Option Pricing Bounds in Discrete Time

Abstract

Upper and lower bounds are derived for call options traded at discrete intervals. These bounds are independent of assumptions on the stock price distribution other than a restriction satisfied by the stock being "non-negative beta." The development of the bounds relies on the single-price law and arbitrage arguments. Both single-period and multiperiod results are produced, and put option bounds follow by extension. The bounds exist as equilibrium values given a consensus on stock price distribution; they are also valid for empirical studies, being adjustable for dividends and commissions. FOLLOWING THE IMPETUS of theBlack-Scholes pricing model, the majority of option analyses have been in continuous time. Such approaches have relied on the formation of perfect, riskless hedges involving the option, its underlying security, and a riskless asset; since the hedge must be continuously revised and maintained, while actual trading opportunities are discrete, this crucial assumption limits the accuracy and applicability of the Black-Scholes and related formulae. In fact, the Black-Scholes formula can only be considered as "a valid approximation to the discrete-time solution" (Merton [8, p. 663]), while the closeness of that approximation has not been ascertained in general.1 Earlier discrete models such as Boness [3] and Samuelson [12] have been succeeded by those of Brennan [5], Cox et al. [6], and Rubinstein [11], in which distributional restrictions are imposed on share returns; usually, the discrete formula converges to the Black-Scholes expression under appropriate limit conditions. Using the Rubinstein [11] approach, we derive upper and lower bounds for option prices with both a general price distribution and discrete trading opportunities. The bounds are functions of the share and exercise prices, the riskless rate of interest, the distribution of the share price change per period, and the number of periods of expiration of the option. The lower bound is tighter than that of Merton [7], derived from more general arbitrage considerations, while being less exact than the results already mentioned for restricted distributions. Thus the bounds form a range of prices in equilibrium derivable for any general distribution; they have the further advantage of being adjustable for dividends and commissions. The generality of assumptions and adjustable range provide important advantages for empirical studies. * Professor and Associate Professor, Faculty of Administration, University of Ottawa. A preliminary version of this paper was presented under a different title at the 4th World Congress of the Econometric Society at Aix-en-Provence, in August 1980. The authors wish to thank a referee of this Journal for helpful advice and comment. 1 See [6], [9], and [10] for pertinent comments on this point. 519 520 The Journal of Finance I. Call Option Bounds We adopt the following notation: S the initial price of the underlying asset, X the exercise price of the option, Y the random price change (AS) per period, F(Y) the probability...

Option Pricing Bounds: Synthesis And Extension

“…This is the lower bound found by Perrakis and Ryan (1984), albeit using a different method. Because R P (S ) ≥ V PR (S ) > V M (S ), it is clear that Perrakis and Ryan's lower bound is tighter than Merton's (1973).…”

“…However, for S w ∈ [0, S f ), using the simpler V 2 (S ) functional form produces an erroneous lower bound for some values of S. In such cases, the longer V 1 (S ) form is used and the maximum operator is dropped. Equation (7) contains as special cases various lower option pricing bounds found by previous researchers through widely differing methods, including those by Merton (1973), Perrakis and Ryan (1984), Levy (1985), and Ritchken (1985). Indeed, I show that these lower bounds differ solely in their particular choice of the V k (S ) function.…”

“…For example, Perrakis and Ryan (1984) improve the lower bounds by placing mild restrictions on the underlying asset price distribution and assuming that investors are risk averse. Perrakis (1986) further tightens the lower bounds by restricting the range of allowable terminal underlying asset prices.…”

Option Valuation in Incomplete Markets: A Discrete‐time, Capm Approach

“…In particular, the formula will underprice American options due to their early exercise features. Call premium calculated using the data taken from Ritchken (1985Ritchken ( , p. 1229) were found to be within Merton (1973), Perrakis-Ryan (1984) and Ritchken's (1985) lower bounds. Chang and Shanker (1987) show (p. 11) that with reasonable values of the Kariables, premiums would only decrease by 80.0007 for a one percent per year increase in R,.…”

Transactions Costs and Option Bid/Ask Spread on the Swiss Options and Financial Futures Exchange (SOFFEX)

“…In such a case, there is no single option price but two option bounds, which provide, under certain assumptions, upper and lower limits on option prices consistent with the observed stock price and riskless rate of interest. These bounds were originally introduced by Perrakis and Ryan (1984) and subsequently extended by Perrakis (1986Perrakis ( , 1988, Ritchken (1985), and Ritchken and Kuo (1988). To the extent that they are tighter than the M-BV bounds, they would, under the binomial assumption and a competitive structure of the market for financial intermediation, form the option bid and ask prices.…”