Transaction costs and option prices

Abstract

Option prices are characterized by wide bid-ask spreads, much wider than the spreads of their underlying assets. This note discusses the various attempts to rationalize and link the two markets’ spreads with each other and explains why their failures are mainly due to the inability to accommodate trading frictions in theoretical asset pricing models. It also presents in summary form the partial results from the stochastic dominance approach to option pricing under proportional transaction costs that provide bounds for the spreads for index call options.

Keywords

Transaction costs option pricing asset allocation stochastic dominance option bounds

1. Introduction: “Will the true equilibrium option price please stand up?”1

The statement that introduces the subject of this paper is taken from one of the most detailed early empirical option market studies involving all reported trades and quotes in the 30 most active CBOE option classes during the mid-1970s. The frustration of the study’s author, Mark Rubinstein, is understandable: he wanted to study simultaneous trades in the option and its underlying asset in order to verify whether the Black–Scholes–Merton (BSM, 1973 [2,22]) model was good enough for trading purposes. What he noticed was that during several time periods when the underlying asset had not moved at all, the call options had traded at two or three different price levels, corresponding to significantly different implied volatilities. As an example he cited a call option on an underlying equity value of $37.5 that had traded at three equidistant prices of 3.25, 3.375 and 3.5.

It is easy to see why these option trades were taking place at different prices for the same underlying value: they were all lying within the option bid-ask spread. The spread is probably the most important component of the costs of trading in the option market that include also brokerage fees and execution costs and are collectively known as transaction costs. In turn, these costs imply a range of indeterminacy of the “true” price of the options. In the example cited by Rubinstein this indeterminacy implied that the “true” trading price of the call option at observation time t, around noon on January 5, 1977, could be defined at best up to an error of 7.4% with respect to its midpoint. Its theoretical and empirical consequences for option market studies form the topic of this essay.

In the next section we provide some empirical evidence about the magnitude of the problem using data from the most liquid and actively traded options, those on the S&P 500 index. In Section 3 we discuss the (mostly failed) attempts to incorporate the existence of the bid-ask spread through the traditional arbitrage-equilibrium approaches introduced by BSM and followed by most option researchers. Section 4 reviews the (rather skimpy) alternative literature on transaction costs and the empirical studies that underlie it. Section 5 concludes and suggests avenues for future research.

2. The size and consequences of the option bid-ask spread

The 7.4% interval of option trading prices is a major understatement of actual conditions in option markets, given that the width of the quoted bid and ask prices is a function of the option’s maturity and degree of moneyness $\frac{K}{S_{t}}$ , the ratio of the strike price K to that of the underlying asset $S_{t}$ . Table 1 shows the average recorded value of the spread for the S&P 500 index options over the period January 1991–February 2013, equal to $\frac{2 (C_{a} - C_{b})}{(C_{a} + C_{b})}$ and $\frac{2 (P_{a} - P_{b})}{(P_{a} + P_{b})}$ for call and put prices C and P respectively, where the subscripts a and b represent the ask and bid prices, for two short maturities of 28 and 14 days, observed at 2 PM Standard Central Time, one hour before option markets close. The spread is shown as a function of the degree of moneyness separately for calls and puts, for in-, at-, and out-of-the-money (ITM, ATM and OTM respectively), defined as $\pm 100 \frac{S_{t} - K}{S_{t}}$ for calls and $\pm 100 \frac{K - S_{t}}{S_{t}}$ for puts.

Table 1
The table shows the average difference of ask and bid prices of call and put options on the S&P 500 index as a percentage of the spread midpoint as recorded on every trading day at 2 PM, for two maturities in days and three different degrees of moneyness. ITM and OTM moneyness corresponds to $\pm 100 \frac{S_{t} - K}{S_{t}}$ respectively for calls and $\pm 100 \frac{K - S_{t}}{S_{t}}$ for puts, with $S_{t}$ denoting the index value and K the strike price

T 2.5% ITM ATM 2.5% OTM

A: Calls

28 4.8 6.8 12.1

14 5.6 8.7 19.7

B: Puts

28 5.5 6.8 10.2

14 6.2 8.6 14.6

T	2.5% ITM	ATM	2.5% OTM
	A: Calls
28	4.8	6.8	12.1
14	5.6	8.7	19.7
	B: Puts
28	5.5	6.8	10.2
14	6.2	8.6	14.6

Observe how quickly the size of the spread increases for OTM options, even though this data refers to very near the money options. On October 2, 2008, at around noon, the spread was around 10% ATM for the November options, but rose to 20% for 6.4% OTM options. Further, these spreads show no sign of having decreased over time: in two different dates distant by about 21 years, between February 1992 and February 2013 when the index went from 411 to 1616, the spread for 28-day 2.5% OTM calls went from 14.3% to 23.3% and for the same 2.5% OTM 14-day calls from 25.6% to 41.9%. Observe also that these option spreads are far, far above the corresponding ones for the underlying asset: although the index itself is not traded, the exchange-traded SPDR fund tracks it very closely, and its observed bid-ask spread rarely exceeds 0.1%. Even if we double this amount it would be more than 100 times smaller than the one corresponding to the 2013 28-day OTM call options.

In the more than thirty years since the Rubinstein article there has been an explosion in theoretical and empirical option market studies whose complexity and level of mathematical sophistication were not even conceivable in those early years. Almost all the theoretical models of asset pricing have assumed a frictionless world, an assumption that yields elegant market equilibrium results, especially when markets are complete and asset dynamics can be represented by diffusion processes in continuous time.2 When markets are incomplete because there are rare events or volatility is stochastic no arbitrage was supplemented by market equilibrium considerations, in which a “representative” investor was assumed to exist, almost always of the constant relative risk aversion (CRRA) class. A variant of this approach is to assume similar asset dynamics in both the underlying asset and the option market and to extract the equilibrium conditions empirically from both markets. The frictionless assumption is very convenient because it offers a one-to-one link not only between option and underlying asset but also between options on the same underlying with different characteristics. From the observed option prices one can estimate the risk neutral distribution $f (K)$ of the index return at option expiration time by the well-known relation that exists under no arbitrage and in the absence of frictions, $\frac{\partial^{2} C}{\partial^{2} K} = \frac{\partial^{2} P}{\partial^{2} K} = e^{- r T} f (K)$ , r being the riskless rate.

Applying these elegant results empirically involves choosing an observed variable that represents the option price. Most studies have chosen the midpoint of the observed spread and assumed that the frictionless economy conditions such as put-call parity held. This creates problems, since applying parity to the observed midpoint put and call prices for every strike gives rise to other types of violations of the conventional no arbitrage relations, which must be removed by suitably “purging” the data since otherwise no option pricing model is conceivable. These adjustments are major: as a recent study points out,3 less than 1% of the observed S&P 500 option midpoints during 2006–2007 were arbitrage-free. Even the less drastic and more reasonable approach that seeks a vector of option prices for all strike prices in each cross section that is arbitrage-free and lies within the bid-ask spread yielded results that were less than fully satisfactory. The same study mentioned above shows that over 2006–2007 about 30% of the S&P 500 option cross section failed the test, implying that for these cases there was no arbitrage-free price lying within the observed bid and ask prices of both puts and calls. Similar results with a slightly different methodology were also found in another study that used a larger data base and more extensive tests.4 Last, there are persistent empirical findings within mainstream frictionless asset pricing models that are treated as anomalies within the models, such as the well-known “overpricing” of S&P 500 ATM and OTM index put (but not call) options, an implicit admission that put-call parity is violated, as is normal in the presence of the bid-ask spread.5 In the following sections we review the literature that has recognized this spread and attempted to deal with it.

3. No arbitrage and market equilibrium under transaction costs

In studying financial markets for any asset the determination of the spread is generally modeled by assuming that there are dealers or market makers who hold diametrically opposite positions to those of investors: the quoted ask (bid) price at which investors can buy (sell) the asset is the lowest (highest) price at which market makers are willing to sell (buy) the asset. In the case of S&P 500 options market makers hold positions on several options in a given cross section as well as an inventory of underlying asset such as tracking fund or futures contract, since this is the cash equivalent that will be delivered upon exercise or expiration of the options. Since the inventory size cannot be determined in the absence of a pricing model that recognizes frictions, there is a chicken and egg problem in trading derivative assets: market making cannot be done efficiently without a good pricing model, and such a model cannot be formulated without taking into account the bid-ask spread.

Such modeling in a no arbitrage context with diffusion asset dynamics was introduced by Leland [19] and subsequently extended in a binomial model by Merton [23] and Boyle and Vorst [5]. These articles applied variants of the dynamic option replication policy of frictionless no arbitrage to a universe that admits proportional transaction costs in trading the underlying asset. Market makers hedge their positions perfectly by constructing portfolios using the underlying and the riskless asset whose final payoffs at maturity equals those of the long and short options respectively, and deliver them to the option holders at expiration. As Merton [23] pointed out, in a continuous time model this would imply a continuous rebalancing of the portfolio to reflect the new underlying price and incur an infinite volume of trade over the lifetime of the option. In a binomial model it would imply restructuring the portfolio at every node of the binomial tree, thus making the bid-ask spread a function of the time subdivisions till option expiration.

The consequences of such restructuring are more clearly evident in the binomial model. As Boyle and Vorst [5] showed, the portfolio replicating the long option tends to a BSM-type model as the time partition $Δ t \to 0$ , with an adjusted volatility equal to $σ^{2} (1 + \frac{2 k}{σ \sqrt{Δ t}})$ , where σ is the volatility and k the transaction cost rate, assumed equal for long and short positions; the equivalent convergence for the short option replication has adjusted volatility $σ^{2} (1 - \frac{2 k}{σ \sqrt{Δ t}})$ . It is easy to see that these expressions yield the trivial no arbitrage bounds at the continuous time limits. Attempts to extend this approach, by allowing the replicating portfolio to remain the same when it was cheaper not to restructure it while still requiring the final position to be equal to that of the option at expiration, also lead nowhere.6

4. Stochastic dominance bounds: Theory and empirical applications

Since the no arbitrage approach fails in the presence of transaction costs, it was perhaps natural to turn to joint equilibrium in the underlying and option markets to search for an alternative. The first step in market equilibrium models is portfolio selection in the presence of transaction costs, for which the literature is rather sparse. To our knowledge asset allocation models involving one risky and one riskless asset are the only ones for which usable results are available. Their main features are reviewed briefly before linking them to the option market.

4.1. Asset allocation under proportional transaction costs

Consider a class of traders who invests only in a risky asset and a riskless bond, a condition that will be assumed throughout this section. Each trader makes sequential investment decisions in the primary assets at the discrete trading dates $t = 0, 1, \dots, T^{'}$ , where $T^{'}$ is the terminal date and is finite. A bond with price one at the initial date has price $R, R > 1$ at the end of the first trading period, where R is a constant; without loss of generality bond trades are assumed to be traded without frictions. On the other hand, the risky asset’s trades incur proportional transaction costs charged to the bond account. At each date t, the trader pays $(1 + k_{1}) S_{t}$ out of the bond account to purchase one share of stock and is credited $(1 - k_{2}) S_{t}$ in the bond account to sell (or, sell short) one share of the risky asset. It is assumed that $0 ⩽ k_{1} < 1$ and $0 ⩽ k_{2} < 1$ , and we also assume that there are no dividends within the option’s time to expiration $T < T^{'}$ .7

The trader enters the market at date t with dollar holdings $x_{t}$ in the bond account and $y_{t} / S_{t}$ shares of stock and increases (or, decreases) the dollar holdings in the stock account from $y_{t}$ to $y_{t}^{'} = y_{t} + υ_{t}$ by decreasing (or, increasing) the bond account from $x_{t}$ to $x_{t}^{'} = x_{t} - υ_{t} - max [k_{1} υ_{t}, - k_{2} υ_{t}]$ . The decision variable $υ_{t}$ is constrained to be measurable with respect to the information up to date t. The bond account dynamics is $\begin{array}{l} x_{t + 1} = {x_{t} - υ_{t} - max [k_{1} υ_{t}, - k_{2} υ_{t}]} R, \\ (1) & t ⩽ T^{'} - 1, \end{array}$ and the stock account dynamics is $\begin{matrix} (2) & y_{t + 1} = (y_{t} + υ_{t}) \frac{S_{t + 1}}{S_{t}}, t ⩽ T^{'} - 1 . \end{matrix}$ At the terminal date, the stock account is liquidated, $υ_{T^{'}} = - y_{T^{'}}$ , and the net worth is $x_{T^{'}} + y_{T^{'}} - max [- k_{1} y_{T^{'}}, k_{2} y_{T^{'}}]$ . At each date t, the trader chooses investment $υ_{t}$ to maximize the expected utility of net worth, $E [u (x_{T^{'}} + y_{T^{'}} - max [- k_{1} y_{T^{'}}, k_{2} y_{T^{'}}]) | S_{t}]$ .8 We make the plausible assumption that the utility function, $u (\cdot)$ , is increasing and concave, and is defined for both positive and negative terminal net worth.9

This problem was examined in a seminal paper by Constantinides [8], which derived the properties of the value function $V (x_{t}, y_{t}, t)$ , defined recursively as follows $\begin{array}{l} (3) & \begin{matrix} V (x_{t}, y_{t}, t) \\ = max_{υ} E [V ({x_{t} - υ - max [k_{1} υ, - k_{2} υ]} R, \\ (y_{t} + υ) \frac{S_{t + 1}}{S_{t}}, t + 1)] \end{matrix} \end{array}$ for $t ⩽ T^{'} - 1$ and $\begin{array}{l} V (x_{T^{'}}, y_{T^{'}}, T^{'}) \\ (4) & = u (x_{T^{'}} + y_{T^{'}} - max [- k_{1} y_{T^{'}}, k_{2} y_{T^{'}}]) . \end{array}$ It was shown under conditions more general than the ones adopted here that $V (x, y, t)$ is increasing and concave in $(x, y)$ , properties inherited from the monotonicity and concavity of the utility function $u (\cdot)$ , given that the transaction costs are quasi-linear. Further, it was also shown that there exists a no trade (NT) zone such that $υ_{t}^{*}$ , the optimal investment decision at date t corresponding to the portfolio $(x_{t}, y_{t})$ , is equal to 0 if $(x_{t + 1}, y_{t + 1})$ lie within NT, while otherwise it is restructured to the nearest NT boundary if it falls outside NT. This fundamental result, by reducing the necessary investor transactions within the interval till option expiration, unblocked the impasse of continuous trading of the no arbitrage models and allowed the derivation of useful results for options.10

The key issue in attempting to solve this problem for a specific utility function, even for the simple CRRA class, is the assumption about the asset dynamics governing the risky asset return $R_{S, t} \equiv \frac{S_{t + 1}}{S_{t}}$ as the time partition tends to 0. Surprisingly, given the length of time since this problem was formulated, solutions for this problem exist only for the simple diffusion case, and have been extended only very recently to jump diffusion.11 Conspicuously absent are stochastic volatility dynamics, which play such a major role in the frictionless asset pricing models. For this reason the option bounds are derived below for unspecified asset dynamics but are in practice applicable only to diffusion and jump diffusion.

4.2. Option bounds under transaction costs

The derivation of the European option bounds combines the investor asset allocation problem under transaction costs with the frictionless stochastic dominance (SD) methodology introduced into option pricing by several studies during the 1980s.12 In that approach the trader considers whether there exist zero net cost portfolios involving long or short positions in a single option, possibly combined with the underlying asset and the riskless bond that can be shown to increase her expected utility without depending on its parameters, the initial wealth or its composition. Alternatively, it adopts the pricing kernel approach to pricing assets and uses linear programming (LP) to determine the kernel that maximizes or minimizes the equilibrium price of a single option given that it depends only on the risky asset’s return distribution.13 For options that are convex with respect to their underlying price both methods derive the same upper and lower option bounds for call and put options, American and European, for index options. These can be extended to equity options if one assumes a linear structure in equity returns similar to the capital asset pricing model (CAPM). In the case of diffusion asset dynamics both bounds have been shown to converge to the BSM price for both index and equity options.14

The extension of SD in the presence of proportional transaction costs is not trivial and yields results only for some types of options. Specifically, we have tight upper bounds for both European and American index and index futures call options, and lower bounds for the same call options, as well as for European and American index and index futures put options, while the upper bounds for put options are less useful and depend on the time partition.15 The key property is the concavity of $V (x, y, t)$ , which yields a pricing kernel that is decreasing in $R_{S, t}$ and is used in the comparison at every time subdivision in the interval $[t, T]$ of the two value functions, with and without the zero net cost option portfolio, but requires the transfer of all proceeds of the portfolio to the risky asset account; such transfers incur transaction costs. For instance, the upper bound $\bar{C} (S_{t}, t)$ of a call option is derived by writing the option and adding the net proceeds $\frac{\bar{C} (S_{t}, t)}{1 + k_{1}}$ to the risky asset account $y_{t}$ . At option expiration, after transfer to the risky asset account, the net payoff becomes $y_{T} + \frac{\bar{C} (S_{t}, t)}{1 + k_{1}} \frac{S_{T}}{S_{t}} - \frac{{[S_{T} - K]}^{+}}{1 - k_{2}}$ , and the shape of this function, combined with the monotonicity of the pricing kernel, guarantees that the value function of the option investor exceeds $V (x_{t}, y_{t}, t)$ for all short option prices above the following European index option upper bound: $\begin{matrix} (5) & \bar{C} (S_{t}, t) = \frac{(1 + k_{1})}{(1 - k_{2}) R_{S, t}^{T - t}} E [{[S_{T} - K]}^{+} | S_{t}] . \end{matrix}$ Similarly, the following lower bound is derived for a put option $\begin{matrix} (6) & \underline{P} (S_{t}, t) = \frac{(1 - k_{2})}{(1 + k_{1}) R_{S, t}^{T - t}} E [{[K - S_{T}]}^{+} | S_{t}] . \end{matrix}$ For the lower bound on a European call option there is no closed form expression and the bound is partition-dependent and derived numerically. Nonetheless, an elegant and tight continuous time limit exists when the underlying asset dynamics are lognormal diffusion as in the BSM model, to which the numerically derived bound converges from below as the partition $Δ t \to 0$ . Specifically, letting $\underline{C} (S_{t}, t)$ denote the lower bound and the BSM model price be $e^{- r T} E^{Q} [{[S_{T} - K]}^{+} | S_{t}]$ , with the superscript Q denoting the risk neutral expectation of the payoff, we have $\begin{array}{l} lim_{Δ t \to 0} \underline{C} (S_{t}, t) \\ (7) & = e^{- r T} E^{Q} [{[\frac{1 - k_{2}}{1 + k_{1}} S_{T} - K]}^{+} | S_{t}] . \end{array}$ Relations (5)–(7) provide some testable implications. For the efficiency of the market making function, assuming weakly risk averse market makers, the bid-ask spread in call options for all underlying assets should lie within the interval $[\underline{C} (S_{t}, t), \bar{C} (S_{t}, t)]$ . For index options only, market efficiency implies that no observed call option bid (ask) price should lie above $\bar{C} (S_{t}, t)$ (below $\underline{C} (S_{t}, t)$ ).16 More broadly, stochastic dominance efficiency in the option market implies that no zero net cost portfolio purchased at observed bid and ask option prices should have a payoff that stochastically dominates that of the index. Yet empirical evidence exists that all three of these implications have been violated in the S&P 500 option market, arguably one of the most liquid derivative markets in the world.

4.3. Empirical tests

To our knowledge no empirical testing of the efficiency of the option market making function exists, although the verification is easy and depends on the assumed distribution of the underlying asset. On the other hand, in verifying the observed price $C_{b}$ in relation to the call upper bound, the equivalent to (5) in S&P 500 index futures options, Constantinides et al. [10] found a pattern of violations ranging from 17% to 53% (depending on the volatility prediction mode) of the dates in a time series of 30-day options over 23 years; at least one violating option was found in each one of those dates. Equally frequent violations had been observed in an earlier study of S&P 500 index options.17 The violations in the 2011 study were all in the form of overpriced call options, in contrast to the “overpriced” put options found in earlier asset pricing empirical studies, as noted above. These in-sample results are sufficiently disturbing on their own, but their out-of-sample verification in the 2011 study were much more serious and the questions they raise about the efficiency of the option market cannot be answered easily.

To test these violations out-of-sample the zero-net-cost portfolios used in deriving the bounds were evaluated at each data point, and their realized payoffs at option expiration were added to the proceeds of investing in a unit of the index, forming a time series of the option trader (OT) returns whose initial cost was equal to that of one unit of the index. After suitable standardization this total OT payoff series was compared to the series of realized returns of one index unit, the index trader (IT) series, in a test of stochastic dominance of one time series over another. The chosen test was by Davidson [16] and Davidson and Duclos [17], in which the null was that of non-dominance of the OT over the IT series, and rejection of the null implies the OT payoffs series dominates the corresponding IT series. The test statistic was evaluated over the entire time series, including the dates where no bound violating option existed, and rejected decisively the non-dominance null. In terms of average OT excess returns, these varied from 0.31% to 0.66% annually, which were risk-adjusted by virtue of the SD relationship; note that the ex-dividend average index return, the IT return, was only 4%.

We elaborate further on the methodology underlying the out-of-sample results arising out of the violations of the call upper bound (5). Whenever a call bid price exceeded (5) the portfolio to exploit the violation adds to the index a short option, yielding a combined OT payoff at option expiration equal to $S_{T} + \frac{C_{b}}{1 + k_{1}} R_{S, t}^{T - t} - \frac{{[S_{T} - K]}^{+}}{1 - k_{2}}$ . By construction, this concave function of $S_{T}$ exceeds $S_{T}$ at low values, intersects it at a single point, and lies below it at high values. Since by definition $C_{b}$ exceeds the RHS of (5) in a violating option, the OT payoff dominates IT since its expectation is non negative. In other words, the OT zero net cost option portfolio shifted positive payoff from the high to the low index return range, thus reducing risk at equal or higher expected returns, consistent with the SD definition of risk-adjusted gain.18 Since this is a gain accessible to any risk averse trader, it constitutes a tradable anomaly.

These results were generalized and extended more recently in an ongoing study of short term (28-, 14- and 7-day) S&P 500 index options. The shifting of positive payoff from high to low index return range is being applied to the OT portfolios in the new study, with the difference that the portfolio composition is not predetermined but found numerically. The algorithm searches among the universe of traded options for zero net cost portfolios of calls and puts, long or short, written or purchased at the appropriate bid or ask prices, such that the payoff at expiration lies above $S_{T}$ at low values, intersects it only once and has a non-negative expectation. Such portfolios were found in almost every date in a long time series of observations and for all three examined maturities, with expected payoffs far higher in most cases than the approximately 0.5% observed in the 2011 study; their standardized time series of returns decisively dominated in all cases the IT portfolio by the Davidson–Duclos [17] test. Further, the OT portfolio composition varied by date and contained both calls and puts, but included short call options in most cases, especially for the two shorter maturities. At the very least these results imply major inefficiencies in the index option market, since investors can realize superior risk adjusted returns by purchasing these short maturity zero net cost portfolios and combine them with the index at least twelve times a year.

5. Conclusions and implications for further research

We started with a reminder of a problem that was pinpointed more than 30 years ago, but that is still present in the overwhelming majority of empirical option pricing studies: the inability to define the appropriate price of an option given the wide spreads at which options are traded in the option market. This problem is more acute for OTM options and does not show any signs of going away with time. Further, the OTM options have, if anything, increased in importance in the recent asset pricing literature, since the development of volatility trading relies overwhelmingly on them.19 We pointed out that neither the observed trading price nor the spread midpoint are satisfactory proxies for this unobservable variable. We also noted that the incorporation of transaction costs into the no arbitrage methodology without changing its basic replication assumptions leads nowhere and can be at best considered as an approximation whose accuracy and cost are a priori unknown.

We then focused on the sparse literature that recognizes frictions in option trading without falling into the continuous trading trap of no arbitrage. This literature brings utility functions explicitly in the modeling of market maker behavior without tying itself to a specific investor utility function, especially its dependence on the (unobservable) risk aversion. Although it has produced comparatively few theoretical results, these make possible the (partial) empirical verification of the efficiency of market making in the index call option markets and the extension of the no arbitrage market efficiency condition by incorporating the “no stochastic dominance” condition into the index option markets. It also introduced rigorous statistical methods of the ex post (out of sample) verification of option market efficiency, by adapting econometric tests developed for different purposes to options violating in-sample efficiency.

How can this empirical literature proceed, beyond verifying efficiency, to option pricing tests? A possible approach would be to proceed beyond the SD bounds, towards a formal derivation of a market maker’s reservation purchase and write prices based on specific utility functions, such that these reservation prices will leave the market maker’s initial utility level unchanged. Although closed form expressions are probably beyond reach, even with a CRRA utility function and jump diffusion asset dynamics, a numerical approach is probably feasible and will at the very minimum allow the determination of the limits on risk aversion implied by the bid-ask spread, as well as on the risk neutral distributions consistent with that spread. More complex asset dynamics need extension of the SD relations to bi-dimensional utility functions.

Footnotes

Acknowledgements

I wish to thank George Constantinides and Michal Czerwonko for their helpful comments. I am the sole responsible for any remaining errors.

Mark Rubinstein [], p. 465

See, for instance, Breeden [].

See Ioffe and Prisman [], p. 310.

See Constantinides, Jackwerth and Perrakis [], Tables 1 and 2.

See, among others, Bollen and Whaley [3], Santa-Clara and Saretto [32], Bondarenko [4], and Constantinides and Lian [].

This approach is also known as super-replication. See Bensaid et al. [1] and Perrakis and Lefoll [].

This assumption can be easily relaxed when dealing with American options; see Constantinides and Perrakis [].

The results extend routinely to the case that consumption occurs at each trading date and utility is defined over consumption at each of the trading dates and over the net worth at the terminal date.

If utility is defined only for non-negative net worth, then the decision variable is constrained to be a member of a convex set that ensures the non-negativity of the net worth. The option bounds apply to this case as well.

See also Constantinides [] for an application of the derivation of the NT zone in diffusion asset dynamics that also illustrate the small number of restructurings in most realistic cases.

See Liu and Lowenstein [21] for a continuous time solution under special conditions and Czerwonko and Perrakis [] for a numerical solution under a more general formulation.

See in particular Perrakis and Ryan [27], Ritchken [28], Levy [20], Perrakis [25], and Ritchken and Kuo [].

This assumption yields a kernel that is monotone decreasing in the risky asset return.

See Oancea and Perrakis [].

See Constantinides and Perrakis [13,].

Unlike the frictionless case, SD cannot be applied to equity options when the CAPM holds since, unlike market makers, individual investors cannot be assumed to hold only the underlying and the riskless asset.

See Constantinides, Jackwerth and Perrakis [].

See Rothschild and Stiglitz [].

See, for instance, Carr and Wu [], p. 1316, equation (5).

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