[PPT] - Pricing American options using martingale bases Jrme Lelong PowerPoint Presentation

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases

Pricing American options using martingale bases

Jérôme Lelong

Grenoble Alpes University

ETH Zürich – May 19, 2016

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 1 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases

Outline

1

American Options

2

An optimization point of view

3

How to effectively solve the optimization problem

4

Numerical experiments

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 2 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases American Options

Framework (1)

◮ Consider a multi–dimensional financial market driven by a

d−dimensional Brownian motion B.

◮ The process (St)t≤T is the underlying asset with values in Rd′, d′ ≤ d. ◮ The discounted payoff process writes

Zt = e−

t

0 rsds φ(St)

t≤T.

Assume E

supt Z2

t

< ∞.

◮ Consider an American option. Its discounted price at time t of the

Bermudan option is given by Ut = esssupτ∈Tt E[Zτ|Ft] where Tt is the set of all F− stopping times with values in [t, T].

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 3 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases American Options

Framework (2)

◮ The Snell envelope process (Ut)0≤t≤T admits a Doob–Meyer

decomposition Ut = U0 + M⋆

t − A⋆ t

where M⋆ is a martingale and A⋆ a predictable increasing process both vanishing at zero and square integrable.

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 4 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases American Options

Dual price (1)

We know from [Rogers, 2002] that U0 = inf

M∈H1

E

sup

0≤t≤T

(Zt − Mt)

= E
sup

0≤t≤T

(Zt − M⋆

t )

◮ This problem admits more than a single solution.

◮ Some of the martingales M attaining the infimum are surely optimal

U0 = sup

0≤t≤T

(Zt − Mt) a.s.

◮ Let τ be an optimal stopping time, then

U0 = inf

M∈H1

E

sup

τ≤t≤T

(Zt − Mt)

= E
sup

τ≤t≤T

(Zt − M⋆

t )

.
J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 5 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases American Options

Dual price (2)

◮ From [Schoenmakers et al., 2013], any martingale satisfying

Var

sup

0≤t≤T

(Zt − Mt)

= 0

is surely optimal.

◮ Let τ be an optimal stopping time, then for any surely optimal

martingale M, (Mt∧τ)t = (M⋆

t∧τ)t.

See [Jamshidian, 2007].

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 6 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases American Options

Dual price (3)

With our square integrability assumption, we can rewrite the minimization problem as U0 = inf X ∈ L2(Ω, FT, P) s.t. E[X] = 0 E

sup

0≤t≤T

(Zt − E[X|Ft])

.

How to approximate L2(Ω, FT, P) by a finite dimensional vector space in which conditional expectations are tractable in a closed form?

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 7 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Wiener chaos expansion (d = 1)

Let Hi be the i − th Hermite polynomial defined by H0(x) = 1; Hi(x) = (−1)i ex2/2 di dxi (e−x2/2), for i ≥ 1.

◮ H′ i = Hi−1 with the convention H−1 = 0. ◮ If X, Y ∼ N(0, 1) and form a Gaussian vector,

E[Hi(X)Hj(Y)] = i! (E[XY])i 1{i=j}. For i ≥ 0, the L2 closure of the space Hi =

Hi

T ftdBt

: f ∈ L2([0, T])
corresponds to the Wiener chaos of order i.
J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 8 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Truncated Wiener chaos expansion (d = 1)

Take a regular grid 0 = t0 < t1 < · · · < tn with step h and consider fi(t) = 1{]ti−1,ti]}(t) √ h ; T fi(t)dBt = Bti − Bti−1 √ h = Gi ∼ N(0, 1). For F ∈ L2(Ω, FT), we introduce the truncated chaos expansion of order p Cp,n(F) =

α∈Ap,n

λα

i≥1

Hαi(Gi) where Ap,n = {α ∈ Nn : α1 ≤ p} with α1 =

i≥0 αi.

In the following we write, Cp,n(F) =

α∈Ap,n

λα Hα(G1, . . . , Gn) with Hα(x) =

i≥1 Hαi(xi).

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 9 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Key properties of the truncated Wiener chaos expansion (d = 1)

◮ Since the Hermite polynomials are orthogonal

λα = E

F

Hα(G1, · · · , Gn)

i≥1 αi!
.

◮ For k ≤ n,

E[Cp,n(F)|Ftk] =

α∈Ak

p,n

λα Hα(G1, . . . , Gn) with Ak

p,n = {α ∈ Nn : α1 ≤ p, αℓ = 0 ∀ℓ > k}.

“Computing E[·|Ftk]” ⇔ “Dropping all non Ftk− measurable terms”

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 10 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Extension to the multi–dimensional case (1)

Take hj

i(t) = 1{]ti−1,ti]}(t)

√ h ej, i = 1, . . . , n, j = 1, . . . , d where (e1, . . . , ed) denotes the canonical basis of Rd. The truncated Wiener chaos of order p ≥ 0 is given by   

d

j=1
Hαj(Gj

1, . . . , Gj n) : α ∈ (Nn)d, α1 ≤ p

   where α1 = n

i=1

d

j=1 αj i.

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 11 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Extension to the multi–dimensional case (2)

With the concise notation

H⊗d

α (G1, . . . , Gn) = d

j=1
Hαj(Gj

1, . . . , Gj n)

∀α ∈ (Nn)d. We introduce the truncated chaos expansion of order p of F ∈ L2(Ω, FT) Cp,n(F) =

α∈A⊗d

p,n

λα H⊗d

α (G1, . . . , Gn)

where A⊗d

p,n =

α ∈ (Nn)d : α1 ≤ p
. With an obvious abuse of notation,

we write, for λ ∈ RA⊗d

p,n ,

Cp,n(λ) =

α∈A⊗d

p,n

λα H⊗d

α (G1, . . . , Gn).

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 12 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Return to the American option price

We approximate the original problem inf X ∈ L2(Ω, FT, P) s.t. E[X] = 0 E

sup

0≤t≤T

(Zt − E[X|Ft])

by

inf λ ∈ RA⊗d

p,n

s.t. λ0 = 0 Vp,n(λ) (1) with Vp,n(λ) = E

max

0≤k≤n(Ztk − E[Cp,n(λ)|Ftk])

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 13 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Properties of the minimization problem (1)

Proposition 1

The minimization problem (1) has at least one solution.

◮ The function Vp,n is clearly convex (maximum of affine funtions). ◮ Not strongly convex but,

Vp,n(λ) ≥ E [(Cp,n(λ))−] ≥ 1 2 E [|Cp,n(λ)|] , E [|Cp,n(λ)|] = |λ| E [|Cp,n(λ/ |λ|)|] ≥ |λ| inf

µ∈RA⊗d

p,n ,|µ|=1

E [|Cp,n(µ)|].

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 14 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Properties of the minimization problem (2)

I(λ, Z, G) = {0 ≤ k ≤ n : the pathwise maximum is attained at time k} .

Proposition 2

Let p ≥ 1. Assume that ∀1 ≤ r ≤ k ≤ n, ∀F Ftk − measurable, F ∈ Cp−1,n, F = 0, ∃ 1 ≤ q ≤ d s.t. P

∀t ∈]tr−1, tr], Dq

t φ(Stk) + F = 0

φ(Stk) > 0
= 0.

Then, the function Vp,n is differentiable at all points λ ∈ RA⊗d

p,n with no zero

component and its gradient ∇Vp,n is given by ∇Vp,n(λ) = E

E
H⊗d(G1, . . . , Gn)
Fti
|i=I(λ,Z,G)
.
J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 15 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Properties of the minimization problem (3)

Vp,n(λ) = E

max

0≤k≤n(Ztk − E[Cp,n(λ)|Ftk])

◮ The maximum is pathwise sub–differentiable. From [Bertsekas, 1973],

Vp,n is sub–differentiable with sub–differential given by

E

 

i∈I(λ,Z,G)

βiE[ H⊗d(G1, . . . , Gn)|Fti]   : βi ≥ 0, βi FT − measurable s.t.

i∈I(λ,Z,G)

βi = 1

.
J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 16 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Properties of the minimization problem (4)

◮ Differentiability is ensured as soon as I(λ, Z, G) is a.s. reduced to a

unique element: purpose of the blue condition.

◮ Alternative approach by [Belomestny, 2013]: use smoothing techniques

instead (see [Nesterov, 2004]). General idea: Replace max

k

ak by p−1 log

k

exp(p ak)

.

◮ Let λ♯ be a solution, Vp,n(λ♯ p,n) = infλ Vp,n(λ). Then ∇Vp,n(λ♯ p,n) = 0.

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ETH Zürich – May 19, 2016 17 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Convergence to the true solution (1)

Proposition 3

The solution of the minimization problem (1), Vp,n(λ♯

p,n), converges to the

price of the American options when both p and n go to infinity and moreover 0 ≤ Vp,n(λ♯

p,n) − U0 ≤ 2 M⋆ T − Cp,n(M⋆ T)2 .

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 18 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Convergence to the true solution (2)

Vp,n(λ♯

p,n) = inf λ Vp,n(λ),

and Cp,n(M⋆

T) = Cp,n(λ⋆ p,n).

0 ≤ Vp,n(λ♯

p,n) − U0 ≤ Vp,n(λ⋆ p,n) − U0

= E

max

k (Ztk − E[Cp,n(λ⋆ p,n)|Ftk]) − max k (Ztk − M⋆ tk)

≤
E
max

k

E

M⋆

T − Cp,n(λ⋆ p,n)

|Ftk

2

≤ 2 M⋆

T − Cp,n(M⋆ T)2

where the last upper–bound ensues from Doob’s inequality.

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ETH Zürich – May 19, 2016 19 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases An optimization point of view

Convergence to the true solution (3)

◮ Non uniform bounds are larger!

0 ≤ Vp,n(λ♯

p,n) − U0 ≤ 2

M⋆

T − Cp,n(λ♯ p,n)

2

Cp,n(M∗

T) minimizes the L2 distance between M∗ T and Cp,n. ◮ Consider a Bermudan option with exercising dates t0, · · · , tn and

discounted payoff (Ztk)k adapted to the discrete time filtration generated by the Brownian increments only. Then, Vp,n(λ♯

p,n) converges to the

price of the Bermudan option when p only goes to infinity.

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 20 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases How to effectively solve the optimization problem

Practically solving the optimization problem (1)

We approximate the solution of Vp,n(λ♯

p,n) =

inf

λ∈A⊗d

p,n

Vp,n(λ) = inf

λ∈A⊗d

p,n

E

max

0≤k≤n(Ztk − E[Cp,n(λ)|Ftk])

by introducing the well–known Sample Average Approximation (see

[Rubinstein and Shapiro, 1993]) of Vp,n defined by Vm

p,n(λ) = 1

m

i=1

max

0≤k≤n

Z(i)

tk − E[C(i) p,n(λ)|Ftk]

.

Note that the conditional expectation boils down to truncating the chaos expansion and hence is tractable in a closed form.

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 21 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases How to effectively solve the optimization problem

Practically solving the optimization problem (2)

For large enough m, Vm

p,N is convex, a.s. differentiable and tends to infinity at

infinity. Then, there exits λm

p,n such that

Vm

p,n(λm p,n) =

inf

λ∈RA⊗d

p,n

Vm

p,n(λ).

Proposition 4

Vm

p,n(λm p,n) converges a.s. to Vp,n(λ♯ p,N) when m → ∞.

The distance from λm

p,n to the set of minimizers of Vp,n converges to zero as m

goes to infinity.

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 22 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases How to effectively solve the optimization problem

Practically solving the optimization problem (3)

Write Mk(λ) = E[Cp,n(λ)|Ftk] for 0 ≤ k ≤ n.

Proposition 5

Assume λ♯

p,n is unique. Then,

1 m

m

i=1
max

0≤k≤n Z(i) tk − M(i) k (λm p,n)

2 − Vm

p,n(λm p,n)2

is a convergent estimator of Var(maxk≤0≤n Ztk − Mk(λ♯

p,n)) and moreover, if

λm

p,n is bounded,

lim

m→∞ m Var

Vm

p,n(λm p,n)

= Var( max

k≤0≤n Ztk − Mk(λ♯ p,n)).

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 23 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases How to effectively solve the optimization problem

The algorithm: bespoke martingales

Define the first time the option goes in the money by τ0 = inf{k ≥ 0 : Ztk > 0} ∧ n. Consider martingales only starting once the option has been in the money Nk(λ) = Mk(λ) − Mk∧τ0(λ). In the dual price, max0≤k≤n can be shrunk to maxτ0≤k≤n. Using Doob’s stopping theorem, we have, for any fixed λ, E

max

τ0≤k≤n(Ztk − Mk(λ))

= E
max

τ0≤k≤n(Ztk − (Mk(λ) − Mτ0(λ)))

The martingales M(λ) or N(λ) lead to the same minimum value.

The set of martingales Nλ is far more efficient from a practical point of view.

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 24 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases How to effectively solve the optimization problem

The algorithm: a gradient descent with line search

Take αℓ = ˜ Vm

p,n(xℓ) − v♯

∇˜

Vm

p,n(xℓ)

2 , see [Polyak, 1987], but with the European price

instead of the American one for v♯. x0 ← 0, k ← 0, γ ← 1, d0 ← 0, v0 ← ∞ ; while True do Compute vk+1/2 ← ˜ Vm

p,n(xk − γαkdk) ;

if vk+1/2 < vk then xk+1 ← xk − γαkdk ; vk+1 ← vk+1/2 ; dk+1 ← ∇˜ Vm

p,n(xk+1) ;

if |vk+1−vk|

vk

≤ ε then return; else γ ← γ/2 ; end end

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 25 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases How to effectively solve the optimization problem

Some remarks on the algorithm

◮ Given the expression of Vm p,n, both the value function and its gradient

are computed at the same time without extra cost. Vp,n(λ) = E

max

τ0≤k≤n

Ztk − E[λ · H⊗d(G1, · · · , Gn)|Ftk]
,

= E[ZtI(λ,Z,G)] − λ · ∇˜ Vp,n(λ).

◮ Checking the admissibility of a step γ costs as much as updating xk. ◮ The algorithm is almost embarrassingly parallel:

◮ Few iterations of the gradient descent are required (≈ 10). ◮ Each iteration is fully parallel: each process treats its bunch of paths. ◮ No demanding centralized computations ◮ Very little communication: a few broadcasts only.

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ETH Zürich – May 19, 2016 26 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases How to effectively solve the optimization problem

Parallel implementation

In parallel Generate (G(1), Z(1)), . . . , (G(m), Z(m)) m x0 ← 0 ∈ RA⊗d

p,n ;

while True do Broadcast xℓ, dℓ, γ, αℓ; In parallel Compute maxτ0≤k≤n(Z(i)

tk − N(i) k (xℓ − γαℓdℓ));

Make a reduction of the above contributions to obtain ˜ Vm

p,n(xℓ+1/2) and

∇˜ Vm

p,n(xℓ+1/2);

vℓ+1/2 ← ˜ Vm

p,n(xℓ − γαℓdℓ) ;

if vℓ+1/2 < vℓ then xℓ+1 ← xℓ − γαℓdℓ ; vℓ+1 ← vℓ+1/2; dℓ+1 ← ∇˜ Vm

p,n(xℓ+1) ;

if |vℓ+1−vℓ|

vℓ

≤ ε then return; else γ ← γ/2 ; end end

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 27 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases Numerical experiments

Basket option in the BS model

p n S0 price Stdev time (sec.) reference price 2 3 100 2.27 0.029 0.17 2.17 3 3 100 2.23 0.025 0.9 2.17 2 3 110 0.56 0.014 0.07 0.55 3 3 110 0.53 0.012 0.048 0.55 2 6 100 2.62 0.021 0.91 2.43 3 6 100 2.42 0.021 14 2.43 2 6 110 0.61 0.012 0.33 0.61 3 6 110 0.55 0.008 10 0.61

TAB.: Prices for the put basket option with parameters T = 3, r = 0.05, K = 100, ρ = 0, σj = 0.2, δj = 0, d = 5, ωj = 1/d, m = 20, 000.

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ETH Zürich – May 19, 2016 28 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases Numerical experiments

Call option on the maximum of a basket

d p m S0 price Stdev time (sec.) reference price 2 2 20, 000 90 10.18 0.07 0.4 8.15 2 3 20, 000 90 8.5 0.05 4.1 8.15 2 2 20, 000 100 16.2 0.06 0.54 14.01 2 3 20, 000 100 14.4 0.06 5.6 14.01 5 2 20, 000 90 21.2 0.09 2 16.77 5 3 40, 000 90 16.3 0.05 210 16.77 5 2 20, 000 100 30.7 0.09 3.4 26.34 5 3 40, 000 100 26.0 0.05 207 26.34

TAB.: Prices for the call option on the maximum of d assets with parameters T = 3, r = 0.05, K = 100, ρ = 0, σj = 0.2, δj = 0.1, n = 9.

J. Lelong (Grenoble Alpes University)

ETH Zürich – May 19, 2016 29 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases Numerical experiments

Geometric basket option

d σj ρ p m price Stdev time(sec) 1−d price 2 0.2 2 5000 4.32 0.04 0.018 4.20 2 0.2 3 5000 4.15 0.04 1.3 4.20 10 0.3 0.1 1 5000 5.50 0.06 0.12 4.60 10 0.3 0.1 2 20000 4.55 0.02 17 4.60 40 0.3 0.1 1 10000 4.4 0.03 1.4 3.69 40 0.3 0.1 2 20000 3.61 0.02 170 3.69

TAB.: Prices for the geometric basket put option with parameters T = 1, r = 0.0488 (it corresponds to a 5% annual interest rate), K = 100, δj = 0, n = 9.

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ETH Zürich – May 19, 2016 30 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases Numerical experiments

Scalability of the parallel algorithm

The tests were run on a BullX DLC supercomputer containing 3204 cores. #processes time (sec.) efficiency 1 4365 1 2 2481 0.99 4 1362 0.90 16 282 0.84 32 272 0.75 64 87 0.78 128 52 0.73 256 34 0.69 512 10.7 0.59

TAB.: Scalability of the parallel algorithm on the 40−dimensional geometric put

ption described above with T = 1, r = 0.0488, K = 100, σj = 0.3, ρ = 0.1,

δj = 0, n = 9, p = 2, m = 200, 000.

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ETH Zürich – May 19, 2016 31 / 32

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American Options An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases Numerical experiments

Conclusion

◮ Purely optimization approach. No need of an optimal strategy. ◮ The problem is in large dimension but convex. ◮ Almost embarrassingly parallel and scales very well. ◮ Can deal with path dependent options

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ETH Zürich – May 19, 2016 32 / 32

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◮ Belomestny, D. (2013).

Solving optimal stopping problems via empirical dual optimization.

Ann. Appl. Probab., 23(5):1988–2019.

◮ Bertsekas, D. P. (1973).

Stochastic optimization problems with nondifferentiable cost functionals.

J. Optimization Theory Appl., 12:218–231.

◮ Jamshidian, F. (2007).

The duality of optimal exercise and domineering claims: a Doob-Meyer decomposition approach to the Snell envelope. Stochastics, 79(1-2):27–60.

◮ Nesterov, Y. (2004).

Smooth minimization of non-smooth functions. Mathematical Programming, 103(1):127–152.

◮ Polyak, B. T. (1987).

Introduction to optimization. Optimization Software.

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ETH Zürich – May 19, 2016 32 / 32

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◮ Rogers, L. C. G. (2002).

Monte Carlo valuation of American options.

Math. Finance, 12(3):271–286.

◮ Rubinstein, R. Y. and Shapiro, A. (1993).

Discrete event systems. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Ltd., Chichester. Sensitivity analysis and stochastic optimization by the score function method.

◮ Schoenmakers, J., Zhang, J., and Huang, J. (2013).

Optimal dual martingales, their analysis, and application to new algorithms for bermudan products. SIAM Journal on Financial Mathematics, 4(1):86–116.

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