ(exp(−∫ T

0λsds)), and value of the firm’s debt is

v(0,T ) = E

{[1{τ≤T}δτ + 1{τ>T}1] exp(−

∫ T0

rsds)

}

The modeler does not see the process A = (At)t≥0, but hasinstead only partial information. How does one model this partialinformation?

There are three main approaches, in general:

1. Duffie-Lando, Kusuoka: Observe A only at discreteintervals, and add independent noise

2. With Kusuoka, a twist is given by introduction of filtrationexpansion

3. Giesecke-Goldberg: The default barrier is a random curve;but A is still assumed to observed continuously

4. Çetin-Jarrow-Protter-Yildirim: Begin with a structuralmodel under G, and then project onto smaller filtration F; Useof cash flows.

• We are interested in the filtration shrinkage approach

• Following Çetin-Jarrow-Protter-Yildiray, consider the cashbalance of the firm.

• X denotes the cash balance of the firm, normalized by themoney market accountdXt = σdWt , X0 = x , where x > 0, σ > 0Z = {t ∈ [0,T ] : Xt = 0}gt = sup{s ≤ t : Xs = 0; gt is the last time before t cashbalance is zeroτα = inf{t > 0 : t − gt ≥ α

2

2 : Xs < 0, all s ∈ (gt−, t)};τα represents the time of potential defaultτ is the time of default;

• Assumeτ = inf{t > τα : Xt = 2Xτα}

• But, the investor does not see the entire cash balance process.

• We are interested in the filtration shrinkage approach• Following Çetin-Jarrow-Protter-Yildiray, consider the cash

balance of the firm.

• X denotes the cash balance of the firm, normalized by themoney market accountdXt = σdWt , X0 = x , where x > 0, σ > 0Z = {t ∈ [0,T ] : Xt = 0}gt = sup{s ≤ t : Xs = 0; gt is the last time before t cashbalance is zeroτα = inf{t > 0 : t − gt ≥ α

2

2 : Xs < 0, all s ∈ (gt−, t)};τα represents the time of potential defaultτ is the time of default;

• Assumeτ = inf{t > τα : Xt = 2Xτα}

• But, the investor does not see the entire cash balance process.

• We are interested in the filtration shrinkage approach• Following Çetin-Jarrow-Protter-Yildiray, consider the cash

balance of the firm.

• X denotes the cash balance of the firm, normalized by themoney market accountdXt = σdWt , X0 = x , where x > 0, σ > 0Z = {t ∈ [0,T ] : Xt = 0}gt = sup{s ≤ t : Xs = 0; gt is the last time before t cashbalance is zeroτα = inf{t > 0 : t − gt ≥ α

2

2 : Xs < 0, all s ∈ (gt−, t)};τα represents the time of potential defaultτ is the time of default;

• Assumeτ = inf{t > τα : Xt = 2Xτα}

• But, the investor does not see the entire cash balance process.

• We are interested in the filtration shrinkage approach• Following Çetin-Jarrow-Protter-Yildiray, consider the cash

balance of the firm.

• X denotes the cash balance of the firm, normalized by themoney market accountdXt = σdWt , X0 = x , where x > 0, σ > 0Z = {t ∈ [0,T ] : Xt = 0}gt = sup{s ≤ t : Xs = 0; gt is the last time before t cashbalance is zeroτα = inf{t > 0 : t − gt ≥ α

2

2 : Xs < 0, all s ∈ (gt−, t)};τα represents the time of potential defaultτ is the time of default;

• Assumeτ = inf{t > τα : Xt = 2Xτα}

• But, the investor does not see the entire cash balance process.

• We are interested in the filtration shrinkage approach• Following Çetin-Jarrow-Protter-Yildiray, consider the cash

balance of the firm.

• X denotes the cash balance of the firm, normalized by themoney market accountdXt = σdWt , X0 = x , where x > 0, σ > 0Z = {t ∈ [0,T ] : Xt = 0}gt = sup{s ≤ t : Xs = 0; gt is the last time before t cashbalance is zeroτα = inf{t > 0 : t − gt ≥ α

2

2 : Xs < 0, all s ∈ (gt−, t)};τα represents the time of potential defaultτ is the time of default;

• Assumeτ = inf{t > τα : Xt = 2Xτα}

• But, the investor does not see the entire cash balance process.

• In ÇJPY the investor sees only when the balances are positiveor negative, and whether or not the cash balances are aboveor below the default threshold.

• Default threshold: 2Xτα

• Yt ={

Xt for t < τα2Xτα − Xt for t ≥ τα

Y defined this way is an F Brownian motion

• τ = inf{t ≥ τα : Yt = 0}. sign(x) ={

1 if x > 0−1 if x ≤ 0

• G is the filtration of sign(Yt)

• In ÇJPY the investor sees only when the balances are positiveor negative, and whether or not the cash balances are aboveor below the default threshold.

• Default threshold: 2Xτα

• Yt ={

Xt for t < τα2Xτα − Xt for t ≥ τα

Y defined this way is an F Brownian motion

• τ = inf{t ≥ τα : Yt = 0}. sign(x) ={

1 if x > 0−1 if x ≤ 0

• G is the filtration of sign(Yt)

• In ÇJPY the investor sees only when the balances are positiveor negative, and whether or not the cash balances are aboveor below the default threshold.

• Default threshold: 2Xτα

• Yt ={

Xt for t < τα2Xτα − Xt for t ≥ τα

Y defined this way is an F Brownian motion

• τ = inf{t ≥ τα : Yt = 0}. sign(x) ={

1 if x > 0−1 if x ≤ 0

• G is the filtration of sign(Yt)

• In ÇJPY the investor sees only when the balances are positiveor negative, and whether or not the cash balances are aboveor below the default threshold.

• Default threshold: 2Xτα

• Yt ={

Xt for t < τα2Xτα − Xt for t ≥ τα

Y defined this way is an F Brownian motion

• τ = inf{t ≥ τα : Yt = 0}. sign(x) ={

1 if x > 0−1 if x ≤ 0

• G is the filtration of sign(Yt)

• In ÇJPY the investor sees only when the balances are positiveor negative, and whether or not the cash balances are aboveor below the default threshold.

• Default threshold: 2Xτα

• Yt ={

Xt for t < τα2Xτα − Xt for t ≥ τα

Y defined this way is an F Brownian motion

• τ = inf{t ≥ τα : Yt = 0}. sign(x) ={

1 if x > 0−1 if x ≤ 0

• G is the filtration of sign(Yt)

• Nt = 1{t≥τ} with G compensator A

•

TheoremAt =

∫ t∧τ0 λsds, and moreover λt = 1{t>τga}

12(t−g̃t−) , 0 ≤ t ≤ τ ,

and where g̃t = sup{s ≤ t : Ys = 0}.

• Nota Bene: We are able to calculate λ explicitly since wehave a formula for the Azéma martingale.

• Use knowledge of λ to calculate quantities of interest. Simpleexample: price of a risky zero coupon bond at time 0:

S0 = exp(−∫ T

0rudu)

{1−

(Q(τα ≤ T )− E (

α/√

2√T − g̃τα

1{τα≤T})

)}.

• Nt = 1{t≥τ} with G compensator A•

TheoremAt =

∫ t∧τ0 λsds, and moreover λt = 1{t>τga}

12(t−g̃t−) , 0 ≤ t ≤ τ ,

and where g̃t = sup{s ≤ t : Ys = 0}.

• Nota Bene: We are able to calculate λ explicitly since wehave a formula for the Azéma martingale.

• Use knowledge of λ to calculate quantities of interest. Simpleexample: price of a risky zero coupon bond at time 0:

S0 = exp(−∫ T

0rudu)

{1−

(Q(τα ≤ T )− E (

α/√

2√T − g̃τα

1{τα≤T})

)}.

• Nt = 1{t≥τ} with G compensator A•

TheoremAt =

∫ t∧τ0 λsds, and moreover λt = 1{t>τga}

12(t−g̃t−) , 0 ≤ t ≤ τ ,

and where g̃t = sup{s ≤ t : Ys = 0}.

• Nota Bene: We are able to calculate λ explicitly since wehave a formula for the Azéma martingale.

• Use knowledge of λ to calculate quantities of interest. Simpleexample: price of a risky zero coupon bond at time 0:

S0 = exp(−∫ T

0rudu)

{1−

(Q(τα ≤ T )− E (

α/√

2√T − g̃τα

1{τα≤T})

)}.

• Nt = 1{t≥τ} with G compensator A•

TheoremAt =

∫ t∧τ0 λsds, and moreover λt = 1{t>τga}

12(t−g̃t−) , 0 ≤ t ≤ τ ,

and where g̃t = sup{s ≤ t : Ys = 0}.

• Nota Bene: We are able to calculate λ explicitly since wehave a formula for the Azéma martingale.

• Use knowledge of λ to calculate quantities of interest. Simpleexample: price of a risky zero coupon bond at time 0:

S0 = exp(−∫ T

0rudu)

{1−

(Q(τα ≤ T )− E (

α/√

2√T − g̃τα

1{τα≤T})

)}.

• Nt = 1{t≥τ} with G compensator A•

TheoremAt =

∫ t∧τ0 λsds, and moreover λt = 1{t>τga}

12(t−g̃t−) , 0 ≤ t ≤ τ ,

and where g̃t = sup{s ≤ t : Ys = 0}.

• Nota Bene: We are able to calculate λ explicitly since wehave a formula for the Azéma martingale.

• Use knowledge of λ to calculate quantities of interest. Simpleexample: price of a risky zero coupon bond at time 0:

S0 = exp(−∫ T

0rudu)

{1−

(Q(τα ≤ T )− E (

α/√

2√T − g̃τα

1{τα≤T})

)}.

Default occurs not at time τα, but at time τ . The default time τis, therefore, less likely than the hitting time τα. The probabilityQ [τα ≤ T ] is reduced to account for this difference.

• The preceding is both artificial and simple. Let us consider amore realistic situation. Use of Markov process theory andhomogeneous regenerative sets (theory of Mémin and Jacod).

• Instead of just using when the cash flow is positive ornegative, we can look at when it crosses a grid of barriers.And instead of looking at just Brownian motion as cash flows,we can consider a diffusion X .

• F denotes the information from the crossings. gt denotes thelast exit time before t that X crosses a level set in ourcollection. Ut = t − gt is the since since last exit. F can bethought of as generated by (Xgt , sign(Xt − Xgt )Ut)t≥0.

• The preceding is both artificial and simple. Let us consider amore realistic situation. Use of Markov process theory andhomogeneous regenerative sets (theory of Mémin and Jacod).

• Instead of just using when the cash flow is positive ornegative, we can look at when it crosses a grid of barriers.And instead of looking at just Brownian motion as cash flows,we can consider a diffusion X .

• F denotes the information from the crossings. gt denotes thelast exit time before t that X crosses a level set in ourcollection. Ut = t − gt is the since since last exit. F can bethought of as generated by (Xgt , sign(Xt − Xgt )Ut)t≥0.

• The preceding is both artificial and simple. Let us consider amore realistic situation. Use of Markov process theory andhomogeneous regenerative sets (theory of Mémin and Jacod).

• Instead of just using when the cash flow is positive ornegative, we can look at when it crosses a grid of barriers.And instead of looking at just Brownian motion as cash flows,we can consider a diffusion X .

• F denotes the information from the crossings. gt denotes thelast exit time before t that X crosses a level set in ourcollection. Ut = t − gt is the since since last exit. F can bethought of as generated by (Xgt , sign(Xt − Xgt )Ut)t≥0.

• We discuss upward and downward excursions, and where theyend up. For each type of excursion there corresponds a Lévymeasure on (0,∞], which we denote by F j±iA (+) (resp. (−)) is for an upward (resp. downward)excursionj = 0 (resp. 1) is for an excursion ending at xi (resp. xi±1).These measures are constructed using the excursion measureni of X at xi .

•

TheoremP almost surely, for all 0 < t < τ ,

At =

{Agt if Xt ≥ x2Agt +

∫ Ut0

F 1−2 (dx)

F−2 [x ,∞)if x1 < Xt < x2

• We discuss upward and downward excursions, and where theyend up. For each type of excursion there corresponds a Lévymeasure on (0,∞], which we denote by F j±iA (+) (resp. (−)) is for an upward (resp. downward)excursionj = 0 (resp. 1) is for an excursion ending at xi (resp. xi±1).These measures are constructed using the excursion measureni of X at xi .

•

TheoremP almost surely, for all 0 < t < τ ,

At =

{Agt if Xt ≥ x2Agt +

∫ Ut0

F 1−2 (dx)

F−2 [x ,∞)if x1 < Xt < x2

• We discuss upward and downward excursions, and where theyend up. For each type of excursion there corresponds a Lévymeasure on (0,∞], which we denote by F j±iA (+) (resp. (−)) is for an upward (resp. downward)excursionj = 0 (resp. 1) is for an excursion ending at xi (resp. xi±1).These measures are constructed using the excursion measureni of X at xi .

•

TheoremP almost surely, for all 0 < t < τ ,

At =

{Agt if Xt ≥ x2Agt +

∫ Ut0

F 1−2 (dx)

F−2 [x ,∞)if x1 < Xt < x2

• If the measure F 1−2 is absolutely continuous with respect toLebesgue measure with density f 1−2 then

λ(t) =

{0 if Xt ≥ x2f 1−2 (Ut)

F−2 [Ut ,∞)if x1 < Xt < x2

is the intensity process (conditional hazard rate), i.e.A(t) =

∫ t0 λ(s)ds.

• Let Yt = E (1{τ≤T}|Ft). We can find an explicit formula forY as well.

• If the measure F 1−2 is absolutely continuous with respect toLebesgue measure with density f 1−2 then

λ(t) =

{0 if Xt ≥ x2f 1−2 (Ut)

F−2 [Ut ,∞)if x1 < Xt < x2

is the intensity process (conditional hazard rate), i.e.A(t) =

∫ t0 λ(s)ds.

• Let Yt = E (1{τ≤T}|Ft). We can find an explicit formula forY as well.

• We can then calculate prices of risky zero-coupon bonds:v(t,T ) is the price at time t of a zero-coupon bond maturingat time T .

• Management value of the bond:

vmgmt(t,T ) = E [{δ1{τ≤T} + (1− 1{τ≤T})}e−∫ Tt rsds |Gt ]

= E [{δ1{τ≤T} + (1− 1{τ≤T})}e−∫ Tt rsds |Gt ]

= 1− [(1− δ)E [1{τ≤T}|Gt ]]e−∫ Tt rsds

= 1− [(1− δ)p(Xt , t)]e−∫ Tt rsds

for t < T ∧ τ , where the last equality follows from the Markovproperty.

• We can then calculate prices of risky zero-coupon bonds:v(t,T ) is the price at time t of a zero-coupon bond maturingat time T .

• Management value of the bond:

vmgmt(t,T ) = E [{δ1{τ≤T} + (1− 1{τ≤T})}e−∫ Tt rsds |Gt ]

= E [{δ1{τ≤T} + (1− 1{τ≤T})}e−∫ Tt rsds |Gt ]

= 1− [(1− δ)E [1{τ≤T}|Gt ]]e−∫ Tt rsds

= 1− [(1− δ)p(Xt , t)]e−∫ Tt rsds

for t < T ∧ τ , where the last equality follows from the Markovproperty.

• Market value of the same bond:

v(t,T ) = [1−(1−δ)]E [1{τ≤T}|Ft ]e−∫ Tt rsds = [1−(1−δ)]Yte−

∫ Tt rsds

• In contrast to the management’s using only Xt and T − t todetermine the price, the market evaluates the price using thefollowing variables: Xgt , Ut , R(Xt), and T − t.

• Market value of the same bond:

v(t,T ) = [1−(1−δ)]E [1{τ≤T}|Ft ]e−∫ Tt rsds = [1−(1−δ)]Yte−

∫ Tt rsds

• In contrast to the management’s using only Xt and T − t todetermine the price, the market evaluates the price using thefollowing variables: Xgt , Ut , R(Xt), and T − t.

Theoretical considerations for Filtration Shrinkage

• Question: If No Free Lunch with Vanishing Risk(NFLVR) holds for (Ω,X ,G,P, G), does it also hold for F?

• Recall that NFLVR holds if and only if there exists Q ∼ Psuch that X is a (Q, G) sigma martingale. If X ≥ 0 a.s., thenit is enough that X be a (Q, G) local martingale.

• Old results of Stricker:

TheoremLet X be a semimartingale for a filtration G and let F be asubfiltration of G such that X is adapted to F. Then X remains asemimartingale for F.

•

TheoremLet X be a positive, G local martingale. Let F be a subfiltration,and assume that X is adapted to F. Then X is an Fsupermartingale, and if X is an F special supermartingale, then Xis an F local martingale.

Theoretical considerations for Filtration Shrinkage

• Question: If No Free Lunch with Vanishing Risk(NFLVR) holds for (Ω,X ,G,P, G), does it also hold for F?

• Recall that NFLVR holds if and only if there exists Q ∼ Psuch that X is a (Q, G) sigma martingale. If X ≥ 0 a.s., thenit is enough that X be a (Q, G) local martingale.

• Old results of Stricker:

TheoremLet X be a semimartingale for a filtration G and let F be asubfiltration of G such that X is adapted to F. Then X remains asemimartingale for F.

•

TheoremLet X be a positive, G local martingale. Let F be a subfiltration,and assume that X is adapted to F. Then X is an Fsupermartingale, and if X is an F special supermartingale, then Xis an F local martingale.

Theoretical considerations for Filtration Shrinkage

• Question: If No Free Lunch with Vanishing Risk(NFLVR) holds for (Ω,X ,G,P, G), does it also hold for F?

• Recall that NFLVR holds if and only if there exists Q ∼ Psuch that X is a (Q, G) sigma martingale. If X ≥ 0 a.s., thenit is enough that X be a (Q, G) local martingale.

• Old results of Stricker:

TheoremLet X be a semimartingale for a filtration G and let F be asubfiltration of G such that X is adapted to F. Then X remains asemimartingale for F.•

TheoremLet X be a positive, G local martingale. Let F be a subfiltration,and assume that X is adapted to F. Then X is an Fsupermartingale, and if X is an F special supermartingale, then Xis an F local martingale.

Theoretical considerations for Filtration Shrinkage

• Question: If No Free Lunch with Vanishing Risk(NFLVR) holds for (Ω,X ,G,P, G), does it also hold for F?

• Recall that NFLVR holds if and only if there exists Q ∼ Psuch that X is a (Q, G) sigma martingale. If X ≥ 0 a.s., thenit is enough that X be a (Q, G) local martingale.

• Old results of Stricker:

TheoremLet X be a semimartingale for a filtration G and let F be asubfiltration of G such that X is adapted to F. Then X remains asemimartingale for F.

•

TheoremLet X be a positive, G local martingale. Let F be a subfiltration,and assume that X is adapted to F. Then X is an Fsupermartingale, and if X is an F special supermartingale, then Xis an F local martingale.

Theoretical considerations for Filtration Shrinkage

• Question: If No Free Lunch with Vanishing Risk(NFLVR) holds for (Ω,X ,G,P, G), does it also hold for F?

• Recall that NFLVR holds if and only if there exists Q ∼ Psuch that X is a (Q, G) sigma martingale. If X ≥ 0 a.s., thenit is enough that X be a (Q, G) local martingale.

• Old results of Stricker:

TheoremLet X be a semimartingale for a filtration G and let F be asubfiltration of G such that X is adapted to F. Then X remains asemimartingale for F.•

TheoremLet X be a positive, G local martingale. Let F be a subfiltration,and assume that X is adapted to F. Then X is an Fsupermartingale, and if X is an F special supermartingale, then Xis an F local martingale.

Theoretical considerations for Filtration Shrinkage

• Question: If No Free Lunch with Vanishing Risk(NFLVR) holds for (Ω,X ,G,P, G), does it also hold for F?

• Recall that NFLVR holds if and only if there exists Q ∼ Psuch that X is a (Q, G) sigma martingale. If X ≥ 0 a.s., thenit is enough that X be a (Q, G) local martingale.

• Old results of Stricker:

TheoremLet X be a semimartingale for a filtration G and let F be asubfiltration of G such that X is adapted to F. Then X remains asemimartingale for F.•

TheoremLet X be a positive, G local martingale. Let F be a subfiltration,and assume that X is adapted to F. Then X is an Fsupermartingale, and if X is an F special supermartingale, then Xis an F local martingale.

What if X is not adapted to F?

Simple results:

•

TheoremLet X be a martingale for a filtration G and let F be anysubfiltration of G. Then the optional projection of X onto F isagain a martingale, for the filtration F.

•

TheoremLet X be a local martingale for a filtration G and let F be anysubfiltration of G. If a sequence of reducing stopping times(Tn)n≥1 for X in G are also stopping times in F, then the optionalprojection of X onto F is again a local martingale, for the filtrationF.

What if X is not adapted to F?

Simple results:

•

TheoremLet X be a martingale for a filtration G and let F be anysubfiltration of G. Then the optional projection of X onto F isagain a martingale, for the filtration F.

•

TheoremLet X be a local martingale for a filtration G and let F be anysubfiltration of G. If a sequence of reducing stopping times(Tn)n≥1 for X in G are also stopping times in F, then the optionalprojection of X onto F is again a local martingale, for the filtrationF.

What if X is not adapted to F?

Simple results:

•

TheoremLet X be a martingale for a filtration G and let F be anysubfiltration of G. Then the optional projection of X onto F isagain a martingale, for the filtration F.

•

TheoremLet X be a local martingale for a filtration G and let F be anysubfiltration of G. If a sequence of reducing stopping times(Tn)n≥1 for X in G are also stopping times in F, then the optionalprojection of X onto F is again a local martingale, for the filtrationF.

What if X is not adapted to F?

Simple results:

•

TheoremLet X be a martingale for a filtration G and let F be anysubfiltration of G. Then the optional projection of X onto F isagain a martingale, for the filtration F.

•

TheoremLet X be a local martingale for a filtration G and let F be anysubfiltration of G. If a sequence of reducing stopping times(Tn)n≥1 for X in G are also stopping times in F, then the optionalprojection of X onto F is again a local martingale, for the filtrationF.

•

TheoremIf X is a G semimartingale, and F is a subfiltration of G, then oXis an F semimartingale, where oX denotes the optional projectionof X onto F.

•

TheoremLet X > 0 be a G supermartingale. Then oX is an Fsupermartingale.

• Before stating the next theorem, we need a result ofProtter-Shimbo:

•

TheoremIf X is a G semimartingale, and F is a subfiltration of G, then oXis an F semimartingale, where oX denotes the optional projectionof X onto F.

•

TheoremLet X > 0 be a G supermartingale. Then oX is an Fsupermartingale.

• Before stating the next theorem, we need a result ofProtter-Shimbo:

•

TheoremIf X is a G semimartingale, and F is a subfiltration of G, then oXis an F semimartingale, where oX denotes the optional projectionof X onto F.

•

TheoremLet X > 0 be a G supermartingale. Then oX is an Fsupermartingale.

• Before stating the next theorem, we need a result ofProtter-Shimbo:

•

TheoremIf X is a G semimartingale, and F is a subfiltration of G, then oXis an F semimartingale, where oX denotes the optional projectionof X onto F.

•

TheoremLet X > 0 be a G supermartingale. Then oX is an Fsupermartingale.

• Before stating the next theorem, we need a result ofProtter-Shimbo:

•

TheoremIf X is a G semimartingale, and F is a subfiltration of G, then oXis an F semimartingale, where oX denotes the optional projectionof X onto F.

•

TheoremLet X > 0 be a G supermartingale. Then oX is an Fsupermartingale.

• Before stating the next theorem, we need a result ofProtter-Shimbo:

TheoremLet M be a locally square integrable martingale such that4M > −1. If

E[e

12〈Mc ,Mc 〉T +〈Md ,Md〉T

]< ∞, (1)

then E(M) is martingale on [0,T ], where T can be ∞.

TheoremLet X > 0 be a local martingale relative to (P, G). Let oX be itsoptional projection onto a subfiltration F. Then oX is asupermartingale, and assume it is special, with canonicaldecomposition oXt = 1 + Mt − At . Moreover assume that 〈M,M〉exists, and that dAt � d〈M,M〉t . Let cs ≡ dAsd〈M,M〉s and assumecs∆Ms > −1, and

E[e

12

∫ T0 c

2s d〈Mc ,Mc 〉s+

∫ T0 c

2s d〈Md ,Md〉s

]< ∞.

Then there exists a probability Q equivalent to P such that oX is a(Q, F) local martingale.

Corollary

Under the hypotheses of the previous theorem, there is NFLVR for(X , F).