Nonparametric Bounds for European Option Prices

Abstract

There is much research whose efforts have been devoted to discovering the distributional defects in the Black-Scholes model, which are known to cause severe biases. However, with a free specification for the distribution, one can only find upper and lower bounds for option prices. In this paper, we derive a new nonparametric lower bound and provide an alternative interpretation of Ritchken’s (1985) upper bound to the price of the European option. In a series of numerical examples, our new lower bound is substantially tighter than previous lower bounds. This is prevalent especially for out-of-the-money (OTM) options where the previous lower bounds perform badly. Moreover, we present that our bounds can be derived from histograms which are completely nonparametric in an empirical study. We first construct histograms from realizations of S&P 500 index returns following Chen, Lin, and Palmon (2006); calculate the dollar beta of the option and expected payoffs of the index and the option; and eventually obtain our bounds. We discover violations in our lower bound and show that those violations present arbitrage profits. In particular, our empirical results show that out-of-the-money calls are substantially overpriced (violate the lower bound).

The financial support of National Science Council, Taiwan, Republic of China (NSC 96-2416-H-006-039-), is gratefully acknowledged.

Appendix 1

In this appendix, we prove Theorem 7.1. Without loss of generality, we prove Theorem 7.1 by a three-point convex function. The extension of the proof to multiple points is straightforward but tedious. Let the distribution be trinomial and the relationship between the pricing kernel M and the stock price S be convex. In the following table, x ≤ 0; y > ε ≥ 0.

Probability	M	S	C = max{S − K, 0}
p 2	M + x	S + y > K	S + y − K
2p(1 − p)	M	S − ε < K	0
(1 − p)2	M − x	S − y < K	0

When ε = 0, the relationship between the pricing kernel M and stock price S is linear, and we obtain equality. When ε > 0, the relationship is convex. We first calculate the mean values:

$$ \begin{array}{l}E\left[M\right]=M-x+2 px\\ {}E\left[S\right]=S-y+2p\left(y-\varepsilon \right)+2{p}^2\varepsilon \\ {}E\left[C\right]={p}^2\left(S+y-K\right).\end{array} $$

The three covariances are computed as follows:

$$ \begin{array}{l}\operatorname{cov}\left[M,S\right]=2p\left(1-p\right)x\left(y+\varepsilon \right(2p-1\left)\right)\\ {}\operatorname{cov}\left[M,C\right]=2{p}^2\left(1-p\right)x\left(S+y-K\right)\\ {}\operatorname{cov}\left[S,C\right]=2{p}^2\left(1-p\right)\left(S+y-K\right)\left(y+ p\varepsilon \right).\end{array} $$

The variance of the stock price is more complex:

$$ \operatorname{var}\left[S\right]=2p\left(1-p\right)z $$

where

$$ z=\left({y}^2+{\varepsilon}^2\left(1-2p+2{p}^2\right)+2\varepsilon y\left(2p-1\right)\right)>0. $$

As a result, it is straightforward to show that

$$ \begin{array}{c}\frac{\operatorname{cov}\left[S,C\right]}{\operatorname{var}\left[S\right]}\operatorname{cov}\left[M,S\right]=\frac{2{p}^2\left(1-p\right)\left(S+y-K\right)\left(y+ p\varepsilon \right)}{2p\left(1-p\right)z}2p\left(1-p\right)x\left(y+\varepsilon \right(2p-1\left)\right)\\ {}=2{p}^2\left(1-p\right)x\left(S+y-K\right)\frac{\left(y+ p\varepsilon \right)\left(y+\varepsilon \left(2p-1\right)\right)}{z}\\ {}=2{p}^2\left(1-p\right)x\left(S+y-K\right)\left[1+\frac{\varepsilon \left(1-p\right)\left(y-\varepsilon \right)}{z}\right]\\ {}\le 2{p}^2\left(1-p\right)x\left(S+y-K\right)=:\operatorname{cov}\left[M,C\right].\end{array} $$

The fourth line is obtained because cov[M,C] < 0 and $1+{\scriptscriptstyle \frac{\varepsilon \left(1-p\right)\left(y-\varepsilon \right)}{z}}>1$. Note that the result is independent of p since all it needs is 0 < p < 1 for $1+{\scriptscriptstyle \frac{\varepsilon \left(1-p\right)\left(y-\varepsilon \right)}{z}}$ to be greater than 1. Also note that when ε = 0 the equality holds.