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2019 | OriginalPaper | Buchkapitel

4. Stochastic Dominance Option Pricing II: Option Bounds Under Transaction Costs

verfasst von : Stylianos Perrakis

Erschienen in: Stochastic Dominance Option Pricing

Verlag: Springer International Publishing

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Abstract

This chapter presents the available theoretical results for the valuation of individual index and index futures options under proportional transaction costs. These include the reservation write (upper) bounds on prices for European and American call options and the reservation purchase (lower) bounds for European and American put option prices, whose violations can generate riskless profits. They also include the reservation purchase (lower) bounds for European and American call options, which are, however, specific to the assumed trading time partition. It is also shown that these call option lower bounds converge to an elegant continuous time limit.

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Vorheriges Kapitel Proportional Transaction Costs: An Introduction

Nächstes Kapitel Stochastic Dominance Option Pricing: Empirical Applications

We elaborate on the precise sequence of events. The trader enters the market at date t with dollar holdings \( {x}_t-{\gamma}_t{y}_t \) in the bond account and \( {y}_t/{S}_t \) cum dividend shares of stock. Then the stock pays cash dividend \( {\delta}_t{y}_t \) and the dollar holdings in the bond account become \( {x}_t \). Thus, the trader has dollar holdings \( {x}_t \) in the bond account and \( {y}_t/{S}_t \) ex dividend shares of stock.

The above argument implies that if the no-sale constraint were to be removed the reservation write price would decrease. Unfortunately removing the constraint would also require modeling the stochastic evolution of the option price till expiration. The required assumptions as to option market equilibrium would severely limit the generality of our results.

See also Constantinides and Perrakis (2007, pp. 83–84).

In expression (4.16) the first term in the RHS exceeds the second one by Jensen’s inequality. The second term is the Merton (1973) lower bound under transaction costs. In the diffusion case that we examine next the first term tends to the second as \( \Delta t\to 0 \). In the numerical work we use only the second term in the algorithm.

Hereafter we use \( f(.) \) for both the return and its random component \( \varepsilon \).

See Perrakis and Czerwonko (2009) for the details of the numerical algorithm.

At t = T − 1 we use the Merton bound in (3.3) with the corresponding value of \( {g}_{T-1}\left({S}_{T-1}\right) \).

As with European options, the trader enters the market at date t with dollar holdings \( {x}_t-{\gamma}_t{y}_t \) in the bond account and \( {y}_t/{S}_t \) cum dividend shares of stock. The trader is informed whether she has been “assigned” or not. If the trader has not been “assigned”, the stock pays cash dividend \( {\gamma}_t{y}_t \) and the dollar holdings in the bond account become \( {x}_t \). Thus, the trader has dollar holdings \( {x}_t \) in the bond account and \( {y}_t/{S}_t \) ex dividend shares of stock. If the trader has been “assigned”, the trader pays \( \left(1+{\gamma}_t\right){S}_t-K \) and not \( {S}_t-K \) because the call is exercised before the stock goes ex dividend. Then the stock pays cash dividend \( {\gamma}_t{y}_t \). Thus, the trader has dollar holdings \( {x}_t-{\gamma}_t{y}_t+{\gamma}_t{y}_t-\left\{\left(1+{\gamma}_t\right){S}_t-K\right\} \) \( =\kern0.5em {x}_t-\left\{\left(1+{\gamma}_t\right){S}_t-K\right\} \) in the bond account and \( {y}_t/{S}_t \) ex dividend shares of stock.

Clearly, any other assumption about the exercise policies of the option holders would result in a larger value function for the option writer and, hence, in a lower reservation write price for the call. Hence, the reservation write price derived under the worst-case scenario is also a reservation price under alternative exercise assumptions.

Naturally, at the maturity date T, \( {\left[\left(1+{\gamma}_T\right){S}_T-K\right]}^{+} \) is a tighter upper bound on the reservation write price of an American call. The upper bound of Eq. (4.45) does not become tighter in the special case \( T={T}^{\hbox{'}} \).

This mixed process has iid returns and can be easily computed by using the tree method developed by Amin (1993).

The reservation purchase price of a put is derived under this constrained policy. As we argued in footnote 9, the reservation price that we derive continues to be valid when the constraint is removed.

See, for instance, Modest and Sundaresan (1983).

See Merton (1982) for a clear statement on this issue.

This property is again the monotonicity condition mentioned in the proof of Theorem 1</InternalRef>, which requires a relatively “small” investment in the option relative to the stockholdings \( {w}_t \).

Amin, Kaushik I. 1993. Jump Diffusion Option Valuation in Discrete Time. Journal of Finance 48: 1833–1863.CrossRef

Constantinides, George M., and Stylianos Perrakis. 2002. Stochastic Dominance Bounds on Derivatives Prices in a Multiperiod Economy with Proportional Transaction Costs. Journal of Economic Dynamics and Control 26: 1323–1352.CrossRef

———. 2007. Stochastic Dominance Bounds on American Option Prices in Markets with Frictions. Review of Finance 11: 71–115.CrossRef

Constantinides, George M., Michal Czerwonko, Jens C. Jackwerth, and Stylianos Perrakis. 2011. Are Options on Index Futures Profitable for Risk Averse Investors? Empirical Evidence. Journal of Finance 66: 1407–1437.CrossRef

Levy, H. 1985. Upper and Lower Bounds of Put and Call Option Value: Stochastic Dominance Approach. Journal of Finance 40: 1197–1217.CrossRef

Liu, J., J. Pan, and T. Wang. 2005. An Equilibrium Model of Rare Event Premia and Its Implications for Option Smirks. Review of Financial Studies 18: 131–164.CrossRef

Merton, R.C. 1973. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science 4: 141–184.CrossRef

Merton, Robert C. 1976. Option Pricing When the Underlying Stock Returns Are Discontinuous. Journal of Financial Economics 3: 125–144.CrossRef

Merton, R.C. 1982. On the Mathematics and Economic Assumptions of Continuous-Time Financial Models. In Financial Economics: Essays in Honor of Paul Cootner, ed. W.F. Sharpe and C.M. Cootner. Englewood Cliffs: Prentice Hall.

Merton, Robert C. 1992. Continuous-Time Finance. Cambridge: Basil Blackwell.

Modest, M., and M. Sundaresan. 1983. The Relationship Between Spot and Futures Prices in Stock Index Futures Markets: Some Preliminary Evidence. Journal of Futures Markets 3: 15–41.CrossRef

Oancea, I.M., and S. Perrakis. 2014. From Stochastic Dominance to Black-Scholes: An Alternative Option Pricing Paradigm. Risk and Decision Analysis 5: 99–112.

Perrakis, S., and M. Czerwonko. 2009. Can the Black-Scholes Model Survive Under Transaction Costs? An Affirmative Answer. Working Paper Available at http://ssrn.com/abstract=1105579

Ritchken, P.H. 1985. On Option Pricing Bounds. Journal of Finance 40: 1219–1233.CrossRef

Titel: Stochastic Dominance Option Pricing II: Option Bounds Under Transaction Costs
verfasst von: Stylianos Perrakis
Verlag: Springer International Publishing
Buch: Stochastic Dominance Option Pricing
Print ISBN: 978-3-030-11589-0

Electronic ISBN: 978-3-030-11590-6

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-11590-6_4