Convex order for path-dependent (American) options

Convex order for path-dependent (American) options: Dynamic programming and Euler schemeGilles Pages
Advances in Financial Mathematics —
PAGES (LPMA-UPMC) Convex order LPMA 1 / 67
Outline 1 Introduction: the origin(s) 2 Convex order for path-dependent European options
Markov dynamics From discrete time. . . . . . to continuous time
Convex order for stochastic integrals 3 Bermuda and American options
Brownian diffusions Jump martingale diffusions
4 Back to co-monotony More on the Gaussian case.
5 Functional co-monotony principle 6 Application to pathwise continuous processes
Continuous Gaussian processes Continuous Markov Processes, Brownian diffusions Quasi-universal bounds for barrier options
7 Functional co-monotony principle for cadlag processes Skorokhod co-mootony Sup-norm co-monotony for PIIs
8 Applications Antiderivative of “co-monotone process” Asian options
Introduction: the origin(s)
Vega of a Vanilla option with convex payoff option
B Dynamics: Let (Xt)t∈[0,T ] be a Black-Sholes dynamics with volatility σ (and interest rate r = 0. . . )
X σ t = xe−
+σW t , x > 0
where W is a standard B.M. defined on a probability space (,A,P).
B Vanilla European Option: : R+ → R+ payoff function.
V (x , σ,T ) = E(X σ T
).
B Vega: Quid of the monotony of σ 7−→ V (x , σ,T )?
Proposition (Vega)
σ 7−→ V (x , σ,T ) = E(X σ T
) is non-decreasing
is a “processus croissant pour l’ordre convexe” ≡ pcoc ≡ peacock .
Proof :
∂V
≥ x EQ ′ r
T
∫ T
or equivalently: The vega of a Euro-Asian option written on

( 1
T
∫ T
with convex payoff is non-negative in a B-S model.
B Further developments
New proof using the Brownian sheet by M. Yor. Kellerer’s Theorem: “peacock ⇐⇒1-martingale”. Hirsch, Roynette, Profeta & Yor’s monography: explicit representations and many extensions. . . Alternative proofs based on functional co-monotony (P. 2012).
T
∫ T
or equivalently: The vega of a Euro-Asian option written on

( 1
T
∫ T
with convex payoff is non-negative in a B-S model.
B Further developments
New proof using the Brownian sheet by M. Yor. Kellerer’s Theorem: “peacock ⇐⇒1-martingale”. Hirsch, Roynette, Profeta & Yor’s monography: explicit representations and many extensions. . . Alternative proofs based on functional co-monotony (P. 2012).
Ok, but. . .
B This suggests many other (new or not so new) questions !
Local volatility models i .e σ = σ(x)(“functional” concex order) ? [Hajek, 1985], [El Karoui-Jeanblanc-Schreve, 1998], [Martini, 1999].
More general path-dependent payoff functions (more peacocks) ? [Ruschendorf, 2008].
American & Bermuda options (“reduite of peacock” ou ”civet de paon”) [Pham 2005], [Ruschendorf, 2008].
Jumpy risky asset dynamics for (X σ t ) (jumpy peacock ou “paon
sautillant”). [Ruschendorf-Bergenthum, 2007].
Ok, but. . .
Ok, but. . .
Ok, but. . .
Ok, but. . .
Ok, but. . .
Convex order for path-dependent European options Markov dynamics
Path-dependent European options & Brownian diffusion
Theorem
Letσ, θ∈ Clinx ([0,T ]× R,R). Let X (σ) and X (θ) be the unique weak solutions to
dX (σ) t = σ(t,X
(·) t )t∈[0,T ] standard B.M.
(a) B If there exists a partitioning function κ∈ Clinx ([0,T ]× R,R) s.t.{ (i) κ(t, .) is convex for every t∈ [0,T ], (ii) 0 ≤ σ ≤ κ ≤ θ.
B F : C([0,T ],R)→ R, convex, with . sup-polynomial growth (hence . sup-continuous). Then
E F (X (σ)) ≤ E F (X (θ)).
(b)Domination: If |σ| ≤ θ = κ (hence convex), the conclusion still holds true.
Path-dependent European options & Brownian diffusion
Theorem
Letσ, θ∈ Clinx ([0,T ]× R,R). Let X (σ) and X (θ) be the unique weak solutions to
dX (σ) t = σ(t,X
(·) t )t∈[0,T ] standard B.M.
(a) B If there exists a partitioning function κ∈ Clinx ([0,T ]× R,R) s.t.{ (i) κ(t, .) is convex for every t∈ [0,T ], (ii) 0 ≤ σ ≤ κ ≤ θ.
B F : C([0,T ],R)→ R, convex, with . sup-polynomial growth (hence . sup-continuous). Then
E F (X (σ)) ≤ E F (X (θ)).
(b)Domination: If |σ| ≤ θ = κ (hence convex), the conclusion still holds true.
Jensen’s Inequality (not really) revisited = Key Lemma
Lemma (Jensen’s Inequality)
B Let : R→ R, such that
∀ u∈ Rd , Q(u) := E ( u Z )
is well defined in R.
If is convex, then Q is convex, reaches its minimum at 0 so that Q is non-decreasing on R+, non-increasing on R−.
B If Z has a symmetric distribution, then Q is an even function and
∀ a∈ R+, sup |u|≤a
Q(u) = Q(a).
Proof. The function Q is clearly convex and by Jensen’s Inequality
Q(u) ≥ ( E(u Z )
) = (0) = Q(0)
Hence Q is convex, reaches its minimum at u = 0 hence is non-increasing on R− and non-decreasing on R+.
Jensen’s Inequality (not really) revisited = Key Lemma
Lemma (Jensen’s Inequality)
B Let : R→ R, such that
∀ u∈ Rd , Q(u) := E ( u Z )
is well defined in R.
If is convex, then Q is convex, reaches its minimum at 0 so that Q is non-decreasing on R+, non-increasing on R−.
B If Z has a symmetric distribution, then Q is an even function and
∀ a∈ R+, sup |u|≤a
Q(u) = Q(a).
Proof. The function Q is clearly convex and by Jensen’s Inequality
Q(u) ≥ ( E(u Z )
) = (0) = Q(0)
Hence Q is convex, reaches its minimum at u = 0 hence is non-increasing on R− and non-decreasing on R+.
Step 1 of the proof: discrete time framework
B Dynamics: (Zk)1≤k≤n be a sequence of independent, centered r.v.
Xk+1 = Xk + σk(Xk) Zk+1,
Yk+1 = Yk + θk(Yk) Zk+1, 0 ≤ k ≤ n − 1, X0 = Y0 = x
where σk , θk : R→ R, k = 0, . . . , n − 1 have linear growth.
B Dynamic programming: We introduce two martingales
Mk = E ( F (X0:n) | FZ
k
Qk()(u) = E(uZk), u∈ R, k = 1 : n.
Mk = E ( F (X0:n) | FZ
k
Qk()(u) = E(uZk), u∈ R, k = 1 : n.
Mk = E ( F (X0:n) | FZ
k
Qk()(u) = E(uZk), u∈ R, k = 1 : n.
A (first) backward induction and the definition of the kernels Qk imply
Mk = Φk(X0:k) and Nk = Ψk(Y0:k), k = 0, . . . , n.
where Φk ,Ψk : Rk+1 → R, k = 0, . . . , n are recursively defined by
Φn := F , Φk(x0:k) := ( Qk+1Φk+1(x0:k , xk+.)
) (σk(xk)), k = 0 : n−1,
Ψn := F , Ψk(y0:k) := ( Qk+1Ψk+1(y0:k , yk+.)
) (θk(yk)), k = 0 : n−1.
B Assume now that all functions σk are ≥ 0 and convex:
) ⇓(
so that
&
(Second) backward induction =⇒ all functions Φk are convex.
(Third) backward induction =⇒ Φk ≤ Ψk , k = 0 : n − 1.
First note that Φn = Ψn = F . If Φk+1 ≤ Ψk+1, then
Φk(x0:k) = ( Qk+1Φk+1(x0,k , xk + .)
) (σk(xk))
) ⇓(
so that
&
Φk(x0:k) = ( Qk+1Φk+1(x0,k , xk + .)
) (σk(xk))
) ⇓(
so that
&
Φk(x0:k) = ( Qk+1Φk+1(x0,k , xk + .)
) (σk(xk))
) ⇓(
so that
&
Φk(x0:k) = ( Qk+1Φk+1(x0,k , xk + .)
) (σk(xk))
) ⇓(
so that
&
Φk(x0:k) = ( Qk+1Φk+1(x0,k , xk + .)
) (σk(xk))
End of discrete times
This time, one shows that:
the functions Ψk are convex, k = 0, . . . , n
Φn ≤ Ψn =⇒ Φk ≤ Ψk , k = 0, . . . , n − 1.
Remark. The discrete time setting has its own interest.
Step 2 of the proof: Back to continuous time
B Euler scheme(s): Discrete time Euler schemes with step T n , starting at
x . For X (σ): for k = 0, . . . , n − 1,
X (σ),n tn k+1
= X (σ),n tn k
⇓
∀ p > 0, sup n≥1
sup t∈[0,T ]
|X (σ),n t |
Step 2 of the proof: Back to continuous time
B Euler scheme(s): Discrete time Euler schemes with step T n , starting at
x . For X (σ): for k = 0, . . . , n − 1,
X (σ),n tn k+1
= X (σ),n tn k
⇓
∀ p > 0, sup n≥1
sup t∈[0,T ]
|X (σ),n t |
From discrete to continuous time
B Interpolation (n ≥ 1)
Piecewise affine interpolator defined by
∀ x0:n∈ Rn+1, ∀ k = 0, . . . , n − 1, ∀ t∈ [tn k , t
n k+1], .
) X (σ),n := in
B F : C([0,T ],R)→ R convex functional (with r -polynomial growth).
∀ n ≥ 1, Fn(x0:n) := F ( in(x0:n)
) , x0:n∈ Rn+1. (1)
Step 1 (Discrete time):
Discrete time result implies: [[ σ(tn
k , .) ≤ [[ κ(tn
k , .) ]] ≤ θ(tn
) .
Step 2 (Transfer): Key = Theorem 3.39, p.551, Jacod-Shiryaev’s book (2nd edition).
X (σ),n L(.sup)−→ X (σ) as n→∞.
E F (X (σ)) = lim n
E F ( X (σ),n
) (idem for X (θ)).
∀ n ≥ 1, Fn(x0:n) := F ( in(x0:n)
) , x0:n∈ Rn+1. (1)
k , .) ≤ [[ κ(tn
k , .) ]] ≤ θ(tn
) .
X (σ),n L(.sup)−→ X (σ) as n→∞.
E F ( X (σ),n
∀ n ≥ 1, Fn(x0:n) := F ( in(x0:n)
) , x0:n∈ Rn+1. (1)
k , .) ≤ [[ κ(tn
k , .) ]] ≤ θ(tn
) .
X (σ),n L(.sup)−→ X (σ) as n→∞.
E F ( X (σ),n
Applications to “functional peacocks”.
dSt = Stσ(t,St)dW t , S0 = s0 > 0, (r = 0)
where σ : [0,T ]× R→ R is a bounded continuous functions. The weak solution satisfies
S (σ) t = s0e
R t 0 σ(s,S
(σ) s )dBs− 1
Theorem (P. 2012)
If there exists a function κ : [0,T ]× R+ → R+ such that x 7→ xκ(t, x) extends into an R+-valued convex function on R satisfying
(a) Partitioning: 0 ≤ σ(t, .) ≤ κ(t, .) ≤ θ(t, .) on R+, t∈ [0,T ],
or
(b) Dominating: |σ(t, .)| ≤ θ(t, .) = κ(t, .) t∈ [0,T ].
If f : R→ R convex function and µ is a (finite) signed measure on [0,T ]
E f
The case of jump diffusions
B Levy process: Let Z = (Zt)t∈[0,T ] be a Levy process with Levy measure ν satisfying∫
|z|≥1 |z |pν(dz) < +∞, p∈ [1,+∞) (hence Zt ∈ L1(P), t∈ [0,T ]).
E Z1 = 0.
Proposition
Let κi : [0,T ]× R→ R, i = 1, 2, be continuous functions, Lipschitz
continuous in x uniformly in t. Let X (κi ) = (X (κi ) t )t∈[0,T ] be the unique
diffusion process solution to
(κi ) t− )dZt , X
(κi ) 0 = x ∈ R, i = 1, 2.
(a) Partitioning: Assume there exists a κ : [0,T ]× R→ R+, κ(t, .) convex, t∈ [0,T ], satisfying
0 ≤ κ1 ≤ κ ≤ κ2.
(b) Dominating: If Z1 L = −Z1 and κ2 is convex in x satisfies
|κ1| ≤ κ2.
Let F : D([0,T ],R)→ R be a convex Sk-continuous functional with
∀α∈ C([0,T ],R), |F (α)| ≤ C (1 + αrsup), 0 < r < p.
Then E F (X (κ1)) ≤ E F (X (κ2)).
Proposition
(κi ) 0 = x ∈ R, i = 1, 2.
0 ≤ κ1 ≤ κ ≤ κ2.
|κ1| ≤ κ2.
∀α∈ C([0,T ],R), |F (α)| ≤ C (1 + αrsup), 0 < r < p.
Then E F (X (κ1)) ≤ E F (X (κ2)).
Proposition
(κi ) 0 = x ∈ R, i = 1, 2.
0 ≤ κ1 ≤ κ ≤ κ2.
|κ1| ≤ κ2.
∀α∈ C([0,T ],R), |F (α)| ≤ C (1 + αrsup), 0 < r < p.
Then E F (X (κ1)) ≤ E F (X (κ2)).
Proof. B Step 1: The discrete time step is . . . identical!
B Step 2: The “genuine” Euler scheme (X (κi ) t )t∈[0,T ] converges in
Lp′(P), p′ < p for the . sup-norm topology (hence the Skorokhod topology as well).
B Step 3: (Jacod’s Lemma) Let αn∈ D([0,T ],R), n ≥ 1. Define the sequence of stepwise constant approximations defined by
αn(t) = αn(tn), t∈ [0,T ].
Then, ( αn Sk−→ α
B Step 4: Conclude like in the pathwise continuous setting.
αn(t) = αn(tn), t∈ [0,T ].
αn(t) = αn(tn), t∈ [0,T ].
αn(t) = αn(tn), t∈ [0,T ].
Counter-example (discrete time setting)
We consider the 2-period dynamics. Let X = X σ,x = (X σ,x 0:2 ) satisfy
X1 = x + σZ1
– Z1:2 L∼ N (0; I2) and σ > 0
– v : R→ R+ is a bounded C2-function s.t. v ′ has strict local maximum at x0 satisfying that v ′(x0) = 0 and v ′′(x0) < −1
⇓ √
Let f (x) = ex . It is clear that
(x , σ) := Ef (X2) = exE ( eσZ1+v(x+σZ1)
) ′σ(x , σ) = exE
1
1
)) .
In particular ′σ(x , 0) = ex+v(x)(1 + v ′(x))E Z = 0
and ′′σ(x , 0) = ex+v(x)
( (1 + v ′(x))2 + v ′′(x))
so that ′′σ(x0, 0) < 0.
Then, there exists σ0 > 0 such that
′σ(x0, σ) < 0, σ∈ (0, σ0].
Convex order for path-dependent European options Convex order for stochastic integrals
Convex order for stochastic integrals (Levy driven)
Theorem (P. 2012, Assumption. . . )
B Let Z = (Zt)t∈[0,T ] be a Levy process, with Levy measure ν satisfying ν(|x |p1{|x |≥1}) < +∞, p > 1.
B Let (Ht)t∈[0,T ] be an (Ft)-predictable process and
B h = (ht)t∈[0,T ]∈ L2([0,T ], dt).
B Let F : D([0,T ],R)→ R be a convex, Sk-continuous with r-polynomial growth, 0 < r < p, for the sup norm.
Convex order for stochastic integrals (Levy driven)
Theorem (P. 2012, Assumption. . . )
B Let Z = (Zt)t∈[0,T ] be a Levy process, with Levy measure ν satisfying ν(|x |p1{|x |≥1}) < +∞, p > 1.
B Let (Ht)t∈[0,T ] be an (Ft)-predictable process and
B h = (ht)t∈[0,T ]∈ L2([0,T ], dt).
B Let F : D([0,T ],R)→ R be a convex, Sk-continuous with r-polynomial growth, 0 < r < p, for the sup norm.
Theorem (. . . Statement)
E F
) .
(b) If (∀ t∈ [0,T ], Ht ≥ ht ≥ 0) P-a.s. & HL2([0,T ],dt)∈ Lr ′ , r< r ′<p,
E F
E F
) .
E F
E F
) .
E F
Non Markovian Doleans martingales
B Same assumptions on (Ht)t∈[0,T ] and h = (ht)t∈[0,T ].
B Let F : C([0,T ],R)→ R convex, . sup-continuous with polynomial growth.
(a) If |Ht | ≤ ht P-a.s. for every t ∈ [0,T ], then
E F
)) .
(b) If Ht ≥ ht ≥ 0 P-a.s. for every t ∈ [0,T ] and
E e (1+ε)HL2([0,T ],dt) < +∞ for an ε > 0, then
E F
E F
)) .
E F
E F
)) .
E F
Yet another counter-example
The theorem fails if the assumption that the dominating process (ht)t∈[0,T ] is deterministic.
B We still consider the 2-period dynamics. Let X = (X0:2) satisfy
X1 = σZ1
X2 = X1 + √
2v(Z1) Z2
B Let f (x) = ex . It is clear that
(σ) := Ef (X2) = E ( eσZ1+v(Z1)
) .
) so that, by a standard one-dimensional co-monotony argument,
′(0) = E ( ev(Z1)Z1
) < E ev(Z1)E Z1 = 0.
As a consequence,
is decreasing on a right neighbourhood [0, σ0] (σ0 > 0) of 0.
B To plug this into a Brownian stochastic integral framework: let σ, σ∈ (0, σ0], σ < σ.
Ht = σ1[0,1](t) + √
2v(W 1)1(1,2](t).
It is clear that 0 ≤ Ht ≤ Ht , t∈ [0, 2] whereas
E (
Bermuda and American options Brownian diffusions
Bermuda options (discrete time)
B Dynamics: Still. . . (Zk)1≤k≤n be a sequence of independent, centered r.v.
where σk , θk : R→ R, k = 0, . . . , n − 1 with linear growth.
B Reduites: Let Fk : Rk+1 → R+, k = 0 : n be “reward” (Borel) functions with linear growth.
u0(x) = sup {
Bermuda options (discrete time)
B Dynamics: Still. . . (Zk)1≤k≤n be a sequence of independent, centered r.v.
where σk , θk : R→ R, k = 0, . . . , n − 1 with linear growth.
B Reduites: Let Fk : Rk+1 → R+, k = 0 : n be “reward” (Borel) functions with linear growth.
u0(x) = sup {
Proposition
B Let Fk : Rk+1 → R+, k = 0, . . . , n be a sequence of non-negative convex (payoff) functions with r-polynomial growth for the sup norm.
B Let F = (Fk)0≤k≤n be a filtration such that Zk is Fk -adapted and Zk
is independent of Fk−1, k = 1, . . . n.
(a) Partitioning: If there exists κk , k = 0, . . . , n, convex, such that
0 ≤ σk ≤ κk ≤ θk , 0 ≤ k ≤ n − 1
then, for every x ∈ R, u0(x) ≤ v0(x).
(b) Domination: If( Zk L = −Zk , k = 1, . . . , n
) & ( |σk | ≤ θk = κk , k = 0, . . . , n − 1
) then the same conclusion holds true.
Proposition
0 ≤ σk ≤ κk ≤ θk , 0 ≤ k ≤ n − 1
) & ( |σk | ≤ θk = κk , k = 0, . . . , n − 1
Proposition
0 ≤ σk ≤ κk ≤ θk , 0 ≤ k ≤ n − 1
) & ( |σk | ≤ θk = κk , k = 0, . . . , n − 1
Proof (Sketch of)
) 0≤k≤n
} and
} .
un = Fn & uk(x0:k) = max (
)) k = 0, . . . , n − 1.
Propagation of the convexity: (u, v) 7→ max(u, v) is non-decreasing in u and v and convex owing to the “revisited” Jensen Inequality.
Proof (Sketch of)
) 0≤k≤n
} and
} .
un = Fn & uk(x0:k) = max (
)) k = 0, . . . , n − 1.
Propagation of the convexity: (u, v) 7→ max(u, v) is non-decreasing in u and v and convex owing to the “revisited” Jensen Inequality.
. . .
B Idem for vk : Rk+1 → R in connection with the (P,F)-Snell envelope V .
B Note that uk+1 convex still implies
ξ 7−→ ( Qk+1uk+1(x0:k , xk + .)
) (ξ) is non-decreasing on R+
B Comparison Principle (0 ≤ σk ≤ θk , σk): Backward induction to prove uk ≤ vk , k = 0 : n (obvious if k = n).
Assume uk+1 ≤ vk+1, k + 1 ≤ n. For every x0:k ∈ Rk+1
uk(x0:k) ≤ max ( Fk
E U0 = u0(x) ≤ v0(x) = E V0.
. . .
B Idem for vk : Rk+1 → R in connection with the (P,F)-Snell envelope V .
B Note that uk+1 convex still implies
ξ 7−→ ( Qk+1uk+1(x0:k , xk + .)
) (ξ) is non-decreasing on R+
B Comparison Principle (0 ≤ σk ≤ θk , σk): Backward induction to prove uk ≤ vk , k = 0 : n (obvious if k = n).
Assume uk+1 ≤ vk+1, k + 1 ≤ n. For every x0:k ∈ Rk+1
uk(x0:k) ≤ max ( Fk
E U0 = u0(x) ≤ v0(x) = E V0.
Back to continuous time
B Snell envelope of the Euler scheme of X and Y
U(n) = P-Snell ( Fk(X
) .
B Convergence: In the case of Brownian diffusions, it is a classical result (with convergence rates in fact see e.g . Bally-P. (’03)) that max
0≤k≤n |U(n)
k − UX tn k |
tn k |
Theorem
Under partitioning or dominating assumptions on σ and θ, F (t, .) convex on C([0,T ],R) and F continuous, etc, one has
u0(x) = E UX (σ),x
0 = v0(x).
Jump martingale diffusions: what makes problem?
B Discrete time step: Identical.
B From discrete to continuous time: Still the Euler scheme. But we have to make the Snell envelopes converge. . . How to proceed?
Filtration enlargement argument/trick
Let (Ft)t∈[0,T ] be a filtration and let Y be an (Ft)t∈[0,T ]-adapted cadlag process defined on a probability space (,A,P) so that
∀ t∈ [0,T ], FY t ⊂ Ft
We introduce the so-called H-assumption (on the filtration (Ft)t∈[0,T ]):
(H) ≡ ∀H∈ FY T , bounded, E
( H | Ft
Theorem (Lamberton-P., 1990, IHP B)
B Let (X n)n≥1 be a sequence of quasi-left cadlag processes defined on a probability spaces (n,Fn,Pn) of (D)-class and satisfying the Aldous criterion. Let (τ∗n )n≥1 be a sequence of
( FX n
,Pn)-optimal stopping times. If (X n)n≥1 is uniformly integrable and satisfies
X n L(Skor)−→ X , P X
= P probability measure on (D([0,T ],R),DT ).
B Non-degeneracy of (τ∗n )n≥1: every limiting value Q of L(X n, τ∗n ) on D([0,T ],R)× [0,T ] satisfies the (H) property [. . . ], then
lim n
0 = EP UX 0 .
B If the optimal stopping problem related to (X ,Q,Dθ) has a unique
solution in distribution, say µ∗τ∗ , not depending on Q, then τ∗n [0,T ]−→ µ∗τ∗ .
Bermuda and American options Jump martingale diffusions
Theorem (P. 2012)
Under the usual partitioning assumptions on κi , i = 1, 2, and F (convexity), etc, the reduites associated to F and X (κi ),x , i = 1, 2, satisfy
u(κ1)(x) ≤ u(κ2)(x).
All the efforts are focused on showing that the filtration enlargement assumption (H) is satisfied by limiting distributions Q.
Back to co-monotony
Classical 1D-co-monotony principle
Another example of the transfer from discrete to continuous time of a useful property.
BProposition [One dimensional co-monotony principle] Let X : (,A,P)→ R be a r.v. and let f , g : R→ R betwo monotone functions with the same monotony.
(a) If f (X ), g(X ), f (X )g(X )∈ L1(P), then
E f (X )g(X ) ≥ E f (X )E g(X ).
Furthermore, equality holds iff f (X ) or g(X ) is P-a.s. constant.
(b) If f and g have the same constant sign with opposite monotony, then
f (X )g(X )1 ≤ f (X )1g(X )1.
Back to co-monotony
BDefinition [Markov chain and co-monotony]
(a) A Markov transition (P(x , dy))x∈R is monotony preserving if,
∀ f : R→ R bounded and monotone =⇒ Pf has the same monotony.
(b) Two Borel functions Φ, Ψ : Rd → R are componentwise co-monotone if,
∀ i ∈ {1, . . . , d}, ∀ (x1, . . . , xi−1, xi+1, . . . , xd)∈ Rd−1,
xi 7−→ Φ(x1, . . . , xi , . . . , xd) and xi 7−→ Ψ(x1, . . . , xi , . . . , xd)
have the same monotony, not depending
on (x1, . . . , xi−1, xi+1, . . . , xd) nor on i .
(c) If Φ and −Ψ are co-monotone, Φ and Ψ are said to be anti-monotone.
Back to co-monotony
B Proposition (a) Let X = (Xk)0≤k≤n be an R-valued Markov chain defined on (,A,P) with monotony preserving transitions
Pk−1,k(x , dy) = P(Xk ∈ dy |Xk−1 = x), k = 1, . . . , n
Then for every pair of componentwise co-monotone functions Φ, Ψ : Rd → R such that Φ(X ), Ψ(X ), Φ(X )Ψ(X )∈ L1(P),
E Φ(X )Ψ(X ) ≥ E Φ(X )E Ψ(X ).
(b) If the random variables X0, . . . ,Xn are independent, the conclusion remains true under the weaker co-monotony assumption: There exists τ ∈ Σ({0, . . . , n}) s.t. ∀ i ∈ {0, . . . , n}, ∀ (x0, . . . , xi−1, xi+1, . . . , xn)∈ Rn, xi 7→ Φ(x0, . . . , xi , . . . , xn) and xi 7→ Ψ(x0, . . . , xi , . . . , xn) have the same monotony, possibly depending on (xτ(0), . . . , xτ(i−1)).
Back to co-monotony
Proof (sketch of)
Proof. (a) Induction on n∈ N.
B n = 0 follows from the scalar co-monotony principle applied to X0 (with distribution µ0).
B (n) =⇒ (n + 1): Assume w.l.g. that Φ and Ψ are bounded and componentwise non-decreasing. Put FX
n = σ(X0, . . . ,Xn). It follows from the Markov property that
E ( Φ(X0:n+1) | FX
) = E
( Φ(n)(X0:n)
Back to co-monotony
Proof (sketch of)
B Let x0:n∈ Rn+1. The one dimensional co-monotony principle applied with Pn,n+1(xn, dy) to Φ(x0:n, .) and Ψ(x0:n, .) yields
(Φ Ψ)(n)(x0:n) = Pn,n+1
E (Φ Ψ)(X0:n+1) = E ( (Φ Ψ)(n)(X0,n)
) ≥ E
( Φ(n)(X0:n)Ψ(n)(X0:n)
) .
B For every i ∈ {0, . . . , n − 1}, xi 7→ Φ(n)(x0, . . . , xn) is non-decreasing since Pn,n+1 is a nonnegative operator.
Back to co-monotony
B Now let xn, x ′n∈ R, xn ≤ x ′n. Then
Pn,n+1
( Φ(x0, . . . , xn, .)
′ n, .) ) (x ′n)
where: – the first inequality follows from the non-negativity of the operator Pn,n+1
– the second inequality follows from its monotony preserving property of Pn,n+1 since xn+1 7→ Φ(x0:n+1) is non-decreasing.
The function Ψ(n), defined likewise, shares the same properties.
B One concludes by the induction assumption applied to the Markov chain (Xk)0≤k≤n:
E (
Back to co-monotony
B Counter-example. Let X = (X0,X1) ∼ N (
0; [ 1 ρ ρ 1
]) , ρ∈ (−1, 0).
P0,1(f )(x0) := E(f (X1) |X0 = x0) = Ef ( ρ x0 +
√ 1− ρ2 Z
P0,1 is monotony. . . inverting as soon as ρ < 0.
In particular we have E X0X1 = ρ < 0 = E X0 E X1.
In fact
(X0,X1) satisfies the co-monotony principle if and only if ρ ≥ 0.
Back to co-monotony More on the Gaussian case.
Pitt’s Theorem
Let X = (X1, . . . ,Xd) ∼ N ( 0; Σ
) with Σ = [σij ]1≤i ,j≤d .
B Σ characterizes the distribution of X , so it characterizes whether or not X shares a co-monotony property.
Ok! But can we read it easily on Σ?
A necessary condition for co-monotony is obviously that
∀ i , j ∈ {1, . . . , d}, σij = Cov(Xi ,Xj) = E XiXj − E Xi E Xj ≥ 0.
B Theorem [Pitt, AoP, 1982] A Gaussian random vector X with covariance matrix Σ = [σij ]1≤i ,j≤d satisfies a componentwise co-monotony principle iff
∀ i , j ∈ {1, . . . , d}, σij ≥ 0.
Extensions. See [Pitt et al., AoP, 1983]. Typically, if Z ∼ N (0; Id) and if h : Rd → R satisfies appropriate regularity and integrability assumptions, one has( ∀ x ∈ Rd ,
∂2h
Remark. Another natural criterion for co-monotony – theoretically straightforward although not easy to “read” in practice on the covariance matrix itself – is to make the assumption that
∃A = [aij ]1≤i≤d ,1≤j≤r , r ∈ N∗, aij ≥ 0 s.t. Σ = AA∗.
Then, the component wise co-monotony property follows from
X L∼ AZ , Z
L∼ N (0; Ir ).
Surprisingly, this criterion is not a characterization in general:
– if d ≤ 4, symmetric matrices Σ with nonnegative entries can always be decomposed as Σ = AA∗ where A has nonnegative entries.
– But if d ≥ 5, this is no longer true. The negative answer is inspired by a former counter-example – originally due to Horn –
To be precise, the nonnegative symmetric 5× 5 matrix Σ (with rank 4) defined by
Σ =
1 0 0 1/2 1/2 0 1 3/4 0 1/2 0 3/4 1 1/2 0
1/2 0 1/2 1 0 1/2 1/2 0 0 1
has nonnegative entries but cannot be written AA∗ with A = [aij ], aij ≥ 0.
The Euler scheme. . .
. . . is an (important) example of Markov chain.
B Let X = (Xt)t∈[0,T ] be a Brownian diffusion assumed to be solution to the stochastic differential equation
SDE ≡ dX x t = b(t,X x
t )dt + σ(t,X x t )dWt , t∈ [0,T ], X0 = x .
Its Euler scheme with step h = T/n and Brownian increments is entirely characterized by its transitions
Pk,k+1(f )(x) = E f (
x + hb(tn k , x) + σ(tn
k , x) √
n T , k = 0, . . . , n.
B If b is Lipschitz continuous in x , uniformly in t∈ [0,T ] and if σ(t, x) = σ(t)∈ L2([0,T ], dt), then, for large enough n, the Euler transitions Pk,k+1 are monotony preserving.
Functional co-monotony principle
B Definition A random variable X whose paths take values in a partially ordered normed vector space (E , . E ,≤E ) (1) satisfies a co-monotony principle on E if, for every bounded, co-monotone, P
X -a.s. continuous,
measurable functionals F , G : E → R,
E F (X )G (X ) ≥ E F (X )E G (X ).
B Examples: • E = C([0,T ],R), D([0,T ],R).
• E = Lp([0,T ], µ), p ∈ [1,+∞).
1For every α, β, γ∈ E and every λ∈ R+, α ≤ β ⇒ α + γ ≤ β + γ and λ ≥ 0⇒ λα ≤ λβ.
Application to pathwise continuous processes
Continuous processes: (E , .E ) = (C([0,T ],R), . sup)
) .
B Proposition Let X = (Xt)t∈[0,T ] be a pathwise continuous process defined on (,A,P) sharing the following finite dimensional co-monotony property:
∀ n ≥ 1, ∀(t1, . . . , tn)∈ [0,T ]n, 0 ≤ t1 < t2 < · · · < tn ≤ T ,
(Xtk )1≤k≤n, satisfies a componentwise co-monotony principle.
Then X satisfies a functional co-monotony principle on C([0,T ],R).
Proof. Assume w.l.g. that F and G are both non-decreasing for the natural order on C([0,T ],R). Let n ≥ 1.
B Canonical linear n-interpolator of α:
α(n)(t) := tn k+1 − t
tn k+1 − tn
k , t n k+1], k = 0, . . . , n−1.
still with tn k = kT
n , k = 0, . . . , n.
α− α(n)sup ≤ w(α,T/n) −→ 0 as n→∞. where w(α, .) denotes the uniform continuity modulus of αso that
X (n) − Xsup −→ 0, P-a.s. as n→∞.
B Canonical linear n-interpolator of α of x = x0:n∈ Rn+1:
χn(x , t) := tn k+1 − t
tn k+1 − tn
n k+1], k = 0, . . . , n − 1
and Fn(x) := F (χn(x , .)).
B It is clear that
x , x ′∈ Rn+1, x ≤ x ′ (componentwise sense) =⇒ Fn(x) ≤ Fn(x ′)
since χn(x , .) ≤ χn(x ′, .) as functions. On the other hand
X (n) = χ ( (Xtn
k )0≤k≤n, .
) = F (X (n)).
B The sequence (Xtn k )0≤k≤n satisfies a componentwise co-monotony
principle. Hence if F and G are bounded,
∀ n ≥ 1, E F (X (n))G (X (n)) ≥ E F (X (n))E G (X (n)).
Letting n→ +∞ yields the conclusion since F and G are continuous with respect to the sup-norm.
Application to pathwise continuous processes Continuous Gaussian processes
Continuous Gaussian processes
B Theorem [Functional Pitt’s Theorem] Let X = (Xt)t∈[0,T ] be a continuous Gaussian process with a covariance operator C
X . X satisfies a
functional co-monotony principle iff
(s, t) ≥ 0.
Proof. For every n ≥ 1, (X n tn k )0≤k≤n is a Gaussian random vector
satisfying a componentwise co-monotony principle since
Σn = [ C
X (tn
Application to pathwise continuous processes Continuous Gaussian processes
Examples
Brownian bridge: CX (s, t) = s ∧ t − st T ≥ 0,
Wiener integrals ∫ t
0 f 2(u)du ≥ 0,
Fractional Brownian motions BH , H∈ (0, 1]: CBH (s, t) = 1
2
) ≥ 0.
but also Liouville, processes, Wiener integrals depending on a parameter, Pseudo-fractional Brownian motion with Hurst constant H∈ (0, 1], etc.
Application to pathwise continuous processes Continuous Markov Processes, Brownian diffusions
Continuous Markov Processes, Brownian diffusion processes
B Proposition Let X = (Xt)t∈[0,T ] be a pathwise continuous Markov process with transition operators (Ps,t)0≤s≤t≤T satisfying the monotony preserving property. Then X satisfies a functional co-monotony principle on its path space C([0,T ],R).
Proof. The sequence (Xtn k )0≤k≤n is a discrete time Markov chain whose
transition operators Ptn k ,t
n k+1
(SDE ) ≡ dX x t = b(t,X x
t )dt + σ(t,X x t )dWt , X x
0 = x
Comparison principle !
Application to pathwise continuous processes Quasi-universal bounds for barrier options
Quasi-universal bounds for barrier options
B Let S = (St)t∈[0,T ] be a cadlag nonnegative stochastic process modeling the price dynamics of a risky asset on (,A,P) (P needs not to to be risk-neutral). No interest rates
B Down-and-In Call with maturity T .
FD&I (α) = ( α(T )− K )+1{∗αT
≤L}, L∈ (0, S0).
CallD&In(K , L,T ) = E ( FD&I (S)
) .
CallD&In(K , L,T ) ≤ Call(K ,T )P(∗ST ≤ L).
B Down-and-Out Call:
FD&O(α) = ( α(T )− K )+1{∗αT>L}
Parity equation CallD&In(K , L,T ) + CallD&Out(K , L,T ) = Call(K ,T ) yields
CallD&Out(K ,H,T ) ≥ Call(K ,T )P(∗ST > L).
B Lower bounds for Up-and-In and Up-and-Out Calls with barrier L > S0
(and strike K ), namely
and CallU&Out(K , L,T ) ≤ Call(K ,T )P(S∗T ≤ L).
B Down-and-Out Call:
FD&O(α) = ( α(T )− K )+1{∗αT>L}
Parity equation CallD&In(K , L,T ) + CallD&Out(K , L,T ) = Call(K ,T ) yields
CallD&Out(K ,H,T ) ≥ Call(K ,T )P(∗ST > L).
B Lower bounds for Up-and-In and Up-and-Out Calls with barrier L > S0
(and strike K ), namely
and CallU&Out(K , L,T ) ≤ Call(K ,T )P(S∗T ≤ L).
Functional co-monotony principle for cadlag processes Skorokhod co-mootony
Functional co-monotony principle for cadlag processes
B Natural idea = mimick the “transfer Proposition” from pathwise continuous setting by simply replacing (C([0,T ],R), . sup) by ID([0,T ],R) (cadlag functions endowed with the J1 -Skorokhod toplogy). This leads to a non-trivial result:
We introduce the stepwise constant approximation operator define on every function α∈ ID([0,T ],R) by
α(n) = n∑
k=1
k = kT
α(n) Sk−→ α.
Proposition Let X = (Xt)t∈[0,T ] be a pathwise cadlag process sharing the finite dimensional co-monotony property. Then X satisfies a functional co-monotony principle on its path space ID([0,T ],R) for the J1-Skorokhod topology.
Functional co-monotony principle for cadlag processes Skorokhod co-mootony
Example: Any cadlag process with independent increments (PII) (Xt)t≥0
satisfies a functional co-monotony principle on its path space ID([0,T ],R) for the J1-Skorokhod topology.
Proof. A cadlag PII (Xt)t≥0 shares the finite dimensional co-monotony property since (Xtn
k )0≤k≤n is a Markov chain whose transitions
Ptn k−1,t
( xk−1 + Xtn
monotony preserving.
Problem: Let t∈ [0,T ) and let α be not continuous at t.
β 7→ sup s∈[0,t]
|β(s)| is not Skorokhod continuous at α. . .
Does not apply e.g . to PII with fixed discontinuities. . .
Functional co-monotony principle for cadlag processes Sup-norm co-monotony for PIIs
Sup-norm co-monotony for PIIs
Theorem. A cadlag PII satisfies a co-monotony principle on (D([0,T ],R), . sup).
Proof. We consider a cadlag process PII (Xt)t≥0 defined on (,A,P). The key is the Levy-Khintchine decomposition as exposed in [Jacod-Shiryaev], chap. II, section 3. First decompose X as the sum
X = X (1) ⊥⊥ + X (2), X (i) PII, i = 1, 2,
X (1) without fixed discontinuities and X (2) a pure jump PII only jumping at a deterministic sequence (tn)n≥1 of times, namely
X (2) t =
Un1{tn≤t}, t∈ [0,T ],
where (Un)n≥ is a sequence of independent r.v. satisfying the usual assumption of the three series Theorem∑ n
P(|Un|≥1) + EUn1{|Un|≤1} + E ( U2
n1{|Un|≤1}−(E Un1{|Un|≤1}) 2 ) <+∞.
Sup-norm co-monotony for PIIs
Theorem. A cadlag PII satisfies a co-monotony principle on (D([0,T ],R), . sup).
Proof. We consider a cadlag process PII (Xt)t≥0 defined on (,A,P). The key is the Levy-Khintchine decomposition as exposed in [Jacod-Shiryaev], chap. II, section 3. First decompose X as the sum
X = X (1) ⊥⊥ + X (2), X (i) PII, i = 1, 2,
X (1) without fixed discontinuities and X (2) a pure jump PII only jumping at a deterministic sequence (tn)n≥1 of times, namely
X (2) t =
Un1{tn≤t}, t∈ [0,T ],
where (Un)n≥ is a sequence of independent r.v. satisfying the usual assumption of the three series Theorem∑ n
P(|Un|≥1) + EUn1{|Un|≤1} + E ( U2
n1{|Un|≤1}−(E Un1{|Un|≤1}) 2 ) <+∞.
Ingredients of proof.
X ⊥ Y E -valued processes, E Banach, satisfy a co-monotony principle =⇒ X ±Y both satisfying a co-monotony principle. Extends to series of independent processes.
If X has no fixed discontinuities
X L2
X (ε)
where X (ε) is a compound Poisson process with |X (ε) t | ≥ ε.
A compound Poisson process reads
X = ∑ n≥1
1{T−1 +···+T−n ≤t}Vn
where (T + n ,T−n ,Un,Vn)n≥1 are mutually independent (and
Un,Vn ≥ 0).
Use the proposition of componentwise co-monotony in the independent case (claim (b)).
Applications
“Functional antithetic” simulation method
E ⊂ F([0,T ],R) and a process X an E -valued process.
Non-increasing T : E → E i .e. α ≤ β ⇒ T (α) ≥ T (β), α, β∈ E ,
F : E → R monotone and P X
-a.s. continuous on E .
Then
Cov(F (X ),F T (X )) = E ( F (X )F T (X )
) − ( E F (X )
)2 ≤ 0.
so that it is more efficient to simulate F (X ) + F T (X )
2 than F (X ).
In practice this becomes true for approximation schemes of X when close to X .
Applications
Peacocks
Definition. An integrable process (Xλ)λ≥0 is a peacock if for every convex function : R→ R, the (−∞,+∞]-valued function λ 7→ E(Xλ) is non-decreasing.
Remark. C2 with compactly supported ′′ is a characterizing family.
Theorem. [Kellerer] A process is a peacock if and only if there exists a
martingale (Mλ)λ≥0 such that Xλ L∼ Mλ, λ∈ R+.
A martingale is a peacock ⇐= Jensen’s inequalitty.
Applications Antiderivative of “co-monotone process”
Antiderivative of “co-monotone process”
Proposition Let X = (Xt)t≥0 be an integrable cadlag process satisfying a co-monotony principle for the sup norm on every interval [0,T ], T > 0. Let µ be a Borel measure on (R+,Bor(R+)). Assume that sup[0,t] E |Xs | < +∞ for every t > 0 and that t 7→ E Xt is cadlag. Then the process (∫
[0,t] (Xs − E Xs)µ(ds)
is a peacock.
Proof. Assume w.l.g. E Xt = 0. Let ∈ C2(R,R) (with linear growth).
∀ x , y ∈ R, (y)− (x) ≥ ′(x)(y − x)
so that, if t1 < t2
(Yt2)−(Yt1) ≥ Φt1(X )
(∫ [0,t1] α(s)µ(ds)
) is bounded, continuous for (ID([0, t2],R), .sup) and non-decreasing for the pointwise order since ′ is.
Antiderivative of “co-monotone process”
Proposition Let X = (Xt)t≥0 be an integrable cadlag process satisfying a co-monotony principle for the sup norm on every interval [0,T ], T > 0. Let µ be a Borel measure on (R+,Bor(R+)). Assume that sup[0,t] E |Xs | < +∞ for every t > 0 and that t 7→ E Xt is cadlag. Then the process (∫
[0,t] (Xs − E Xs)µ(ds)
is a peacock.
Proof. Assume w.l.g. E Xt = 0. Let ∈ C2(R,R) (with linear growth).
∀ x , y ∈ R, (y)− (x) ≥ ′(x)(y − x)
so that, if t1 < t2
(Yt2)−(Yt1) ≥ Φt1(X )
(∫ [0,t1] α(s)µ(ds)
) is bounded, continuous for (ID([0, t2],R), .sup) and non-decreasing for the pointwise order since ′ is.
Consequently, owing to the functional co-monotony principle, we get
E (
Φt1(X )
∫ (t1,t2]
Applications Asian options
Asian options
Proposition Let X = (Xt)t∈[0,T ] be a (cadlag) PII such that, for every
u∈ R, L(u, t) = E euXt is bounded and bounded away from 0 over [0,T ]. Set
Ψ(u, t) = log E euXt ∈ R
Let F : Lp T
(µ)→ R be a differentiable functional satisfying
(Lip)grad ≡ ∀α, β∈ D([0,T ],R), |F (α)− F (β)| ≤ [F ]Lipα− βLp T (µ).
and with a monotone gradient ∇F defined on [0,T ]× D([0,T ],R).Then the function
f (σ) = E (
∫ T
F (α) = ∫ T
0 α(s)µ(ds) it is Carr et al.’s result.
When µ = δT it is the introductory result.
B Proof (sketch of) Specify X = W and F (α) = ( ∫ T
0 α(t)dt )
2
0 Ssds
)) .
Ψt is non-decreasing for the pointwise order. Let Qt = St .P.
E ( Ψt(S t)St(Wt − σt)
Markov dynamics
Bermuda and American options
Continuous Gaussian processes
Functional co-monotony principle for càdlàg processes
Skorokhod co-mootony

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