polynomial bounds for european option pricingsma.epfl.ch/~anchpcommon/publications/polbound.pdf2...

Polynomial bounds for European option pricing∗

Francesco Statti †

AbstractWe study the problem of computing the price of European options under the as-

sumption that all the moments of the underlying asset price process are available. Oneof the techniques from the literature addressing this problem consists of deriving a se-quence of lower and upper bounds of the option price by solving optimization problems.However, in the works presented so far, the payoff function is always assumed to bepiecewise polynomial, limiting the approach. In this paper, we extend it further to con-sider more general payoff profiles. After establishing the theoretical convergence of ourbounds for some specific cases, we explain how to numerically solve the optimizationproblems, introducing a new algorithmic technique. The methodology is applied in thecontext of polynomial models, for which theoretical and numerical results are provided.

Key words Option pricing, polynomial bounds, semidefinite programming, cutting plane

1 IntroductionWe consider the problem of computing the price of European options in the setting wherethe moments of the underlying asset price process at maturity exist and are available. Thistranslates to computing a quantity of the form

E[f(X)], (1)

where f is a scalar function and X is a multidimensional random variable, for which weassume the availability of moments.

The core idea of the methodology presented in this paper is to use the moments of Xin order to set up two optimization problems whose solutions yield a lower and an upperbound of (1). In particular, for a fixed n ∈ N, an upper bound of (1) can be obtained byminimizing E[p(X)] over the set of all multivariate polynomials p of total degree at most nthat bound f from above, i.e.

infpE[p(X)] | p polynomial s.t. deg(p) ≤ n and p(x) ≥ f(x) ∀x ∈ E, (2)

where E is the state space of X. Similarly, a lower bound is derived by solving

suppE[p(X)] | p polynomial s.t. deg(p) ≤ n and p(x) ≤ f(x) ∀x ∈ E. (3)

Solving the problems (2) and (3) for an increasing sequence of polynomial degrees n yieldsa monotone sequence of upper and lower bounds. Moreover, writing the moments of X ina vector γ allows to write the objective function as the linear function γ>~p, where ~p is the∗The author would like to thank Damir Filipovic, Daniel Kressner, and Stefano Massei for helpful discus-

sions on this paper. The research leading to these results has received funding from the European ResearchCouncil under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agree-ment n. 307465-POLYTE.†Institute of Mathematics and Swiss Finance Institute, EPFL, Route Cantonale, 1015 Lausanne, Switzer-

land. Email: [email protected]

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coordinate vector of the polynomial p. In turn, the optimization problems (2) and (3) arelinear semi-infinite programming problems, in the sense that a linear objective function isminimized/maximized subject to an infinite number of constraints. Having the problems(2) and (3) at hand, two questions arise:

(a) What is the quality of the obtained bounds and what happens as n→∞?

(b) How do we solve the optimization problems (2) and (3) numerically?

Partial answers to these questions can be found in the existing literature, where (2) and(3) have been already studied, together with their associated dual problems. In particular,a very similar approach is considered in [6, 17, 18], in the setting of option pricing as well.There, the underlying assumption is that only a fixed number of moments of X is available,and the required bounds are computed by considering the same optimization problems asdescribed above. The numerical approach pursued in these works is based on rewriting (2)and (3) as semidefinite programming problems, whose numerical solutions can be computedvia standard algorithms. The dual version of the optimization problems (2) and (3) isconsidered in [12, 22], again in the context of option pricing. There, the methodology isextended further to price exotic options, such as Asian or barrier options. Similarly to oursetting, the case where all the moments of X are available is studied, leading to a convergencestudy of the bounds for n → ∞ for some explicit choices of the payoff function f . In [22],the optimization problem is again solved via a semidefinite programming approach, while in[12] a linear programming approach is used.

In the works mentioned above, however, the function f is always assumed to be piece-wise polynomial when considering the problem of pricing European options. This a severerestriction for the application of the methodology. In this paper, we extend the approach inorder to consider settings where f only needs to be upper bounded by a sequence of piece-wise polynomials. This generalization allows us to treat, for example, f(x) = (1− ex)+, thepayoff function of the European put in log-asset formulation. We prove the convergence ofthe bounds for the cases where X is a scalar random variable, under suitable assumptionson f and on the probability distribution of X. This will give us an answer to the question(a). Concerning question (b), we first propose to use again the semidefinite programmingapproach developed in [6, 17, 18]. Second, we introduce a new algorithm for the numericalsolution of (2) and (3). This approach, based on the cutting plane technique, is intuitiveand can be applied to a vast choice of functions f .

An important framework, where this methodology applies, are polynomial (jump-)diffusionmodels. Recently developed in [13, 14], this class of models has become a versatile tool infinancial applications during the last few years. They are used, for instance, for modelingcredit risk [1] and the term structure for dividends and interest rates [15], for developingstochastic volatility models [3], in portfolio theory [11] and in the insurance field [7]. There-fore, there is an interest in developing pricing methodologies for polynomial models andother methods have been introduced, see e.g. [2] for a pricing technique based on orthogonalpolynomial expansions. One of the main properties of polynomial (jump-)diffusions is thatall of their conditional moments are finite and given in closed form, making our methodologysuited for this setting. Moreover, it is worth mentioning that the log-asset price formulationof several widely used stochastic volatility models, as the Stein-Stein model [25], the Hull-White model [20] and the Heston model [19] are polynomial jump-diffusions, as explainedin [2]. In the log-asset setting, payoff functions usually become piecewise exponential, as forexample (1− ex)+ (put option). Our extension applies to these settings.

The rest of the paper is organized as follows. In Section 2 we formalize the optimizationproblems (2) and (3), and we establish the convergence of the bounds for the scalar case. InSection 3 we introduce the two numerical approaches to compute the numerical solutions of(2) and (3). In Section 4 we apply the methodology in the setting of polynomials models,showing also some numerical results for the European put option pricing problem. In Section

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5 we introduce a black-box algorithm for the computation of European option prices, basedon our methodology.

2 Polynomial bounds via optimizationFix a probability space (Ω,F ,Q) together with an E-valued random variable X, where E isa Borel subset of Rd. Let µ be the distribution of X and assume that its support is givenby E. Define the set Poln(E) of d-variate polynomials of total degree at most n (n ∈ N),that is,

Poln(E) := ∑

0≤|k|≤n

αkxk|x ∈ E,αk ∈ R,

where we use multi-index notation: k = (k1, . . . , kd) ∈ Nd0, |k| := k1 + · · · + kd and xk :=xk1

1 . . . xkd

d . The dimension of Poln(E) is given by N :=(n+dn

). Assume that all the moments

of X exist, i.e.E[Xk] <∞

for any multi-index k, and that they are available or can be easily computed. The goal isto find an upper and a lower bound for a quantity of the form

E[f(X)], (4)

where f : E → R is an arbitrary measurable function, which is integrable with respect to µ.The starting point consists of defining two optimization problems whose solutions rep-

resent the desired bounds for (4). More precisely, fix a value n ∈ N for the consideredpolynomial degree. Then, the solutions of the optimization problems

UBn(f) := infp∈Sn(f)

E[p(X)], Sn(f) := p ∈ Poln(E) so that p(x) ≥ f(x), ∀x ∈ E, (5)

LBn(f) := supp∈Cn(f)

E[p(X)], Cn(f) := p ∈ Poln(E) so that p(x) ≤ f(x), ∀x ∈ E. (6)

satisfy

LBn(f) ≤ E[f(X)] ≤ UBn(f).

In other words, solving (5) and (6) corresponds to finding the best upper and lower boundingpolynomials of degree at most n of the function f . Note that

LBn(f) = supp∈Cn(f)

E[p(X)] = − infp∈Cn(f)

E[−p(X)]

= − inf−p∈Sn(−f)

E[−p(X)] = − infp∈Sn(−f)

E[p(X)] = −UBn(−f).(7)

The first goal of this work is to study the convergence of the bounds UBn(f), LBn(f)for n→∞ under suitable conditions on f and µ. In particular, we aim to prove

limn→∞

UBn(f) = E[f(X)], monotonically from above, and (8)

limn→∞

LBn(f) = E[f(X)], monotonically from below. (9)

Due to the relation (7), under the requirement that both f and −f satisfy the neededassumptions, proving (8) is equivalent to proving (9). Therefore, from now on, we restrictour analysis to the minimization problem UBn(f).

The study of the convergence requires the formulation of the dual problem of (5), whichwe review in the next chapter.

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2.1 DualityIn this section we derive the dual problem of the optimization problem (5). For fixed chosenpolynomial degree n, consider a basis Hn := h1, . . . , hN of Poln(E) and

Hn(x) = (h1(x), · · · , hN (x)) ∈ RN .

Consider the vector γ ∈ RN of mixed moments of the distribution µ of X corresponding toH. More precisely, let

π : E → 1, . . . , Nbe an enumeration of the multi-index set E = k ∈ Nd0 | |k| ≤ n. Then, the entries of γare defined as

γπ(k) :=∫E

hπ(k)(x)dµ,

which allows us to rewrite and rename (5) as

Pn := inf~pγ>~p | ~p ∈ RN so that Hn(x)~p ≥ f(x), ∀x ∈ E. (10)

Now, consider an arbitrary finite Borel measure ν on E that satisfies the moment conditions

γπ(k) =∫E

hπ(k)(x)dν, for all k ∈ E . (11)

Then, for all vectors ~p ∈ RN contained in the constraint set of (10) we have∫E

f(x)dν ≤∫E

Hn(x)~pdν = γ>~p, (12)

where the inequality comes from the constraint conditions in (10) and the equality from(11).

Intuitively, the inequality (12) implies that finding a finite Borel measure ν that satisfiesthe moment conditions (11) allows us to find a lower bound for the solution value of (10).Following standard results in duality theory, see e.g. [23], we define the dual problem of Pnas finding the largest lower bound of γ>~p over the set of all finite Borel measures the satisfy(11), i.e. the dual problem of Pn is given by

Dn := supν∫E

f(x)dν | ν ∈ B(E) satisfying∫E

hπ(k)(x)dν = γπ(k), k ∈ E,

where B(E) denotes the set of all finite Borel measures on E.The inequality (12) already shows that weak duality between Pn and Dn holds, i.e.

Dn ≤ Pn whenever the two feasible sets are not empty. In order to find conditions for strongduality to hold, i.e. Pn = Dn, we write Pn and Dn as conic optimization problems, followingthe approach from [23], Chapter 1. Define the convex sets

P (E) := (~p, p0) ∈ RN+1 | Hn(x)~p+ p0f(x) ≥ 0, for all x ∈ E, (13)

C(E) := (γ, γ0) ∈ RN+1 | ∃ν ∈ B(E) s.t. γ0 =∫E

fdν and∫E

hπ(k)(x)dν = γπ(k), k ∈ E.

(14)

Then, Pn and Dn can be rewritten as

Pn = inf~pγ>~p | (~p,−1) ∈ P (E),

Dn = supγ0

γ0 | (γ, γ0) ∈ C(E)).

Finally, standard results of conic duality in convex optimization (see e.g. [9]) yield thefollowing theorem (see also Theorem 1.2 in [23]).

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Theorem 2.1. (Strong duality). If (γ, γ0) ∈ C(E) for some γ0 and there exists ~p ∈ RNsuch that (~p,−1) lies in the interior of P (E), then Pn = Dn and both problems have anoptimal solution, that is, the sup and the inf are attained whenever they are finite.

Proof. See proof of Theorem 1.2 in [23].

2.2 Convergence results for the case d = 1In this section we analyze the convergence of the bounds UBn(f) for n→∞, i.e. we prove(8). In particular, we give sufficient conditions on µ, E and f for which (8) holds. Werestrict the analysis to the one-dimensional case d = 1. As already discussed in Section 1,the convergence of UBn(f) and its dual has already been addressed for some cases in theliterature, see e.g. [22, 23]. In particular, in [22] the convergence of the dual problems isshown in the framework of European option pricing, where f is assumed to be the payofffunction of the European call option, i.e. f(x) = (x − K)+ for a strike price K, and µ isassumed to be moment-determinate, meaning that it is uniquely determined by its moments.

All of the convergence results presented so far in the literature assume, however, apolynomial or piecewise polynomial function f . In this work, we are interested in relaxingthis condition to consider f of a more general form. The need for a non-piecewise polynomialf arises, for example, when one models the log-asset price instead of the price, as alreadydiscussed in Section 1. As a consequence, pricing European options often boils down tocomputing the expectation of piecewise exponential payoff functions, as for example f(x) =(ek − ex)+, see Section 4.2. In this case, the convergence (8) is not guaranteed by existingresults.

We start by assuming, without loss of generality, that E = R. The following results canbe similarly obtained for any Borel subset E of R. For simplicity, we use the monomial basisfor the space Poln(R), i.e.

Hn(x) = (1, x, x2, · · · , xn).Note that the following results are, however, independent of the choice of the basis. Weassume that the considered f and the distribution µ of X satisfy the following conditions:

C1 There exists a sequence qmm∈N of piecewise polynomials of degree m that bound ffrom above, i.e.

f(x) ≤ qm(x) for all x ∈ R and for every m.

C2 The measure µ is moment-determinate.

C3 The measure µ and the sequence qmm∈N are defined such that the condition

limm→∞

E[f(X)− qm(X)] = 0

holds.We define the degree of a piecewise polynomial as the highest degree amongst the degrees ofthe different pieces of polynomials. For each bounding polynomial qm and for an arbitraryfixed r ∈ N we define the optimization problems

D2r,m := supν∫Rqm(x)dν | ν ∈ B(R) satisfying

∫Rxidν = γi, i = 0, · · · , 2r,

where γ is the vector of moments of µ, as defined in Section 2.1.We now show that the ad-hoc Conditions C1-C3 are sufficient to obtain the convergence

(8) of the bounds. We start by showing that D2r,m converges towards E[qm(X)] as r →∞and for every fixed m. For the particular case qm(x) := (x−K)+ the convergence is provenin [22, Theorem 5.2]. However, the arguments in the proof apply more generally to piecewisepolynomial functions qm. This yields the following lemma.

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Lemma 2.2. Under the Condition C2, for any fixed m, one has

limr→∞r≥m

D2r,m = E[qm(X)],

from above, monotonically.

Consider now the dual problem of D2r,m, defined as

P2r,m := inf~pγ>~p | ~p ∈ R2r+1 so that H2r(x)~p ≥ qm(x), ∀x ∈ R. (15)

Using the strong duality result reviewed in Section 2.1 we show that the sequence P2r,mconverges to E[qm(X)] as well.

Lemma 2.3. Under the Conditions C1 and C2 one has

limr→∞r≥m

P2r,m = E[qm(X)]. (16)

Proof. We want to show that the assumptions of Theorem 2.1 hold. The statement followsthen directly from the strong duality combined with Lemma 2.2.

Condition C1 implies, in particular, that f is polynomially bounded over R (i.e. thereexists a polynomial p satisfying p(x) ≥ f(x) for all x ∈ R). This implies the existenceof a ~p such that (~p,−1) lies in the interior of P (R) (see Definition (13)). Also, X is arandom variable with probability distribution µ and all finite moments, which implies that(γ, γ0) ∈ C(R) for γ0 :=

∫Efdµ (see Definition (14)). Hence, the conditions of Theorem 2.1

are satisfied and the strong duality holds.

Combining Lemma 2.3 with the Condition C3 yields the main convergence result.

Theorem 2.4. Assume that the Conditions C1, C2 and C3 hold. Then, there exists asequence an in N that satisfies

liman→∞

infp∈San (f)

E[p(X)− f(X)] = 0,

where San(f) := p ∈ Polan(R) so that p(x) ≥ f(x), ∀x ∈ R.

Proof. Fix an arbitrary ε > 0. Then, Condition C3 implies that there exists m such thatfor all m > m, one has

E[f(X)− qm(X)] < ε

2 . (17)

Additionally, (16) implies that, for any fixed value m, there exists r ∈ N such that for allr > r, we have

P2r,m − E[qm(X)] < ε

2 . (18)

Both relations (17) and (18) imply that there exist some finite values m, r ∈ N such that

P2r,m − E[f(X)] < ε.

Now, Condition C1 implies that the polynomial p?2r,m defined as the solution argument ofP2r,m is in the set S2r(f). This implies that

infp∈S2r(f)

E[p(X)− f(X)] ≤ E[p?r,m(X)− f(X)] < ε.

Since ε was arbitrary chosen, this last relation implies the statement of the theorem.

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In the rest of the chapter we focus on the special case where E is compact, in particularof the form [a, b] for some real values a < b. The convergence result of Theorem 2.4 holdsin this case as well and the Condition C2 is automatically satisfied, since all the measureswith compact support are moment-determinate, see e.g. [5]. Moreover, if we assume f to becontinuous, then the Conditions C1 and C3 directly hold, as well. Indeed, if f is continuousand E is compact, the Weierstrass approximation theorem implies that it is possible to definea sequence qmm∈N of upper bounding polynomials that satisfies the Condition C3. Even ifthe convergence is already guaranteed by Theorem 2.4, the following direct argument givesus more insights on the speed of convergence of the bounds UBn(f) in terms of the modulusof continuity of f . This approach does not need the dual formulation of UBn(f) and usesstandard results from approximation theory.

Lemma 2.5. Let f be a continuous function on E := [a, b] ⊆ R for some a < b, and letn ∈ N be a fixed polynomial degree. Then,

infp∈Sn(f)

(supx∈[a,b]

(p(x)− f(x)))≤ Cω(1/n),

for a constant C which is independent of f and n. Here, Sn(f) is defined as in (5) and ω(δ)is the modulus of continuity of f defined as

ω(δ) := supy,z∈[a,b],|y−z|<δ

|f(y)− f(z)|.

Proof. The proof of this lemma is based on Jackson’s theorem (see e.g. [16]) which statesthat for a continuous function f on [a, b], the sequence of best polynomial approximationsBn(f) satisfies

supx∈[a,b]

|Bn(f)(x)− f(x)| ≤ Cω(1/n), (19)

for a constant C which is independent of f and n. Since Bn(f) is a polynomial of degree nfor every n, (19) implies

infp∈Poln([a,b])

(supx∈[a,b]

|p(x)− f(x)|)≤ Cω(1/n).

Now, we want to obtain the estimate for the infimum taken over Sn(f). Define Bn(f) :=Bn(f)+c ω(1/n). Then, Bn(f) is a polynomial of degree n satisfying Bn ≥ f for all x ∈ [a, b].Hence Bn(f) ∈ Sn(f) and

supx∈[a,b]

(Bn(f)(x)− f(x)) ≤ 2Cω(1/n),

The statement follows by taking the infimum over all polynomials in Sn(f) and definingC := 2C.

We finally propose a convergence result for the case where E is compact and f is con-tinuous, based on the above direct approach.

Theorem 2.6. Let f be a continuous function on E := [a, b] ⊆ R for some a < b and let Xbe an E-valued random variable. Then, (8) holds.

Proof. From Lemma 2.5 we obtain

infp∈Sn

E[p(X)− f(X)] ≤ µ([a, b]) infp∈Sn

(supx∈[a,b]

(p(x)− f(x)))

≤ Cω(1/n).

Since f is continuous, its modulus of continuity goes to 0 for n→∞, implying the statementof the theorem.

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Remark 2.7. The statement of Theorem 2.6 could be proven more directly using Weierstrassapproximation theorem, which states that for any ε > 0 there exists a polynomial p so thatsupx∈[a,b] |f(x)− p(x)| ≤ ε. However, the considered proof allows us to have a rough idea ofthe speed of convergence of the bounds for n→∞ towards E[f(X)]. Indeed, the inequality

infp∈Sn

E[p(X)− f(X)] ≤ Cω(1/n),

implies that the convergence rate is given by the modulus of continuity of f . For example,in the case of the European put option with f(x) = (ek − ex)+ for some log-strike value k(see also Section 4.2), one can show that

ω(1/n) = O(1/n),

so that we expect the convergence rate to be 1/n.

3 Numerical algorithms for the optimization problemsIn Section 2, we have introduced the general optimization problems that allow us to findbounds for the quantity E[f(X)]. Solving them is, however, a difficult task. In this section wepresent two algorithmic techniques that will allow us to compute their solution numerically.

3.1 Semidefinite programming approachIn this subsection we explain the first strategy to numerically solve (5) (and (6)) for a fixedpolynomial degree n. The idea is to rewrite the optimization problem as a semidefiniteprogramming (SDP) problem, which can be numerically solved via standard algorithms.This approach has already been used in the literature, see e.g. [6, 23]. In the following, wereview the main steps of the methodology for an arbitrary dimension d. This approach isdeveloped for solving (5) in the cases where f is piecewise polynomial and the state spaceE can be partitioned in semialgebraic sets (see below for details). Even if we are interestedin solving problems for more general forms of f , this method can be applied to numericallysolve the problems (15), whose solutions tends to E[f(X)] (for d = 1) for n → ∞, seeTheorem 2.4. It is therefore worth reviewing it.

Consider the problem formulation (10). The first step consists of rewriting the constraintset such that both sides of the inequality are polynomials on suitable specified subsets of E.In particular, define a disjoint partition Ej , j = 1, · · · , k of E such that f can be writtenas

f(x) =k∑j=1

fj(x)1Ej, x ∈ E,

for some polynomials fj , j = 1, . . . , k. Then, (10) can be rewritten as

inf~pγ>~p | ~p ∈ RN so that Hn(x)~p− fj(x) ≥ 0, ∀x ∈ Ej , j = 1, · · · , k. (20)

In order to obtain the SDP formulation of (20), we aim to find an equivalent characterizationfor the non-negativity of the polynomials Hn(x)~p− fj(x) on the corresponding sets Ej . Inthe case that the sets Ej ’s are semialgebraic, this characterization can be given in terms ofsum of squares polynomials, whose definition and properties are reviewed in the following.Note that these results can be mainly found in [23].

Definition 3.1. A polynomial p ∈ Pol(Rd) is a sum of squares (in short s.o.s.) if it can bewritten as

p(x) =∑i∈I

pi(x)2, x ∈ Rd,

for some finite family of polynomials pi : i ∈ I, where I is an index set.

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Remark 3.2. Note that the degree of p must be even and the degree of the polynomials piis necessarily bounded by half of that of p.

An equivalent characterization for a polynomial on Rd to be a sum of squares is given inthe following lemma.

Lemma 3.3. Let Hn(x) be a basis vector of Poln(Rd). Then, a polynomial p ∈ Pol2n(Rd)is s.o.s. if and only if there exists a real symmetric positive semidefinite matrix Q ∈ RN×Nsuch that p(x) = Hn(x)QHn(x)>,∀x ∈ Rd.

Proof. The proof of Proposition 2.1 in [23] shows the statement for the monomial basisvector Bn(x) = (1, x1, · · · , xd, x2

1, x1x2, · · · , xn−1xn, xn1 , · · · , xnd ). For a general basis vector

Hn(x) the statement follows by an appropriate change of basis of the form Hn(x) = LBn(x)for some transformation matrix L ∈ RN×N .

Being s.o.s. is clearly a sufficient condition for a polynomial p ∈ Pol(Rd) to be non-negative on Rd. On the other side, the non-negativity property does not necessarily implythat p can be written as a sum of squares. More specifically, for an arbitrary d > 1, one canconstruct a non-negative polynomial which is not s.o.s., see Example 3.5. For the case d = 1,instead, being s.o.s. is equivalent to being non-negative, as stated in the next theorem.

Theorem 3.4. For any polynomial p ∈ Pol(R) the relation

p(x) ≥ 0,∀x ∈ R ⇐⇒ p(x) is s.o.s.

holds.

Proof. See proof of Theorem 2.5 in [23].

Example 3.5. Consider the polynomial

p(x1, x2, x3) = x21x

22(x2

1 + x22 − x2

3) + 6x63.

Then, p is non-negative on R3 but it cannot be written as a sum of squares.

Proof. We first show that p is non-negative on R3. It is clear that p is always positive inthe region x ∈ R3 | x2

1 + x22 ≥ x2

3. Let us consider all values of x such that x21 + x2

2 ≤ x23.

Then,

x21 + x2

2 ≤ x23 =⇒ x2

1 + x22 + x2

3 ≤ 2x23 =⇒ (x2

1 + x22 + x2

3)3 ≤ 8x63,

which implies8x6

3 ≥ 6x21x

22x

23.

Therefore,

p(x1, x2, x3) = x21x

22(x2

1 + x22 − x2

3) + 6x63 ≥ x4

1x22 + x2

1x42 + 14

3 x63 ≥ 0.

Hence, p is non-negative. We now show that p can note be written as a s.o.s. polynomial.Since all the monomials of p are of total degree 6, we attempt to write p in the form∑n

(Anx31+Bnx2

1x2+Cnx21x3+Dnx1x

22+Enx1x2x3+Fnx1x

23+Gnx3

2+Hnx22x3+Inx2x

23+Jnx3

3)2.

Now, since there is no x61 nor x6

2 term in p, we must have An = Gn = 0. A similar argumentapplied to the monomials x4

1x23, x

42x

23, x

21x

43, x

22x

43 yields Cn = Hn = Fn = In = 0. Hence

p(x1, x2, x3) =∑n

(Bnx21x2 +Dnx1x

22 + Enx1x2x3 + Jnx

33)2.

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The last equation implies that the coefficient of the monomial term x21x

22x

23 is given by∑

n

E2n ≥ 0,

which is a contradiction since the the coefficient of x21x

22x

23 in p is −1. Hence, p is not a s.o.s.

polynomial.

As already mentioned, a s.o.s. polynomial is non-negative on Rd . We are now interestedin finding sufficient conditions (in terms of s.o.s. polynomials) for a polynomial to be non-negative on an arbitrary semialgebraic set S ⊆ Rd of the form

S = x ∈ Rd | gj(x) ≥ 0, j = 1, · · · ,m, (21)

for some polynomials gj(x).

Lemma 3.6. Let S ⊆ Rd be a semialgebraic set of the form (21). Then, a polynomialp ∈ Pol(Rd) is non-negative on S if it can be written in the form

p(x) = h0(x) + h1(x)g1(x) + · · ·+ hm(x)gm(x), x ∈ Rd,

for some s.o.s. polynomials hj(x), j = 1, · · · ,m.

Proof. For simplicity, let us assume m = 1. A more general proof will follow directly. Wefirst impose the following inequality on p

p(x) ≥ h1(x)g1(x), x ∈ Rd,

for a s.o.s. polynomial h1. This inequality will ensure us that p will be non-negative on S.Next, we just use the fact that a s.o.s. polynomial is non-negative on the whole space Rd.Hence, we impose

p(x)− h1(x)g1(x) = h0(x), x ∈ Rd,

for a second s.o.s. polynomial h0. The statement follows.

For the case d = 1, the following two theorems give a characterization in terms of s.o.s.polynomials for p to be non-negative on intervals of the form [a, b], [a,∞) or (−∞, b] forsome a < b.

Theorem 3.7. Let p ∈ Poln(R). Let g(x) := (x − a)(b − x) for some real values a < b.Then p ≥ 0 on [a, b] if and only if

p(x) = f(x) + g(x)h(x),

for some s.o.s. polynomials f(x) and g(x), with both summands of degree less than n.

Proof. See proof of the Theorem 2.6 in [23].

Theorem 3.8. Let p ∈ Pol(R) be non-negative on [a,∞) for some real value a. Then

p(x) = f(x) + (x− a)h(x),

for two s.o.s. polynomials f(x) and h(x). Similarly, if p ∈ Pol(R) is non-negative on (−∞, b]for some real value b, then

p(x) = g(x) + (b− x)l(x),

for two s.o.s. polynomials g(x) and l(x). In both cases the degree of both summands isbounded by the degree of p(x).

Proof. See proof of the Theorem 2.7 in [23].

10

Assume now that each set Ej of the partition Ej , j = 1, · · · , k of E is a semialgebraicset of the form

Ej = x ∈ Rd | gji (x) ≥ 0, i = 1, · · · ,mj.

Moreover, assume that n is even, of the form n = 2r for some r ∈ N. Finally, by exploitingthe previous lemmas and theorems, we rewrite the optimization problem (20) as

inf~p∈RN γ>~p, such that

Hn(x)~p− f1(x) = Hr(x)Q0,1Hr(x)> + g11Hr(x)Q1,1Hr(x)> + · · ·+ gm

1

1 Hr(x)Qm1,1Hr(x)>,...

Hn(x)~p− fk(x) = Hr(x)Q0,kHr(x)> + g1kHr(x)Q1,kHr(x)> + · · ·+ gm

k

k Hr(x)Qmk,kHr(x)>,Qi,j 0, for all i, j.

(22)

This defines an SDP problem. Note that, depending on d and on the specific form of thesets Ej , the problem (22) can be either equivalent to (20), or a relaxation of it, since thenon-negativity conditions we impose for the polynomials Hn(x)~p− fj(x) are only sufficientin some cases, see Lemma 3.6.

In order to numerically solve (22), one first needs to rewrite the equality constraints bygetting rid of the x variable. This is done by comparing the coefficients of all basis elementsstored in the basis vectors Hr(x) and Hn(x). In [6] this operation is explicitly performed forsome specific forms of f , for instance f(x) = (x−K)+ (payoff of the European call option).The solution of (22) can then be numerically computed by standard SDP solvers.

To conclude this part of the first numerical approach, we would like to comment furtheron the s.o.s. conditions that we impose for replacing the non-negativity conditions in (20).

Remark 3.9. How do the the bounds change if we replace the non-negativity conditions bys.o.s. conditions? Theorems 3.7 and 3.8 imply that non-negativity and s.o.s. conditions areequivalent in the case d = 1, for Ej of the form [a, b], [a,∞) or (−∞, b]. For the multivariatecase d > 1, Theorem 2.4 in [23] states that the space of s.o.s. polynomials is dense in thespace of non-negative multivariate polynomials. This implies that, even if the form (22)is a relaxation of (10) due to the fact that we impose only some sufficient conditions fornon-negativity, we still expect to obtain sharp bounds for E[f(X)] when solving (22).

3.2 Cutting plane algorithmThe second way we propose to numerically solve (5) and (6) (for a fixed n) is based onthe cutting plane (CP) technique. This algorithm is more direct and intuitive, and doesnot require us to rewrite the constraint set of the optimization problem. A description of ageneral cutting plane algorithm can be found, for example, in [8]. Note that an algorithmbased on the cutting plane strategy has been used in the context of European option pricingin [17]. There, however, the strategy is applied to an SDP formulation of (5) for d = 1 anda specific choice of f . Here, we do not use any SDP formulation. Instead, we design thealgorithm directly for solving the problem (10). In principle, this technique can be appliedfor any choice of f and d, making it very tractable and suitable for our extension.

The algorithm is iterative and at iteration l we perform the following steps.

1. Define a finite discrete subset El = xl,1, · · · ,xl,m of E and impose the inequalityconstraint in (10) only in El. We obtain a linear program (LP) of the form

min~pγ>~p | ~p ∈ RN so that H~p ≥ f, (23)

where H := (Hn(xl,1)| · · · |Hn(xl,m))> ∈ Rm×N contains the basis vector Hn(x) eval-uated in all points of El, and f = (f(xl,1), · · · , f(xl,m))>.

11

2. Solve the LP problem (23) using standard techniques. Denote by p?l the resultingoptimal coefficient vector.

3. Find the point xv ∈ E defined as xv = argminx∈E(p?l (x)− f(x)) where the inequalityconstraint is violated the most.

4. If p?l (xv) − f(xv) < 0, then insert xv in El, defining a new finite discrete subsetEl+1 := El ∪ xv and restart from Step 1 with El+1. If p?l (xv) − f(xv) ≥ 0 stop theiteration and return γ>p?l .

This algorithm is in principle very intuitive and easy to implement. However, some stepsrequire attention. For example, the existence of a solution of (23) or of the minimizationproblem argminx∈E(p?l (x) − f(x)) is not guaranteed a priori. These points need to beaddressed for the specific type of application. In Section 4.2 we analyze the algorithm forthe example of pricing European put options.

4 Application: European option pricing in polynomialmodels

We now apply the methodology developed in the previous sections to a concrete example. Inparticular, we consider the problem of pricing European options in a specific type of models:polynomial models. We recall that evaluating at initial time t = 0 the price of a Europeanoption expiring at time T > 0 consists of computing an expression of the form

e−rTE[f(XT )], (24)

where (Xt)0≤t≤T is a d-dimensional stochastic process modelling the price of some financialassets of interest over the time interval [0, T ], f : Rd → R is the payoff function, and rrepresents a fixed deterministic interest rate.

4.1 Polynomial modelsWe provide a short summary of polynomial models and refer to [13] (for polynomial diffu-sions) and [14] (for polynomial jump-diffusions) for the mathematical foundations.

Fix a filtered probability space (Ω,F ,Ft,Q), where Q denotes a risk neutral pricingmeasure. In this framework, we model the price of the financial assets of interest via apolynomial (jump-)diffusion (Xt)0≤t≤T defined on a time horizon [0, T ] for T > 0 and takingvalues in a state space E ⊆ Rd. One of the properties of polynomial (jump-)diffusions isthat the infinitesimal generator G of (Xt)0≤t≤T maps the space Poln(E) to itself, i.e.

GPoln(E) ⊆ Poln(E). (25)

As a consequence, its conditional moments are given in closed form and can be computed viathe so-called moment formula. This formula is the property we use for applying our methodsince it will allow us to compute the vector γ of mixed moments as defined in Section 2.1,on which our methodology is based. We now explain how to derive it. Property (25) allowsus to define the matrix representation of G with respect to the chosen polynomial basis Hnand restricted to Poln(E), which we denote by Gn, satisfying

Gp(x) = Hn(x)Gn~p,

for any p ∈ Poln(E) with coordinate vector ~p ∈ RN with respect to Hn. The momentformula, introduced in [13] for polynomial diffusions (Theorem 3.1) and in [14] for polynomialjump-diffusions (Theorem 2.4), is then given by

E[p(XT )|Ft] = Hn(Xt)eGn(T−t)~p. (26)

12

As a consequence, in the case of a deterministic initial value X0 ∈ Rd, formula (26) impliesthat all the mixed moments of XT (γ from Section 2.1) exist and can be computed as

γ> = Hn(X0)eGnT . (27)

Therefore, the setting of polynomial models represents a perfect framework for applying themethodology presented in the previous sections, in oder to compute quantities of the form(24), which now represent prices of European options.

In this section, we apply our approach to price European options for two specific models,the Black-Scholes model and the Jacobi stochastic volatility model, developed in [3]. Inthese cases, we model the log-asset price Xt of a single asset, meaning that the asset priceSt is of the form St = eXt . More specifically, we use the following polynomial diffusions:

1. One dimensional Black-Scholes model: The log-asset price (Xt)0≤t≤T is given by

dXt = (r − σ2

2 )dt+ σdWt,

where σ is the volatility, r ≥ 0 the risk-free interest rate and Wt is a one-dimensionalBrownian motion. The state space is E = R.

2. Jacobi stochastic volatility model: The log-asset price (Xt)0≤t≤T and the squaredvolatility process (Vt)0≤t≤T are given by

dVt = κ(θ − Vt)dt+ σ√Q(Vt)dW1t,

dXt = (r − Vt/2)dt + ρ√Q(Vt)dW1t +

√Vt − ρ2Q(Vt)dW2t,

whereQ(v) = (v − vmin)(vmax − v)

(√vmax −√vmin)2 ,

for some 0 ≤ vmin < vmax. Here, W1t and W2t are independent standard Brownianmotions and the model parameters satisfy the conditions κ ≥ 0, θ ∈ [vmin, vmax], σ > 0,r ≥ 0, ρ ∈ [−1, 1]. The state space is E = [vmin, vmax]× R.

It is shown in [3] that the process (Xt, Vt)0≤t≤T in Model 2 is a polynomial diffusion.Moreover, Lemma 2.2 in [13] allows us to show that Model 1 is polynomial as well. In orderto compute γ as in (27), we need to construct the matrix Gn with respect to the chosenpolynomial basis Hn. For Model 1, Gn can be easily constructed following the general uppertridiagonal structure for one-dimensional polynomial diffusions. More precisely, for a scalarpolynomial diffusion of the form dXt = (b+ βXt) dt+

√a+ αXt +AX2

t dWt the matrix Gnwith respect to the monomial basis 1, x, · · · , xn is given by

Gn =

0 b 2a2 0 · · · 0

0 β 2(b+ α

2)

3 · 2a2 0...

0 0 2(β + A

2)

3(b+ 2α2

) . . . 0

0 0 0 3(β + 2A2

) . . . n(n− 1)a2... 0

. . . n(b+ (n− 1)α2

)0 . . . 0 n

(β + (n− 1)A2

)

∈ R(n+1)×(n+1).

For Model 2, Gn is block triangular and it can be constructed, for example, as explained in[21] for the two-dimensional monomial basis 1, x, v, x2, xv, v2, · · · , xn, xn−1v, · · · , vn. For

13

instance, for n = 2, G2 is explicitly given by

G2 =

0 r κθ 0 −ρσvmaxvminS −σ

2vmaxvminS

0 0 2r κθ 0− 1

2 −κ 1 r + ρσ(vmax+vmin)S 2κθ + σ2(vmax+vmin)

S

0 0 0−1 −κ 00 − 1

2 −ρσS −2κ− σ2

S

,

where S := (√vmax −√vmin)2.

4.2 European put option - Theoretical settingIn the framework of the two polynomial models of Section 4.1, we apply the techniquedeveloped above to the specific case of the European put option. In particular, consider thepayoff function f(x) = (ek− ex)+ for a log-strike value k. The goal is to compute, or better,to find polynomial bounds for the quantity

e−rTE[f(XT )].

We first discuss the convergence of the polynomial bounds in Section 4.2.1. Then, in Section4.2.2 we explain how to set up the optimization routines. Finally, we show some numericalresults in Section 4.3. Again, we restrict our analysis to the upper bounds (5).

4.2.1 Convergence

Showing the convergence (8) of the upper bounds boils down for this particular case toshowing

limn→∞

infp∈Sn(f)

E[p(X)− (ek − eX)+] = 0, (28)

where Sn(f) := p ∈ Poln(R) so that p(x) ≥ (ek − ex)+, ∀x ∈ R. Note that we omit thediscounting factor e−rT and we write X instead of XT , in line with the notation of Section 2.

Without loss of generality, we assume k = 0. The goal is to apply the Theorem 2.4.Hence, we first aim to construct a sequence of piecewise polynomials qmm∈N that satisfythe Condition C1 in Section 2.2. First, we consider the Taylor series T2m−1(x) of order2m− 1 around 0 of the function 1− ex, which are explicitly given by

T2m−1(x) = −2m−1∑k=1

xk

k! . (29)

Then, we define q2m−1 for m ∈ N (we consider only odd polynomial degrees) as the positivepart of T2m−1(x), i.e. qm := T2m−1(x)+. The following lemma summarizes some propertiesof (29) and, in particular, shows that the Condition C1 is satisfied for our choice q2m−1m∈N.

Lemma 4.1. Let T2m−1(x) be defined as in (29) for some m ∈ N. Then,

1. T2m−1(x) ≥ 1− ex, for all x ∈ R,

2. T2m−1(x) has only one real zero, which is given by x = 0.

In particular, it follows that T2m−1(x)+ ≥ (1− ex)+ for all x ∈ R and every m.

Proof. 1. Expanding the exponential in 1− ex gives us the expression

T2m−1(x)− (1− ex) =∞∑

k=2m

xk

k! ,

14

which clearly implies T2m−1(x) ≥ 1− ex for all x ≥ 0.To get the inequality on the negative axis, consider again the difference

s2m−1(x) := T2m−1(x)− (1− ex) = ex − 1 + T2m−1(x).

A simple computation shows that the l-th derivative of s2m−1(x) is given by s(l)2m−1(x) =

ex − 1 + T2m−1−l(x) for 0 < l < 2m − 1, and s(2m−1)2m−1 (x) = ex − 1 which implies

s(l)2m−1(0) = 0 for 0 < l ≤ 2m − 1. Moreover, s(2m)

2m−1(x) = ex. We now show thats2m−1(x) has no real negative zeros, arguing by contradiction. Assume that s2m−1(x)has a real negative zero in some point x1 < 0. Then, since s2m−1(0) = 0, Rolle’s the-orem implies that there exists a point y1 ∈ (x1, 0) where s(1)

2m−1(y1) = 0. Now, sinces

(1)2m−1(0) = 0, we can again apply Rolle’s theorem to find a point y2 ∈ (y1, 0) so thats

(2)2m−1(y2) = 0. Applying inductively the same argument, we get a point y2m < 0 sat-

isfying s(2m)2m−1(y2m) = 0, which is clearly a contradiction since s(2m)

2m−1(x) = ex. Hence,T2m−1(x) doesn’t cross 1 − ex on the negative real axis. Moreover, since the leadingcoefficient of T2m−1(x) is negative, one has

limx→−∞

sm(x) =∞.

It follows that s2m−1(x) > 0, and hence, T2m−1(x) > 1− ex for x < 0.

2. From equation (29) one can easily see that 0 is a zero of T2m−1(x). Since all thecoefficients of T2m−1(x) are negative, the polynomial can not have strictly positivezeros. Finally, statement 1 tells us that T2m−1(x) is larger than 1 − ex, which isstrictly positive on the negative real axis. This implies that T2m−1(x) cannot havenegative real zeros. Therefore, 0 is the only real zero of T2m−1(x).

We now address the Condition C3. Rather then showing that this condition holds forthe chosen polynomial models, we first give some sufficient condition on the distribution µof X so that it holds.

Lemma 4.2. Let X be a R-valued random variable whose distribution µ satisfies

liml→∞l∈N

∫ 0

−∞

x2l

(2l)!dµ = 0. (30)

Then,limm→∞m∈N

E[T2m−1(X)+ − (1− eX)+] = 0.

Proof. Property 2 of Lemma 4.1 implies∣∣E[T2m−1(X)+ − (1− eX)+]∣∣ =

∣∣E[(T2m−1(X)− (1− eX))Ix≤0]∣∣ .

Then, since all derivatives of 1− ex are upper bounded by 1 and lower bounded by 0 on thenegative real axis, the Lagrange form of the Taylor remainder T2m−1(x)− (1− ex) implies

∣∣E[(T2m−1(X)− (1− eX))Ix≤0]∣∣ ≤ ∣∣∣∣∫ 0

−∞

x2m

(2m)!dµ∣∣∣∣ =

∫ 0

−∞

x2m

(2m)!dµ,

which goes to 0 for m→∞ thanks to the assumption (30).

Now that we have found conditions on µ such that the Condition C3 is satisfied, wecan give explicit sufficient conditions so that the convergence (28) holds. The followingconvergence result comes as a corollary of Theorem 2.4.

15

Corollary 4.3. Let X be an R-valued random variable and assume that all of its momentsare finite. Assume that the distribution µ of X is moment-determinate and satisfies (30).Then, there exists a sequence an in N that satisfies

liman→∞

infp∈San

E[p(X)− (1− eX)+] = 0,

where San:= p ∈ Polan

(R) so that p(x) ≥ (1− ex)+, ∀x ∈ R.

We further address the Conditions C2 and C3 in order to relate them to the choice ofthe asset price models, in particular to the ones chosen in Section 4.1. In the case of theBlack-Scholes model (Model 1), the log-asset price Xt is normally distributed for any time tand it can be shown that the condition (30) is satisfied. Indeed, for an arbitrary even l = 2k(for some positive integer k) and for any a > 0 one has

1l!

∫ ∞0

xle−ax2dx = 1

l!

∫ 0

−∞xle−ax

2dx = (2k − 1)!!

2k+1ak(2k)!

√π

a= 1

2k+1ak(2k)!!

√π

a,

which clearly converges to 0 as k (or l) tends to infinity. Moreover, as mentioned in [22],the normal distribution is moment-determinate. This implies that the Conditions C1-C3 aresatisfied in the case of the Black-Scholes model. Hence, the convergence (28) is guaranteed.

In the following, we give a further sufficient condition on X such that (30) holds.

Lemma 4.4. Let X be a R-valued random variable satisfying

E[e|X|] <∞. (31)

Then, condition (30) is satisfied.

Proof. Property (31) together with the dominated convergence theorem allows us to write

∞∑k=0

E[|X|k]k! = E[e|X|] <∞,

which in turn impliesE[|X|k]k! → 0, as k →∞,

and consequentlyE[|X|2l]

(2l)! → 0, as l→∞. (32)

Finally, the inequalities

0 ≤∫ 0

−∞

x2l

(2l)!dµ ≤E[|X|2l]

(2l)!

together with (32) imply the statement of the lemma.

It turns out that if X satisfies (31), then its distribution µ is also moment-determinate,satisfying the Condition C2 as well. Indeed, for any measure µ on R, the so-called Cramercondition (see e.g. [26]) states that if

E[ec|X|] <∞ for some c > 0,

then µ is moment determined. Therefore, if X satisfies (31), then (28) is guaranteed. Notethat for the case when XT is defined as in the Jacobi model, Theorem 3.1 in [3] can be usedto show that XT satisfies (31), depending on the model parameters. Hence, for both chosenmodels the convergence (28) can be shown.

16

4.2.2 Formulation of the optimization problems

Since the payoff function f is not a piecewise polynomial in this setting, we can not directlyapply the SDP method developed in Section 3.1. Instead, we apply it to T2m−1(x)+. Theconvergence result of the previous subsection guarantees that if we solve the correspondingSDP for T2m−1(x)+ and we let m and n going to infinity, the bounds will still convergetowards E[f(X)].

Fix a polynomial T2m−1(x)+ for some m, and an even natural number n = 2r for thefixed polynomial degree such that n > (2m− 1). Then, consider the optimization problem

inf~pγ>~p | ~p ∈ R2r+1 so that Hn(x)~p ≥ T2m−1(x)+, ∀x ∈ R, (33)

where γ is computed using the moment formula (27) for polynomial models. In particular,note that for the Black-Scholes model, γ is directly given by γ = Hn(X0)eGnT . For theJacobi model, however, the latter formula returns the vector of mixed moments in bothvariable x and v. We need only the moments in the x variables. Therefore, before defin-ing our optimization problem we define γ containing only the entries of Hn(X0, V0)eGnT

corresponding to the moments in x.In order to apply the SDP technique we rewrite the optimization problem as explained

in Section 3.1. In particular we derive an SDP of the form (22) which is equivalent to (33).Consider the equivalence

p(x) ≥ (T2m−1(x))+, ∀x ∈ R ⇐⇒ p(x) ≥ T2m−1(x), ∀x ∈ R and p(x) ≥ 0, ∀x ∈ R.

Then, according to the Theorem 3.4 and the Lemma 3.3 we can rewrite the non-negativityconditions and we get an SDP formulation given by

inf~p∈RN γ>~p, such that

Hn(x)~p = Hr(x)Q0,1Hr(x)>Hn(x)~p− T2m−1(x) = Hr(x)Q0,2Hr(x)>,Qi,j 0, for all i, j.

This is the problem that it is solved in practice, to obtain the required upper bounds forthe European put options in the setting of polynomial models.

For the cutting plane approach explained in Section 3.2, there is no need to approximatethe payoff function because the method does not require a piecewise polynomial payofffunction. Hence, we directly solve the problem

UBCPn (f) := inf~pγ>~p | ~p ∈ R2r+1 so that Hn(x)~p ≥ (ek − ex)+, ∀x ∈ R. (34)

However, it is useful to show that all the steps of the algorithm are well defined, undercertain conditions. This will allow the CP routine to output the optimal solution when thestopping criterion is satisfied. This particular example of European put option allows us toexploit the structure of the payoff function to make the CP routine computationally moreefficient. In particular, we modify the general CP algorithm of Section 3.2 by taking care ofthe following points.

• In the LP problem (23) we include the linear condition

γ>~p ≥ 0 (35)

in the constraint set. Hence, at each iteration l we solve

min~pγ>~p | ~p ∈ RN so that H~p ≥ f and γ>~p ≥ 0, (36)

17

Algorithm 1 CP routine for upper bounding European put option pricesInput: Model and payoff parameters, tolerance tol, maximal number of iterations maxiterOutput: Upper bound (34) of the option price

1: Construct Gn and γ according to (27)2: Define a random discrete set E0 of E3: Build H and f as defined in (23)4: l = 15: while l ≤ maxiter do6: Solve LP (36) and get p?l (using linprog.m)7: Solve (37) and get xv1 (using fmincon.m)8: Solve (38) and get xv2 (using roots.m)9: if p?l (xv1)− f(xv1) ≥ tol and p?l (xv2)− f(xv2) ≥ tol then

10: Break11: end if12: El = El ∪ xv1, xv213: Update H and f14: l = l + 115: end while16: UBCPn (f) = γ>p?l

where H and f are defined as in (23). This will make the LP problem well defined, aswe see below in Lemma 4.5. Moreover, note that this condition makes sense from theeconomical point of view. Indeed, (35) implies that the obtain upper bounding priceis non-negative.

• The minimization problem argminx∈E(p?l (x) − f(x)) is often seen, as mentioned in[28], as the bottleneck of the CP routine since it might be too expensive. For ourcase, this can be efficiently performed by splitting the problem on the two regionsE1 := x ∈ R | x ≤ k and E2 := x ∈ R | x ≥ k. More precisely we solve

argminx≤k (p?l (x)− (ek − ex)), (37)argminx≥k p?l (x), (38)

and we add both resulting points to the discrete set El. Problem (37) is well definedwhenever the leading coefficient of p?l is positive and n is even, as seen in Remark 4.6,and can be efficiently computed using, for example, a standard Newton method sincethe first derivative of the objective function is easily computable. Problem (38) canbe efficiently solved since the objective function is a polynomial.

The final complete CP algorithm is summarized in Algorithm 1. Note that the value tol inthe algorithm describes a tolerance value that controls the stopping criterion. In a standardCP algorithm this value is set to tol = 0. However, in order to make the computation moreefficient one can set, for example, tol = −10−5, allowing the constraints to be satisfied upto the tolerance tol. Also, in Algorithm 1 we have indicated the Matlab functions we canuse for solving the involved optimization problems.

In the following we prove the needed results to show that the CP routine is well definedand it produces an optimal solution of (34) whenever it stops. Note that we assume thatthe obtained bounding polynomial p?l (x) has a positive leading coefficient at every step l ofthe CP routine.

Lemma 4.5. The optimization problem (36) is well defined.

Proof. The goal is to show that the function γ>~p has a global minimum on the constraint

18

set A := ~p ∈ RN so that H~p ≥ f and γ>~p ≥ 0. We distinguish two different cases toprove the statement.

If A ∩ ~p ∈ RN so that γ>~p = 0 6= ∅, then the global minimum is given by 0.If A ∩ ~p ∈ RN so that γ>~p = 0 = ∅, then

lim||~p||→∞,~p∈A

γ>~p = +∞.

Hence, the objective function is continuous, coercive and bounded from below on the closedset A. The existence of a global minimum follows from standard results on coercive functions,see e.g. Theorem 2.32 in [4].

Remark 4.6. In each step of the CP routine we have to solve the minimization problem

argminx≤k p(x)− (ek − ex).

It is easy to show that the function p(x) − (ek − ex) restricted to (−∞, k] has a globalminimum, whenever p is of even degree and the leading coefficient is positive. Therefore,the above minimization problem is well defined.

Lemma 4.7. The sequence γ>p?i |i = 1, · · · of solution values of the CP steps is boundedand monotone increasing towards the solution value of (34). Moreover, suppose that Algo-rithm 1 stops after l iterations. Then, γ>p?l is the solution value of the original problem(34) up to a tolerance tol, i.e.

|UBCPn (f)− γ>p?l | ≤ tol. (39)

Proof. Let F be the feasible set of (34) and let Fl the feasible set of the LP problem (36).Moreover, let p? be the optimal solution of (34). Then, since we add a new linear constraintstep by step in the CP routine, we have

F ⊆ · · · ⊆ Fl ⊆ · · · ⊆ F1

which implies that γ>p?i |i = 1, · · · is monotone increasing and upper bounded by γ>p?.Using the tolerance value tol makes us solving the perturbed problem

inf~pγ>~p+ tol | ~p ∈ R2r+1 so that Hn(x)~p ≥ (ek − ex)+, ∀x ∈ R

This observation directly implies inequality (39).

Remark 4.8. Since the payoff function is piecewise exponential, we have to use, for example,a Newton type algorithm (as in fmincon.m) to perform Step 7 of Algorithm 1. Unfortunately,the algorithm can return the argument value of a local minimum, depending on the startingpoint. However, the numerical results presented in the Section 4.3 show that this does notaffect the final results for our application. A more practical way to avoid this problem isto use T2m−1(x)+ instead of (ek − ex)+, as in the SDP approach. In this case, Step 7 ofAlgorithm 1 could be also performed using a rootfinder (for instance roots.m) ensuring theargument of the global minimum on the region x ≤ k.

4.3 European put option - Numerical experimentsIn this section we finally show some numerical results for the computation of European putoption prices in the Black-Scholes model and in the Jacobi model. All algorithms have beenimplemented in Matlab and run on a standard laptop (Intel Core i7, 2 cores, 256kB/4MBL2/L3 cache). We used the toolbox YALMIP [24] together with SDPT3 [27] to model andsolve the SDP problems, while the LP problems and the minimization problems arising inthe cutting plane algorithm have been solved via Matlab’s built-in functions linprog.m

19

(for LP), fmincon.m (for minimizing non-polynomial functions) and roots.m (for findingthe critical points of polynomials and, consequently, their minima).

For both models we set the payoff parameters to

x0 = 0, T = 1, k = −0.1, 0, 0.1,

so that we consider out-of-the-money, at-the-money and in-the-money put options. For themodel parameters, we set

σ = 0.2, r = 0.01

for the Black-Scholes model, while for the Jacobi model we consider the model parameters

v0 = 0.04, σ = 0.15, κ = 0.5, θ = 0.04, ρ = −0.5, vmin = 10−4, vmax = 0.08, r = 0.

For the CP algorithm (Algorithm 1) the tolerance value is set to tol = −10−5 and theinitial discrete set E0 contains 50 uniformly distributed random points in [−4, 4]. For theSDP approach we solve the optimization problems for the sequence of upper boundingpolynomials defined in Section 4.2, i.e. Tn−1(x)+, where n is the considered polynomialdegree in the optimization problem. The vector of moments γ is computed according tothe moment formula (27). In the Black-Scholes model we compute the reference price usingthe Black-Scholes formula, while for the Jacobi model the reference price is computed usingthe polynomial expansion method developed in [3] with truncation value N = 50, and viaput-call parity. As done in the rest of the analysis, also in the numerical experiments weonly consider upper bounds. The quality of the lower bounds is expected to be similar, inline with the numerical results obtained in, for example, [6, 22].

In Table 1 we show the obtained upper bounds for the option prices for different valuesof n, together with the corresponding absolute errors computed with respect to the referenceprice. We show the results for both the SDP and the CP approach, in the Black-Scholesmodel. Similarly in Table 2 for the Jacobi model. In Figure 1 we show the obtained upperbounding polynomials of degree n = 20 in the Jacobi model, while in Figure 2 we show thebounding polynomials obtained in the Black-Scholes model for n = 20, multiplied by theprobability density function of XT , which is explicitly known in this model. This will giveus more insight on the quality of the bounds with respect to the correct integrating measure.Finally, in Figures 3 and 4 we show the obtained upper bounding implied volatilities (Black-Scholes implied volatility) for the Black-Scholes model and the Jacobi model, respectively.

We can observe that both methods yield tight upper bounds for both models, in linewith the numerical results obtained also in [6, 22]. In particular, the CP algorithm producesbetter upper bounds for small polynomial degrees (n = 2, · · · , 8), while for higher degreesthe SDP approach becomes more accurate. This is due to the fact that the SDP approachrequires the polynomial approximation of the payoff function, which is poor for n small,while the CP algorithm is directly applied to the piecewise exponential payoff function.For n large, however, the SDP approach is able to better capture the shape of the payofffunction, as shown in Figure 1 and in Figure 2.

5 A black box algorithm for European option pricingIn real-world applications, one usually needs option pricing algorithms that are able tooutput an option price, given some model and payoff parameters as input. In other words,the result of running the pricing algorithm must be given by a single number: the price. Thetechnique described in this work, as well as the approaches developed in [6, 17, 18, 22, 12], isable to compute a lower bound and an upper bound for the option price of interest. However,it does not give any a priori information on the quality of the bounds or on the price itself.Moreover, even if in theory it can be shown that for n→∞ the bounds converge to the realprice, it is not known a priori how to choose the value of n (number of moments) requiredto get a satisfactory accuracy.

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Algorithm 2 European option pricing in polynomial models based on optimizationInput: Model and payoff parameters, tolerance εOutput: Approximation Price of E[f(X)]

1: n = 22: Gap = 13: while Gap > ε do4: Set up the optimization problems (5) and (6)5: Solve them numerically to get a lower bound LB and an upper bound UB6: Gap = UB− LB7: n = n+ 28: end while9: Price = (UB + LB)/2

In order to overcome these obstacles, we can design an algorithm which computes thebounds incrementally (by increasing n) and stops after that the gap between them goesbelow a certain tolerance ε. Such a black-box algorithm would then require model and payoffparameters as input, and would return a ε approximation of the exact price in output, asdesired. Assuming that both upper and lower bounds converge to the exact price, such analgorithm can be defined as in Algorithm 2. Note that the same idea has been proposed in[21], where the bounds were computed through the approach developed in [22].

The Step 5 of Algorithm 2 can be performed by employing one of two numerical tech-niques developed in Section 3. Also, in order to perform Step 4 and efficiently updatethe vector γ of moments in the setting of polynomial models, one can use the algorithmincexpm.m developed in [21]. This algorithm exploits the block triangular structure of thenested sequence G0, G1, · · · of matrices in the moment formula (27) in order to incrementallyupdate the vector γ in an efficient way, for n increasing. We refer to [21] for more details.

6 Summary and future workIn this paper we have considered the problem of finding bounds for a quantity of the formE[f(X)], where all the moments of X exist and are available. After setting up two optimiza-tion problems whose solutions yield a lower and an upper bound for the quantity of interest(Section 2), we have reviewed some duality properties (Section 2.1) which have allowed usto investigate the convergence of the bounds for the one-dimensional case d = 1 and forsome suitable choices of f , E and X (Section 2.2). In the second part, we have proposed twoalgorithmic techniques to solve the optimization problems. The first one, based on semidef-inite programming and previously proposed in the literature, has been reviewed in Section3.1, while a new cutting plane algorithm has been introduced in Section 3.2. We have thenapplied the methodology in the setting of polynomial models (reviewed in Section 4.1) to theproblem of pricing European options. In particular, we have considered the European putoption, for which we have shown convergence and numerical results (Section 4.2 and Section4.3) in the Black-Scholes model and in the Jacobi model. The results obtained suggest that,for this particular example, the cutting plane algorithm is to be preferred for n small, whilethe SDP approach works better for n large. Finally, in Section 5 we have explained how toefficiently design a black box algorithm able to take model and payoff parameters as input,and return the option price in output. This algorithm is based on an efficient computationof the moment sequence.

In the field of mathematical finance, considering the dual problem Dn allows us to extendthe methodology to price exotic options, as Asian and barrier options, see [22, 12]. On theother side, considering the primal optimization problems in the form (5) and (6) allows foran extension to the American option pricing problem. Indeed, it can be shown (see e.g. [10])

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that the price at time t = 0 of an American option with payoff function f maturing at timeT can be upper bounded by solving the problem

inf~pe−rT γ>~p | ~p ∈ RN s.t. e−r(T−t)Hn(x)eGn(T−t)~p ≥ f(x), ∀(x, t) ∈ E × [0, T ]. (40)

where we make use of the moment formula (26). Basically, the idea is to upper bound thepayoff function f at any time t ∈ [0, T ] by polynomials. The algorithms presented in Section3 can be in principle adapted to solve (40), opening new opportunities for future research.

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SDP

k = −0.1 k = 0 k = 0.1n UB Abs error UB Abs error UB Abs error2 0.0579 0.0234 0.1041 0.0297 0.1851 0.04634 0.0493 0.0148 0.0942 0.0198 0.1550 0.01636 0.0429 0.0084 0.0833 0.0090 0.1495 0.01078 0.0410 0.0065 0.0833 0.0089 0.1459 0.007210 0.0399 0.0054 0.0802 0.0058 0.1453 0.006612 0.0387 0.0042 0.0801 0.0057 0.1435 0.004714 0.0384 0.0039 0.0786 0.0043 0.1434 0.004716 0.0377 0.0031 0.0786 0.0042 0.1424 0.003618 0.0376 0.0031 0.0778 0.0034 0.1424 0.003620 0.0370 0.0025 0.0777 0.0034 0.1418 0.0030

CP


Table 1: Upper bounds (with absolute errors) for put option prices obtained via SDP andCP approach in the Black-Scholes model. The reference prices are 0.0345 (k = −0.1), 0.0744(k = 0) and 0.1388 (k = 0.1).

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SDP


CP


Table 2: Upper bounds (with absolute errors) for put option prices obtained via SDP and CPapproach in the Jacobi model. The reference prices are 0.0356 (k = −0.1), 0.0736 (k = 0)and 0.1361 (k = 0.1).

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-1.5 -1 -0.5 0 0.5 1 1.5-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9k = -0.1

Payoff

SDP

CP

-1.5 -1 -0.5 0 0.5 1 1.5-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

k = 0

Payoff

SDP

CP

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

k = 0.1

Payoff

SDP

CP

Figure 1: Upper bounding polynomials of degree 20 for the European put option payoff inthe Jacobi model.

26

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14k = -0.1

CP

SDP

Payoff

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.05

0.1

0.15

0.2

0.25k = 0

CP

SDP

Payoff

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4k = 0.1

CP

SDP

Payoff

Figure 2: Upper bounding polynomials of degree 20 multiplied by the density function forthe European put option payoff in the Black-Scholes model.

27

2 4 6 8 10 12 14 16 18 200.15

0.2

0.25

0.3k = -0.1

Ref Impvol

SDP

CP

2 4 6 8 10 12 14 16 18 200.15

0.2

0.25

0.3k = 0

Ref Impvol

SDP

CP

2 4 6 8 10 12 14 16 18 200.15

0.2

0.25

0.3k = 0.1

Ref Impvol

SDP

CP

Figure 3: Upper bounding implied volatilities for different values of n (x-axis) in the Black-Scholes model. For all choices of k the reference implied volatility is 0.2.

2 4 6 8 10 12 14 16 18 200.15

0.2

0.25

0.3k = -0.1

Ref Impvol

SDP

CP

2 4 6 8 10 12 14 16 18 200.15

0.2

0.25

0.3k = 0

Ref Impvol

SDP

CP

2 4 6 8 10 12 14 16 18 200.15

0.2

0.25

0.3k = 0.1

Ref Impvol

SDP

CP

Figure 4: Upper bounding implied volatilities for different values of n (x-axis) in the Jacobimodel. The reference implied volatilities are 0.2034 (k = −0.1), 0.1979 (k = 0) and 0.1929(k = 0.1).

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polynomial bounds for european option pricingsma.epfl.ch/~anchpcommon/publications/polbound.pdf2...

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