Abstract
Abstract. In this article, we consider European options of type h(X 1 T , X 2 T , . . . , X n T ) depending on many underlying assets. We study how such options can be valued in terms of simple vanilla options in different market models. We consider different approaches and derive several pricing formulas for a wide class of functions h : IR n + → IR. We also give multidimensional version of the result of Breeden and Litzenberger [2] on the relation between derivatives of the call price and the risk-neutral density of the underlying asset.
Introduction
Option valuation is one of the most central problems in financial mathematics. However, in many models of interest the option valuation cannot be solved in closed form and thus different approaches have been developed. For instance one can use partial differential equations (PDE) or partial integrodifferential (PIDE) methods, Monte Carlo methods, or tree methods. One approach to value more complicated options is to determine its value in terms of the values of simple options of the underlying such as call options and digital options. In the work of Breeden and Litzenberger [2] the authors showed that if the second derivative of the call option price V C (K) with respect to the strike exists and is continuous, then the price of European option f (X T ) is given by
Introduction
where we treated the short interest rate as 0 for the sake of simplicity. Thus the second derivative of the strike price of the call is the state price density of the underlying asset X T . This result has significant applications especially to static hedging which is a field of active research. For more details, see for instance Carr [4] and references therein. Bick [1] extended the result of Breeden and Litzenberger to a case where either the payoff function or the price of a call has continuous second derivative with respect to its strike price except in a finite set of points (s k ) N k=0 in which the left-and right derivatives exist and are finite. In particular,
Introduction
Bick showed that
Introduction
where B T denotes the bond function, Q is the given pricing measure and ∆ − and ∆ + denotes the jump of the payoff function f . For later studies on the relation between call options and general options, see also Jarrow [9] , who derived a characterisation theorem for the distribution function of the underlying asset, and Brown and Ross [3] , who consider a model with finite state space and showed that a wide class of options are a portfolio of call options with different strike prices. In similar spirit, Cox and Rubinstein [5] introduced a method for approximating continuous functions with piecewise linear functions, which are a portfolio of call options with different strikes. They also considered the pricing error of this approximation, and suggested that one should find approximation which is the best in the sense of maximum absolute difference. However, this may cause problems when considering infinite state space.
Introduction
Recently the results of in [2] and [1] have been extended by the second named author [11] to cover the case where f is only once piecewise differentiable. In particular, in [11] it is shown that
Introduction
where the measure V C (da) always exists since V C (a) is decreasing function in strike. Barrier-type options were also considered in this context. To summarize, there exists vast array of studies in the relation between call and digital option values and values for more general options. However, all the mentioned studies consider market models with bond and one stock and similar results in multidimensional case are not so well-known. In this article we give multidimensional versions of the Breeden-Litzenberger type representation of the state-price density, widely understood. In particular, under some natural assumptions we give a pricing formula for European options h(X T ) = f (X 1 T , X 2 T , . . . , X n T ) for a wide class of payoff functions f , including the rainbow and basket options. In particular, our results cover all continuous functions h for which the partial derivative ∂ σ(1) . . . ∂ σm h exists in the sense of distributions for every m = 1, . . . , n and every permutation σ of integers {1, . . . , n}. For options which are not of this form we consider standard mollifying techniques with respect to Lebesgue measure. The benefit of this is that in this case the resulting smooth function does not depend on the underlying asset X T or the particular choice of the measure Q. We also derive multidimensional version of the Breeden-Litzenberger type representation of the state price density. In other words, if the distribution of the underlying asset X T is absolutely continuous with respect to Lebesgue measure, we show that then the density can be derived from the partial derivatives of the price of rainbow options
Introduction
The benefit of our results is that they cover a wide class of models. In particular, we only assume that at least one pricing measure for X T exists. We do not assume that it is unique. Moreover, we consider general underlying assets X T . Hence our results are valid in models which may be complete or incomplete, or discrete or continuous in time or the state space. The problem of inferring the state-price density from observed prices of the derivatives can also be regarded as an inverse problem. One plausible approach would be interpreting to pricing functional a rather general integral operator Φ(f, Q) = f dQ which may be invertible on the latter coordinate if a sufficiently wide class of payoff functions f is included. Instead of inverting the operator forcibly, e.g. by discretizing the operator and then inverting the resulting matrix numerically, we will apply some subtle properties of the payoff function class in question. The rest of the paper is organised as follows. In section 2 we introduce our notation, assumptions and we present our main results. In subsection 2.2 we consider approximation of more general payoff functions with mollifiers and subsection 2.3 is devoted to multidimensional version of Breeden and Litzenberger result. In section 3 we give a result related to uniqueness of the pricing measure Q.
Main results
Let S k t denote the stock price processes and X k t the underlying assets of an option. As examples, X k t can be a functional of S k t :s like the average
Main results
Throughout the article, B t denotes the bond given by an non-decreasing deterministic function with B 0 = 1 (all the results can be extended to stochastic interest rate models, with obvious changes in theorems). A vector (x 1 , . . . , x n ) is denoted by x. Similarly, X t denotes the vector X t = (X 1 t , . . . , X n t ). We assume that our model is, to some extent, free of arbitrage which means that there exists at least one pricing measure Q such that for each claim C, the discounted value at time t is given by IE Q [B −1 T C|F t ]. For more details on mathematics of arbitrage, see [6] and [10] and references therein. In the notation, we usually omit the dependence on Q and IE stands for expectation with respect to Q. We also assume that for given maturity T we have X k T ∈ L 1 (Q) and X k T ≥ 0 almost surely. Moreover, the price of a European option with payoff profile h(X 1 T , . . . , X n T ) is denoted by V h . We present our result for prices only i.e. values at time t = 0. However, our results could be extended to cover values at arbitrary time t with obvious changes. Note also that we assume the maturity T , but omit it on the notation.
Main results
Definition 2.1. For a function f : IR + → IR, we denote f ∈ Π Q (X T ) if the following conditions are satisfied:
Main results
(1) f is continuously differentiable except on at most countable set of points 0 ≤ s 0 < s 1 < . . . < s α < . . . (α < µ countable ordinals) in which f and f ′ have jump-discontinuities,
Main results
Definition 2.2. We denote by µ c,− and µ c,+ the counting measures, so that for a given function f : IR + → IR, we have
Main results
The jump from the left at 0 is defined as∆ − f (0) = 0.
Main results
We will also need the following counting measures. Definition 2.3. We denote by |µ| c,− and |µ| c,+ the counting measures such that for a given function f : IR + → IR, we have
Main results
For a given measure Q and underlying process X t we consider the following class of payoff functions.
Definition 2.4. For a function h : IR
n + → IR, we write h ∈ Π n Q (X T ) if the following conditions are satisfied:
Definition 2.4. For a function h : IR
(1)
Definition 2.4. For a function h : IR
where for every k = 1, . . . , m and j = 1, . . . , n we have
Definition 2.4. For a function h : IR
. . , n, and every permutation σ = (σ (1), . . . , σ(n)) of integers 1, . . . , n we have
Definition 2.4. For a function h : IR
In particular, h(X T ) ∈ L 1 (Q). (3) for every k = 1, . . . , m and every i = 1, . . . , n
Definition 2.4. For a function h : IR
We note that all polynomials are of form (2.6). We also need some operators for further use. For the rest of the paper ∂ k denotes the usual partial derivative with respect to variable x k . We find it convenient to use multi-indices to formulate the main results. Recall that a multi-index is a vector a ∈ {0, 1, 2, . . .} n which encodes the order of each pure multiple partial derivative in a mixed higher order partial derivative. In what follows all multi-indices a satisfy a := max i a i ≤ 1, being binary sequences, which means that they can also be regarded as subsets of {1, . . . , n}. Recall the following standard notation: |a| := a 1 + . . . + a n .
Definition 2.5. For a function h : IR
We will apply the multi-index notation for the operators 0 k as well. So, for example
Definition 2.5. For a function h : IR
and b = (0, 0, 1, 1). One may also consider e 1 , e 2 ∈ a; e 3 , e 4 ∈ b; e 1 , e 2 , e 3 , e 4 ∈ a + b.
Definition 2.5. For a function h : IR
Definition 2.6. Let h ∈ Π n Q (X T ) and let z, d, r, and l (short for 'zero', 'derivative', 'right jump' and 'left jump', respectively) be multi-indices such that |z + d + r + l| = n and z + d + r + l = 1. We consider a functional A z,d,r,l : Π n Q (X T ) → IR (implicitly depending on Q and X T ) given by
Definition 2.5. For a function h : IR
We will also require a similar positive functional:
Definition 2.5. For a function h : IR
For further use we also consider restrictions of operators A z,d,r,l and
Definition 2.5. For a function h : IR
and |A| z,d,r,l is defined similarly. The definition of the operators admittedly appears complicated at first sight. However, it is rather natural, we simply start with a function h and choose |z| variables which we set to zero. Next we choose |d| variables and compute partial derivatives with respect to these variables. Next we choose |r| variables and consider right jumps with respect to these variables and for the remaining |l| variables we consider left jumps. Finally, we weight the resulting function with probability where for partial derivatives and right jumps we consider strict tails and for left jumps we consider tail probabilities of the form Q(X m T ≥ y m ), and integrate over these variables. The functional A z,d,r,l computes this and later on we may sum over every possible permutation. Moreover, if |A| z,d,r,l is finite, then also A z,d,r,l is finite and well-defined. Now we are ready to formulate some of our main theorems.
Definition 2.5. For a function h : IR
for each combination of multi-indices z, d, r, and l such as above, then the price of a European option with payoff h(X T ) is given by
Definition 2.5. For a function h : IR
We note that according to the above result we may price option of given type as follows: for every variable y k we either set it to zero, take partial derivative or consider jump from right or left and then integrate with respect to corresponding measure. The price is obtained by summing over all possible combinations. As a result we obtain 4 n terms. However, usually payoff functions are continuous at least with respect to some of the variables. Hence many of the terms vanish.
Definition 2.5. For a function h : IR
Example 2.1. As an example set n = 2 and consider up-and-in Barrier call option with strike K and barrier H given by
Definition 2.5. For a function h : IR
where X T = sup 0≤u≤T S u . The price of this option is given by
Definition 2.5. For a function h : IR
This result is already established in [11] .
Definition 2.5. For a function h : IR
The proof of Theorem 2.1 is based on the following lemma which is an extension of results in [11] . The proof is presented in the appendix.
Definition 2.5. For a function h : IR
and (2.14) lim
Definition 2.5. For a function h : IR
(2.15)
Definition 2.5. For a function h : IR
Proof of Theorem 2.1. By linearity it is sufficient to consider function
Definition 2.5. For a function h : IR
We put Y 1 = n−1 k=1 f k (X k T ) and apply Lemma 2.1 for payoff f n (X n T )Y 1 to obtain
Definition 2.5. For a function h : IR
and applying Lemma 2.1 for
Definition 2.5. For a function h : IR
Indeed, assumption (2.7) implies that (2.14) is satisfied for every Y i . Moreover, (2.9) implies that assumptions (2.11)-(2.13) are satisfied for every Y i . Hence, by proceeding similarly and applying Lemma 2.1 repeatedly, we obtain the result.
Definition 2.5. For a function h : IR
In many practical cases the payoff function is continuous but not of form (2.6). By taking a sequence of functions h k ∈ Π n Q (X T ) we obtain similar results for limiting functions having enough smoothness. For discontinuous functions the jump parts may cause problems. However, we can approximate discontinuous functions with continuous ones. This is the topic of the next subsection.
Pricing with distributions.
Recall that for each continuous functions h : IR n + → IR all the mixed partial derivatives ∂ β h exist in the sense of distributions, see [8] for discussion. Therefore for every continuous function g with compact support there exists a sequence of smooth (test) functions h n , obtained by applying the Stone-Weierstrass Theorem on compact sets, such that 
Pricing with distributions.
Proof. Assume first that h ∈ C ∞ 0 (IR n + ) and let N be a number such that supp(h) ⊂ [0, N ] n . From real analysis we know that in a compact set [0, N + δ] n we can approximate h with a sequence of polynomials T n (x) uniformly such that the partial derivatives of T n convergence to partial derivatives of h also. Now, setting T n (x) = 0 outside [0, N + δ] n , by using suitable coordinate-wise convolution, we have T n (x) ∈ Π n Q (X T ) for every n. Hence the claim for h ∈ C ∞ 0 (IR n + ) follows. Assume next that h is merely continuous. By assumption (2.16) there exists a compact set K ⊂ IR n−i + , a finite union of suitable smaller compact sets
Pricing with distributions.
Since h is continuous, we can take a sequence h k ∈ C ∞ 0 such that h k converges to h uniformly on compact sets and all the partial derivatives converge in the space of distributions D ′ (IR n + ) accordingly. Thus
Pricing with distributions.
as k → ∞ for all multi-indices |z + d| = 1, z + d = 1. Hence the claim follows as ǫ > 0 was arbitrary. y) is continuous, f (0, 0) = f y (0, y) = 0, and f x (x, 0) = 1. Moreover, f xy (x, y) exists in the sense of distributions and equals −δ x (y), where δ x (y) is the Dirac delta function at x. Hence we obtain
Pricing with distributions.
Example 2.3. Consider a payoff function f (x, y) = 1 x≥y . Now f is not continuous nor of form (2.6). However, we have
Pricing with distributions.
Hence, by the previous example and the dominated convergence theorem, we may calculate formally as follows
Pricing with distributions.
Another way to price such options is to use mollifiers as in next subsection.
2.2.
Approximation with smooth functions. Our main theorem explains how options with sufficient smoothness can be priced. In this section we consider general integrable functions h and consider how to apply our main theorems for pricing such options. We use mollifiers with respect to the Lebesgue measure. The benefit is that this mollifier does not depend on X, and hence not on the particular choice of Q. We use the standard mollifier given by
2.2.
where | · | denotes the standard Euclidean norm and c is a constant such that
2.2.
Now we have ρ ∈ C ∞ 0 . Let now h be an arbitrary function. Formally we set
2.2.
A standard result of real analysis states that for sufficiently small ǫ, h ǫ is infinitely differentiable on compact subsets. Moreover, if h is continuous, then h ǫ → h uniformly on compact subsets. We also recall the following fact from real analysis (cf. [13] ).
2.2.
Lemma 2.2. Let µ be a positive Radon measure on IR n . For any h ∈ L 1 (IR n , µ) and any ǫ > 0, there exists a function ϕ ∈ C 0 (IR n ) such that ||h − ϕ|| µ < ǫ.
2.2.
We now proceed to consider our case.
2.2.
Then the following are equivalent:
2.2.
for everyǫ there exists a compact set K and a constant η > 0 such that
2.2.
Proof. 
2.2.
Moreover, by Lemma 2.2 we can take continuous ϕ such that
2.2.
We obtain
2.2.
Since ϕ is continuous, the second term is bounded byǫ 4 for sufficiently small ǫ. To finish the proof we obtain by continuity of ϕ, compactness of K and assumption (2.23) that
2.2.
(2) ⇒ (1): this implication is obvious. Note that condition (3) is closely related to the notion of uniform integrability with respect to pair (B, h), a notion which was introduced in [12] in one-dimensional case. In many practical case the Condition (2.22) is satisfied. The next result gives an easily verifiable sufficient condition, under which we have (2.22). If we now have IE[p(X T )] < ∞, then we also have (2.23).
2.3.
Absolute continuity with respect to Lebesgue measure. In a general model the measure Q is not necessarily absolutely continuous with respect to Lebesgue measure (more precisely, the law of X T under Q). Yet, if the payoff function has enough smoothness as it does in many practical cases, one may apply our Theorem 2.1. However, typically the state-price density is absolutely continuous with respect to the Lebesgue and then we have nice representations for it. It was shown by Breeden and Litzenberger [2] that in one dimensional case the risk-neutral density can be obtained by taking the second derivative of the strike price in the call's price functional. In this section we derive similar result for multidimensional case. We omit the interest rate for simplicity. Let 0 < p < ∞. We define a function h p : IR
2.3.
We will denote by V p (K, K) the corresponding price of European rainbow option wit payoff h p (X T , K, K). We will also consider function h ∞ given as a limit
2.3.
and the corresponding price given by V ∞ . We begin with some results on relation between prices V p with different values of p.
2.3.
for all K 1 , . . . , K n . The similar conclusion holds for the corresponding payoff functions with limits taken pointwise.
2.3.
Proof. It is easy to see that the statement about the payoff functions holds in the sense that the limits are taken pointwise. Let now K 1 ⊂ . . . K n ⊂ . . . be a sequence of compact sets such that K n ↑ IR n . Now (2.26) follows by considering expectations IE[h p ]1 X T ∈Kn and applying monotone convergence theorem.
2.3.
Proof. Let dQ dm be the Radon-Nikodym derivative of Q. Observe that
2.3.
According to Lebesgue's differentiation theorem the right hand side is differentiable with respect to K 1 for m 1 -a.e. K 1 ≥ 0. The set of tuples (K 1 , x 2 , . . . , x n ), where this differentiability fails is m-null, so that we may disregard it. By proceeding in this manner and differentiating n times we obtain the statement.
2.3.
Proof. Denote by 1 = (1, 1, . . . , 1, 0) ∈ IR n+1 . Here we apply the elementary fact that in the case with continuous partials the directional derivative can be calculated by taking the inner product of a gradient and a direction vector. It is easy to see that the above partials are continuous in an open subset of the state space IR n + with m-null complement. The directional derivative of the payoff function satisfies
2.3.
when defined. This limit on the left hand side is both defined and uniform on compact subsets of
2.3.
which has clearly Q-null complement. By the uniform convergence and the fact that Q is a Radon measure we get
2.3.
On the other hand, 1 − IE(− max i 1 X i
2.3.
The argument is finished by Lemma 2.4.
Proposition 2.2. The following equality holds
Similarly,
Proposition 2.2. The following equality holds
Proof. The proof is similar to that of Theorem 2.4. It is easy to see that the right derivative in the latter statement coincides with −1 X j T ≥K j . In fact,
Proposition 2.2. The following equality holds
as ǫ → 0 + and even uniformly so in compact subsets of
Proposition 2.2. The following equality holds
for 0 < p ≤ ∞.
Proposition 2.2. The following equality holds
Proof. We observe that for a given K the limit
Proposition 2.2. The following equality holds
exists and is uniform on compact subsets of
Proposition 2.2. The following equality holds
whose complement is m-null. Hence,
Proposition 2.2. The following equality holds
since Q is a Radon measure. By using the σ-additivity of Q we obtain that lim
Proposition 2.2. The following equality holds
Thus, by keeping the definition of h p in mind, it is easy to see that
Proposition 2.2. The following equality holds
The argument is finished similarly as in the proof of Theorem 2.4 and the order of taking the limit can be changed according to the monotone convergence theorem.
Proposition 2.2. The following equality holds
We note that for n = 2 and p = 1 the above state price density can be expressed in an alternative form due to the fact that
On the uniqueness of arbitrage-free prices
The main theme in this paper has been deducing the pricing kernel from observed prices of European style options with several underlying assets.
On the uniqueness of arbitrage-free prices
If there is an explicit formula for the pricing measure, then, of course, the measure must be unique. One may ask if the price information about some class of derivatives sufficiently determines the measure, although no explicit formula may not be available (cf. [3] ).
On the uniqueness of arbitrage-free prices
The following fact is probably evident to the specialists in the field and it can be obtained rather immediately from the considerations in the previous section.
On the uniqueness of arbitrage-free prices
Proof. This fact is based on disintegrating the measure similarly as in the proof of Lemma 2.4.
On the uniqueness of arbitrage-free prices
Next we will attempt to see how some partial information obtained from the prices of a class of derivatives translates to a kind of partial uniqueness of the pricing measure. The information accumulation can be conveniently encoded in terms of sub-σ-algebras, as is customary in the probability theory.
On the uniqueness of arbitrage-free prices
The following result roughly states that derivatives obtained by multiplying European call options determine the pricing measure on the sub-σ-algebra they generate.
On the uniqueness of arbitrage-free prices
Theorem 3.1. Let F be a set of non-negative Borel functions on the state space IR n + considered as European style payoff. Let Q be a set of Borel probability measures on the same state space. Assume that IE Q n i=1 (f i − K i ) + exists and does not depend on the particular choice of Q ∈ Q for K 1 , . . . , K n ∈ IR + and f 1 , . . . , f n ∈ F. Suppose that g is a σ(F)-measurable payoff function. Then g has Q-expectation if and only it has expectation with respect to all the measures in the family. Moreover, the value IE Q g, when defined, does not depend on the particular choice of Q ∈ Q.
On the uniqueness of arbitrage-free prices
Recall that σ(F) is the smallest σ-algebra containing the sets f −1 (U ) for f ∈ F and U ⊂ IR open and the statement that g is Σ-measurable means that g −1 (U ) ∈ Σ for each open U ⊂ IR.The payoff profiles f i and g can be seen as random variables with respect to the different measures Q ∈ Q. For example putting g = IE Q 0 (h|σ(F)) gives a typical σ(F)-measurable random variable. We will apply Dynkin's lemma which is well suited for analyzing the uniqueness of measures, see [7] .
On the uniqueness of arbitrage-free prices
Proof. We denote by D the collection of all Borel sets A such that Q(A) does not depend on the particular choice of Q. Note that D is closed with respect to taking complements, since Q are probability measures. By the σ-additivity of the measures we observe that D is closed with respect to taking countable unions of disjoint subsets. Thus D is a Dynkin system. We note that σ(F) is generated by the sets f ≥ K with f ∈ F, K ≥ 0. In what follows we restrict our attention to σ(F), i.e. the measures and measurable functions are considered with this σ-algebra. We aim to show that σ(F) ⊂ D. This suffices in order to obtain the statement for σ(F)-simple functions g. Indeed, then it follows that the measures in Q restricted to σ(F) coincide. If g should be integrable with respect to a measure Q 0 ∈ Q, then IE Q 0 (g) can be approximated by expectations of simple functions IE Q 0 (g n ) with g n ր g Q 0 -a.s. as n → ∞. It follows that the values IE Q (g n ) coincide for different choices of Q. Note that g n ր g Q-a.s. as n → ∞ by the equivalence of the measures. Thus, IE Q (g) = IE Q 0 (g) by the Monotone Convergence Theorem. Next we invoke Dynkin's Theorem, which yields that if the sets of the form
On the uniqueness of arbitrage-free prices
are included in D, then also σ(F) ⊂ D. Indeed, the collection of sets in (3.1) are closed with respect to taking finite intersections and they also σ-generate sets M < f i < K and consequently the σ-algebra σ(F) as well.
On the uniqueness of arbitrage-free prices
Recall that an indicator function 1 M i ≤f i <K i can be written as 1 M i ≤f i −1 K i ≤f i and that
On the uniqueness of arbitrage-free prices
This means that n i=1 1 M i ≤f i <K i can be written as lim ǫ→0 + g ǫ where
On the uniqueness of arbitrage-free prices
Note that by the assumptions and the linearity of taking expectations, the value IE Q (g ǫ ), when defined, does not depend on Q. Since n i=1 1 M i ≤f i <K i = lim ǫ→0 + g ǫ we obtain by applying the Monotone Convergence Theorem for each measure Q separately that 

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