Binomial options pricing has no closed-form solution

We set a lower bound on the complexity of options pricing formulae in the lattice metric by proving that no general explicit or closed form (hypergeometric) expression for pricing vanilla European call and put options exists when employing the binomial lattice approach. Our proof follows from Gosper’s algorithm. The binomial model as heralded by Cox et al. (1979), has become a well-known approach to pricing options due to its simplicity as well as flexibility. Recently, Dai et al. (2008) demonstrated how combinatorial techniques can be applied to address the slow convergence issue in pricing options via the CRR approach. In particular, they provided a pricing algorithm that runs in linear time for a variety of options. Combinatorially, pricing options via this binomial lattice or binomial tree method can be viewed as an extended problem in lattice path counting. Costabile (2002) and Lyuu (1998)1 demonstrated beautifully, successful applications of lattice path counting techniques in valuing options. In practice, enumerating lattice paths often leads to summation formulae involving binomial coefficients, factorials and rational functions. These types of summation formulae give rise to a large class of sums, namely, hypergeometric ones2. Hypergeometric sums, in turn, have been extensively studied in the 1Both authors incorrectly attribute the reflection principle to the French mathematician Désiré André. André’s (1887) original solution to the famous ballot problem is based on a purely explicit counting argument without geometric insight. As a result, employing André’s original argument for the development of the combinatorial identities at stake should be left as an exercise for the interested reader. Note that Renault (2008) provides an English translation of André’s work. 2See section 4.4 of Petkovšek et al. (1996) for a precise definition of hypergeometric term. mathematics literature and as we will see, some powerful tools have been developed for them. One such tool involves an algorithm that solves the problem of whether or not a given hypergeometric sum has a closed form. It turns out that our binomial options pricing formula can be expressed as such a sum and thereby analyzed in this setting. Thus, addressing the question of whether an explicit expression exists for our summation becomes natural not only from a computationally theoretic perspective that explores inherent limitations of the discrete metric but also from a practical point of view that strives for efficient computation. To this end, Petkovšek et al. (1996)3 developed a completely algorithmic solution. Practitioners can harness the power of PWZ’s work, through a software implementation as proposed by Paule and Schorn (1994) or Zeilberger (1991). Without loss of generality, our definition of closedform or explicit expression is that of Petkovšek et al. (1996). A function f(n) is said to be of (hypergeometric) closed form, if it is expressible as a linear combination of a fixed number, z, say, of hypergeometric terms in n. For instance, ∑n i=1 n+1 i(n−i+1) ( 2i−2 i−1 )( 2n−2i n−i ) (where n ≥ 1) is not a closed form expression, but its 3The interested reader should consult Chapter 5 in particular. 2158-5571/11/$27.50 c © 2011 – IOS Press and the authors. All rights reserved 14 Evangelos Georgiadis / Binomial options pricing has no closed-form solution closed form exists, and can be derived with the use of Gosper’s algorithm to be ( 2n n ) .4 Note that there seems to be no references in the finance literature that address the problem of formula complexity stemming from an inherently discrete approach. This paper helps to fill this gap. In order to settle the question of whether a closed form expression exists for the vanilla European options when priced on the discrete binomial lattice, we first set the scene by establishing the formula in question. For simplicity, we consider options on stocks. A vanilla option gives the holder the right to buy or sell the underlying stock for price X defined in the option contract at the maturity date. A call option permits the owner of the option to buy the underlying stock for X dollars at time T , whereas a put options allows the owner of the option to sell the underlying stock for X at time T . The payoff functions as well as their pricing formulae are folklore. For convenience, we state the payoff function of a vanilla option as follows. max(δS(T )− δX, 0) { for call option, we have δ = 1, for put option, we have δ = −1. The theoretical option value is the expected value of the the payoffs discounted with the risk-free rate r, namely, exp (−rT )E(payoff). Thus, the theoretical price of a vanilla option on an n-time step binomial lattice is,