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Is there an informationally passive benchmark for option pricing incorporating maturity?

2007; Taylor & Francis; Volume: 7; Issue: 1 Linguagem: Inglês

10.1080/14697680601011438

1469-7696

Vicky Henderson, David Hobson¶, Tino Kluge¶,

Financial Markets and Investment Strategies

Abstract Figlewski proposed testing the incremental contribution of the Black–Scholes model by comparing its performance against an “informationally passive” benchmark, which was defined to be an option pricing formula satisfying static no-arbitrage constraints. In this paper we extend Figlewski's analysis to include options of more than one maturity. Once maturity has been included in the model, any “informationally passive” call pricing function is consistent with some “active” model. In this sense, the notion of a passive model cannot be extended to pricing formulas incorporating option maturity. We derive the index dynamics of the active model implicit in Figlewski's implied G example. These dynamics are far more complicated than the dynamics of the Samuelson–Black–Scholes or Bachelier models. The main implication of our analysis is that an appropriate benchmark for assessing option pricing models should in fact have simple dynamics, such as those of Bachelier or the Black–Scholes models. This is despite the fact that the maturity extension of Figlewski's model gives as good a fit as the Black–Scholes model. Acknowledgments We would like to thank Gurdip Bakshi, Nikunj Kapadia and Robert Tompkins for kindly sharing their data sets. We also thank seminar participants at Stanford University and Columbia University, and Steve Figlewski for comments on a previous version of this paper titled “Extending Figlewski's option pricing formula”. The first author acknowledges partial support from the NSF via grant DMI 0447990. The second author is supported by an Advanced Fellowship from the EPSRC. The third author acknowledges partial financial support from DAAD, EPSRC and KWI. Notes ¶Emails: dgh@maths.bath.ac.uk (DH) ¶tino@statslab.cam.ac.uk (TK) †As Figlewski (Citation2002, p. 90) states: “No” effort has been made to tweak the model in any way to improve its performance.' ‡Modified implied G. †In defense of Figlewski's (Citation2002) original model, although the IG model misprices a put with zero strike (whereas MIG prices it at zero) over the range of traded options, the differences between the two models are very small. To this extent, the fact that IG admits static arbitrage can be viewed as a theoretical problem that has little impact in practice. Provided that the IG model is only used “locally” then no mispricing problems arise. Indeed, we also find many circumstances in which IG provides a better fit to data than MIG. ‡There is good reason to believe that both of the models IG and MIG would fit option price data with a symmetric smile, such perhaps as currency option data, better than index option data for which implied volatilities display a skew. We discuss this reasoning in section 7, when we attempt to explain why the IG and MIG models give such a good fit to data. §It seems that entries in some of the tables in Figlewski (Citation2002, Exhibit V) have been accidentally reversed. ¶Throughout we assume that S has been adjusted for dividends. ∥In order to even state these properties it is necessary to assume a certain amount of regularity. For instance, we need to assume that St is a Markov process. †Note that if we think instead of the call price as a function of current time and the index level St , then property (viii) does not hold for the IG model. We have ‡The problem with a model with call prices given by the function IG is that it is consistent with a price process which can go negative. §Figlewski (Citation2002) interprets the parameter G in his model in this way. However, the parameters G and g might more correctly be interpreted as analogs of implied squared volatility. See also footnote on p. 5, where takes the place of a constant multiple of σ. †More generally we could have used any increasing functions of time to maturity, but our aim is to give the simplest possible extension to the time varying case. For an at-the-money option, under our choice (T−t)G, whereas in the Black--Scholes model to leading order. Hence our choice of linear scaling in time to maturity is the most natural. ‡The assumption that the underlying follows a diffusion process is a combination of two assumptions, firstly that the underlying is a Markov process, and secondly that it is continuous. Both of these assumptions can be challenged as unrealistic, and both are clearly idealizations. However, making these assumptions leads to what, in many cases, is the simplest possible model consistent with the given call prices. §Compare the dynamics for the index under the two time-extended models IGT and MIGT. Notice that for the IGT model, whereas for the MIGT model, In particular, in the IGT model, when the index hits zero its diffusion coefficient is non-zero and the price process can and does go negative. Conversely, in the modified model, MIGT, when the index first hits zero, the diffusion coefficient is also zero and the process stops. This explains why the IG model gives positive value to put options with zero strike, whereas MIG correctly gives a zero value to these options. †RMSE is defined as where and are the predicted model price and the observed market price, respectively, of the ith option. ‡The terminology O/M/D refers to one volatility per option/maturity/day. §Note that the results for the Bachelier and modified Bachelier models are indistinguishable to the given level of accuracy. One needs to take 10 decimal places before the numerical performance of the models differs. This is because for market parameters, the event that the stock price hits zero is several standard deviations from the mean, and therefore the probability of this event is so small as to leave the model fits unaffected. †The exception is that the RMSE for out-of-the-money calls is very small. The reason for this is that these options have a very small price (much less than a dollar), so that even a large pricing error in relative terms translates to a small error in absolute terms. This effect also explains why the performance of IG and MIG improves markedly for far out-of-the-money calls. ‡Figlewski (Citation2000) obtains the opposite relationship, in that he finds that for both puts and calls errors decrease with strike. The fact that our conclusions are contradictory was one of the main motivations for our attempt to explain this relationship. The fact that we are able to give a plausible explanation in terms of stylized facts such as skew and smile gives us confidence that our numerical results are the correct ones, and that the entries in Figlewski's Exhibit V have indeed been mislabelled. Additional informationNotes on contributorsDavid HobsonFootnote¶ ¶Emails: dgh@maths.bath.ac.uk (DH) Tino KlugeFootnote¶ ¶tino@statslab.cam.ac.uk (TK)