filtration shrinkage and credit risk second princeton credit risk conference, may...
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Filtration Shrinkage and Credit RiskSecond Princeton Credit Risk Conference, May
2008
Philip Protter, Cornell University
May 23, 2008
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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.
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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.
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TheoremLet X > 0 be a G supermartingale. Then oX is an Fsupermartingale.
• Before stating the next theorem, we need a result ofProtter-Shimbo:
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•
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:
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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 ∞.
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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.
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Corollary
Under the hypotheses of the previous theorem, there is NFLVR for(X , F).
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