Arbitrage violations and implied valuations: the option market

Abstract

The ideas presented in this paper are those of the authors and not necessarily reflect the views of the National bank of Canada. Both authors thank the National Bank of Canada and the SSHRC of Canada for their help. Thanks are also due to Professor Y. Tian for his comments, and for participating, together with students of the Financial Engineering program at York University, in the data preparation and the execution of the Matlab programs. In this paper, we propose a necessary and sufficient condition for bid and ask prices of European options to be free of arbitrage, and derive from it an efficient numerical methodology to determine its satisfaction by a given set of prices. If the bid and ask prices satisfy the no-arbitrage (NA) condition, our methodology produces a vector of NA prices that lie between the bid and ask prices. Otherwise, our methodology generates a vector of arbitrage-free prices that is as close as possible, in some sense, to the bid–ask strip. The arbitrage-free prices detected by our methodology render the commonly used practice of using mid-points and then ‘cleaning’ arbitrage from them as unnecessary. Moreover, a vector of ‘cleaned’ prices obtained from mid-point prices may be outside the bid–ask spread even in an arbitrage-free market and, hence, in this case will not be representative of the current market. A new procedure of estimating implied valuation operators is also suggested here. This procedure is rooted in the economic properties of put and call prices and is based on Phillips and Taylor's approximation of a convex function. This approach is superior to common estimation techniques in that it produces an analytical expression for the implied valuation operator and is not data intensive as some other studies. Empirical findings for the new methods are documented and their economic implications are discussed.

Keywords:

• estimation
• model construction and estimation
• contingent pricing
• futures pricing
• asset pricing
• trading volume
• bond interest rates

Notes

Necessary and sufficient condition for arbitrage free markets in the presence of proportional transactions costs were developed utilizing duality theory by Garman and Ohlson (1981).

See again Rubinstein (1994), Aït-Sahalia and Lo (1998), Aït-Sahalia and Duarte (2002).

If a risk-free asset is used as one of the assets in the economy, one of the rows of the payoff matrix A is [1, 1, …, 1, 0 , and the corresponding element of the price vector P is the present value of a dollar received at maturity of the options considered.

These arguments can be proved by applying to duality in Linear Programming or, more precisely by theorem of the alternative (a variant of Farka's Lemma). The latter method can show that the exact formulation for the NA condition should require that, at the optimum, each of the constraints are satisfied as equalities. This consequently implies that the optimal d in Equation (3) is strictly positive.

If a risk-free rate, r, is used as one of the assets in the economy, an additional equation will appear in the above system. Namely, , where T is the time (in years) from the observation date to the maturity of the options.

The vector d is, in fact, the vector of shadow prices of problem (2). The result presented here essentially identifies a discretization of [0, ∞) to J discrete points, so that the NA holds in this discrete economy with J+1 states of nature, if and only if, the NA in the real world with infinite states of nature is satisfied. If the discretization of states of natures does not include all states identified here, misleading results may be produced. The electronic appendix elaborates on this point.

A similar method was used by Bowie and Carr (1994) for a static hedging of a digital option. The resulting expression for d _j is a discrete approximation of the well-known second derivative result of Breeden and Litzenberger (1978) which is valid when options with a continuum of strikes are tradable. The Breeden and Litzenberger's result is used in the ‘Implied Valuation Operators’ section of this paper.

Nevertheless, while the data still exhibit arbitrage, a valuation operator or, equivalently, a risk-neutral density is obtained in some studies. This is the case, for example, if the RND is solved for via a discretization that does not satisfy the NA condition developed above. See the discussion in the appendix for NA on [0, ∞) and discretizations.

The can be replaced with a general function that is some measure of the deviation of the resultant pricing from the bid–ask strip, e.g. the Euclidean norm. However, numerically, the choice of f does not make much difference, while the above is a choice that allows using the efficient simplex method to solve the linear optimization problem.

The estimation procedure utilizing call prices is stipulated in the appendix. Comparing the procedures using call prices and put prices, it can be verified that the approximation with put prices requires few constraints. This might be the explanation to the fact that better results are obtained using put prices.

If a few pairs of put and call exist, the interest rate imputed from each pair must be the same since these prices support the NA condition. In these cases, the calls are converted into puts.

Choosing M is equivalent to deciding where to trunkate the interval [0, ∞), as is done in all the estimations of the RND.