Access the full text.

Sign up today, get DeepDyve free for 14 days.

Risks
, Volume 11 (2) – Jan 17, 2023

/lp/multidisciplinary-digital-publishing-institute/a-generalized-model-for-pricing-financial-derivatives-consistent-with-dHFhC8K641

- Publisher
- Multidisciplinary Digital Publishing Institute
- Copyright
- © 1996-2023 MDPI (Basel, Switzerland) unless otherwise stated Disclaimer Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. Terms and Conditions Privacy Policy
- ISSN
- 2227-9091
- DOI
- 10.3390/risks11020024
- Publisher site
- See Article on Publisher Site

risks Opinion A Generalized Model for Pricing Financial Derivatives Consistent with Efﬁcient Markets Hypothesis—A Reﬁnement of the Black-Scholes Model Jussi Lindgren Ministry of Finance, 00023 Helsinki, Finland; justicus1982@gmail.com Abstract: This research article provides criticism and arguments why the canonical framework for derivatives pricing is incomplete and why the delta-hedging approach is not appropriate. An argument is put forward, based on the efﬁcient market hypothesis, why a proper risk-adjusted discount rate should enter into the Black-Scholes model instead of the risk-free rate. The resulting pricing equation for derivatives and, in particular, the formula for European call options is then shown to depend explicitly on the drift of the underlying asset, which is following a geometric Brownian motion. It is conjectured that with the generalized model, the predicted results by the model could be closer to real data. The adjusted pricing model could partly also explain the mystery of volatility smile. The present model also provides answers to many ﬁnance professionals and academics who have been intrigued by the risk-neutral features of the original Black-Scholes pricing framework. The model provides generally different fair values for ﬁnancial derivatives compared to the Black-Scholes model. In particular, the present model predicts that the original Black-Scholes model tends to undervalue for example European call options. Keywords: options pricing; financial derivatives; efficient market hypothesis; martingale; Feynman-Kac; Black-Scholes Citation: Lindgren, Jussi. 2023. A Generalized Model for Pricing 1. Introduction Financial Derivatives Consistent with Efﬁcient Markets Hypothesis—A Pricing of ﬁnancial derivatives was revolutionized in 1973, when the famous Black- Reﬁnement of the Black-Scholes Scholes framework was introduced (Black and Scholes 1973). The value of, say a European Model. Risks 11: 24. https:// call option, is given by a linear parabolic partial differential equation, and an explicit doi.org/10.3390/risks11020024 formula is available to compute the value of the option, given parameters. The explicit formulas are obtained by transforming the Black-Scholes partial differential equation (PDE) Academic Editors: into a constant coefﬁcient PDE and using Fourier methods, for example. The Black-Scholes Mogens Steffensen and PDE can also be seen as a Hamilton-Jacobi-Bellman equation for a certain stochastic control Tianyang Wang problem (Lindgren 2020). The parameters needed are the risk-free rate, volatility, exercise Received: 5 September 2022 price, and time to maturity. In empirical terms, the Black-Scholes model does not predict Revised: 24 October 2022 true market values of options, so as a scientiﬁc model, it performs rather poorly. However, Accepted: 13 January 2023 it has been thought that it gives a reasonable benchmark for traders and ﬁnancial markets Published: 17 January 2023 professionals, as well as for risk managers. One classical narrative against the assumptions of the model comes from (Bergman 1982) and (Musiela and Rutkowski 2005), where it is argued that the hedging portfolio is not self-ﬁnancing in the ﬁrst place in the original model. The present approach goes further and claims that the whole premise of the model Copyright: © 2023 by the author. is too narrow and in particular the argument related to delta-hedging is almost irrelevant Licensee MDPI, Basel, Switzerland. to any real speculant, hedger, or investment professional. This article is an open access article One of the hardest concepts to understand intuitively within the Black-Scholes model distributed under the terms and is indeed the fact that, in the Black-Scholes formulas, the fair price of the option does conditions of the Creative Commons not depend on the drift of the underlying asset. This is a result of the framework, even Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ though the geometric Brownian motion for the underlying asset is assumed to have some 4.0/). non-zero drift. The prediction of the model is rather counterintuitive—one would assume Risks 2023, 11, 24. https://doi.org/10.3390/risks11020024 https://www.mdpi.com/journal/risks Risks 2023, 11, 24 2 of 5 that placing one’s bet on an option with a higher positive drift would affect the value of the respective call option in a positive manner. The Black-Scholes model in effect works as if the market was risk-neutral as a whole, or in other words, as if the drift of the asset was equal to the risk-free rate. The mathematical reason why the true drift is irrelevant in the Black-Scholes model is nevertheless a direct consequence of the hedging portfolio approach and the reason is the following: in the hedging portfolio, one is shorting the underlying asset by an amount which is the delta of the option. In terms of the hedging portfolio, if the asset has a higher positive drift, the call option goes up in terms of value, but the short position goes down in terms of value and the effects cancel each other out exactly. That is the reason why the hedging portfolio should be instantaneously risk free and yield the risk free rate. This is the core of the Black-Scholes reasoning. The value of the option is determined as a kind of residual in order to keep the hedging portfolio locally risk-free. There are, however, at least two main problems with this approach. First, the fair price of a call option given by the Black-Scholes model is thus the fair value merely for the hedging portfolio holder, and this is the main problem with the model. If a randomly chosen market participant buys a call option on say a stock of a blue-chip company, it is unreasonable to assume that he or she holds a short position on the company exactly an amount corresponding to the delta of the call option. If a speculant holds a long call option, his portfolio is probably not completely offset by shorting the respective underlying asset. On top of this conceptual problem, there is the well known technical challenge in terms of self-ﬁnancing portfolios. The key assumption in the Black-Scholes model is the assumed self-ﬁnancing of the hedging portfolio, in other words, it is assumed that there is no net ﬂows of funding in or out of the hedging portfolio. A self-ﬁnancing delta-hedging portfolio in this case means that the holdings of the option and underlying are not changed, i.e., the changes in the value of the portfolio come purely from changes in the value of the option and the underlying asset. However, this seems to be false and is quite clearly argumented in (Bergman 1982) and (Bartels 1995). This possibly fundamentally ﬂawed assumption of a self-ﬁnancing hedging portfolio might mean in itself that the rate of return on the hedged portfolio is not truly riskless. If this is indeed the case, the Black-Scholes pricing model assumptions are fundamentally problematic. Then again, the false assumption of a self-ﬁnancing portfolio does not destroy the Black-Scholes model as such, as is argued in (Bana 2007). On top of this, naturally, the assumption of geometric Brownian motion for the underlying asset is itself perhaps not fully correct, but is less of concern. Whether a geometric Brownian motion is a proper model for asset price dynamics is an important issue, but it does not affect qualitatively the reasoning within the Black-Scholes model. In the literature concerning imperfect hedging portfolios due to market incompleteness, the value of the contingent claim is shown to lie within some hedging bounds, see for example (Hao 2008). On the other hand, costly short-selling has been shown to affect the bid-ask-spreads of options, see (Atmaz and Basak 2019). An equal risk pricing rule in incomplete markets was developed in (Guo and Zhu 2017), and further developed in (Marzban et al. 2022). For equal risk pricing using deep learning, see (Carbonneau and Godin 2021). These approaches do not however consider the fundamental problem of delta-hedging, so that usually holding a long call option, the portfolio is probably not completely offset by shorting the respective underlying asset. 2. Properly Anticipated Prices in the Options Pricing Framework—A General Framework for Pricing Derivatives without the Hedging Portfolio The present model argues that the canonical framework for options pricing based on Black-Scholes pricing is to be generalized to be consistent with the efﬁcient markets hypothesis as put forward by Paul Samuelson (Samuelson 1965; Samuelson 1973). The traditional approach of delta-hedging is therefore not suitable in general; instead, we need to consider ﬁrst what the appropriate discount rate of an efﬁcient ﬁnancial market is. The Risks 2023, 11, 24 3 of 5 underlying asset following geometric Brownian motion is not a martingale due to its drift as such, but properly discounted it is. To require the martingale property from the discounted price process is in line with (Samuelson 1973) and is a mathematical consequence of the efﬁcient market hypothesis (Fama 1965; Fama 1970). The key assumption thus is that the properly discounted process must be a martingale in an efﬁcient market. Consider an asset such as a common stock. Suppose that an asset price X follows geometric Brownian motion: dX = mX dt + sX dW , (1) t t t t with some drift of instantaneous return m > 0 and volatility s > 0. W is a standard Brownian motion, and we consider valuing a generic ﬁnancial derivative written on the asset. In line with the formulation of the efﬁcient market hypothesis by Samuelson (1973), see also the review article (LeRoy 1989), we now require that the discounted price process is a martingale. Given that for a geometric Brownian motion, the expectation for the price at future time T > 0 is given by: m(Tt) E X = X e , (2) ( ) t T t where E is an expectation operator with respect to the probability measure generated by the Brownian motion. We see, immediately, that we need to introduce a discount factor m(Tt) e for the market in order to obtain: m(Tt) X = E e X (3) t t T This requirement of market efﬁciency can be interpreted as follows: the expected discounted price of an asset at future time must be equal to the current price of the asset. In other words, the risk aversion preferences of the market as a whole are reﬂected in the discount factor and all relevant information is already reﬂected in the current price of the asset. Consider now a ﬁnancial derivative written on the asset. The ﬁnancial derivative has a payoff at terminal time T and a respective payoff function j(X ). In line with above, we are looking the fair value independent of individual risk preferences of market participants, instead we discount the derivative payoff using the market discount function above and evaluate the expectation according to the physical or real probability measure: m(Tt) C(x(t), t) = E e j(X ) , (4) t T where C(x(t), t) is the fair value of the ﬁnancial derivative at time t when the asset has some known price x(t). It is straightforward to use the Feynman-Kac formula (Pavliotis 2014) to write down the partial differential equation describing the evolution of the value of the ﬁnancial derivative: ¶C ¶C 1 ¶ C 2 2 (x, t) + mx (x, t) + s x (x, t) mC(x, t) = 0. (5) ¶t ¶x 2 ¶x With the initial condition C(x, T) = j(X ). Notice that the only difference to the canonical Black-Scholes partial differential equation is that the risk-free rate is replaced with the discount rate reﬂecting the efﬁciency of the ﬁnancial market. As an instruc- tive example, consider now pricing a plain vanilla European call option. The payoff is j(X ) = max(X K, 0), where K > 0 is the exercise or strike price of the call option T T maturing at time T > t. As everything else is the same as in the Black-Scholes pricing PDE, Risks 2023, 11, 24 4 of 5 we can deduce the price for a European call option easily by using the well-known formulas and by just replacing the risk-free rate in the formulas with the drift of the underlying asset: m(Tt) C(X , t) = N(d )X N(d )Ke (6) t 1 t 2 where N is the cumulative distribution function of the standard normal variable, and 1 X 1 d = p Log + m + s (T t) (7) K 2 s T t d = d s T t. (8) 2 1 The obtained pricing formula is similar as presented already by James Boness in (Boness 1964). It is instructive to note that now the price of the European call depends explicitly on the drift of the asset, as it should intuitively be and the Rho of the option for the call is positive so that an increase in the drift increases the value of the European long call. The only difference with the canonical Black-Scholes model is therefore that the risk-free rate is replaced by the drift of the underlying asset performing geometric Brownian motion. As the higher discount rate effectively affects the strike price in (6), the present approach could also alleviate the volatility smile phenomenon as described in (Derman and Miller 2016). 3. Discussion and Conclusions It is argued that based on the assumption that in an efﬁcient market the risk-adjusted discounted expected price of an asset following geometric Brownian motion should be the current price, it is deduced that the correct risk-adjusted discount rate should be the drift of the asset process. These assumptions will lead to derivatives pricing, where the fair value of the ﬁnancial derivative can be evaluated as a conditional expectation, discounted at the above rate reﬂecting market efﬁciency. The fair value for an option can then be solved by using the Feynman-Kac formula, leading to a modiﬁed Black-Scholes PDE, where the drift of the underlying process is explicitly present. The results thus suggest that the approach based on the hedging portfolio is too limited. The fair price of an option in the Black-Scholes approach is based on the idea of a hedging portfolio. The value of the ﬁnancial derivative is forced to be such that the hedging portfolio yields exactly the risk-free rate. In the present approach, it is argued that the delta-hedge approach is not sufﬁcient in general, as it implicitly requires that the option holders have that delta-hedged portfolio. For an investor with the delta-hedging portfolio, it is indeed true that the drift of the underlying asset does not make any difference. If the underlying goes up, the short position on the underlying goes down and the call option gains in value; these effects cancel each other out. However, for a general investor, there is no perfectly hedging portfolio, and the increase in the drift of the asset has an effect on the value of the call option, for example. In this article, it is shown mathematically, based on the theory of efﬁcient markets, why the drift should in fact matter when pricing ﬁnancial derivatives. The present model predicts that for a generic call option holder, the price of a European call option depends explicitly on the drift of the underlying asset. The results might lead to better empirical results when comparing the actual prices of options in the markets and the theoretical prices predicted by the present model. Furthermore, volatility smile (Derman and Miller 2016) should be examined through the lens of this extended model. The higher discount rate compared to the original Black-Scholes model implies that the fair value of the ﬁnancial derivative depends in general on the drift of the underlying asset performing geometric Brownian motion. Therefore, if an option holder is long in a European call option, the value of the call is higher for underlying assets, which have higher instantaneous return or drift. The reason is simply that the fair value of the option is not valued in terms of a hedging portfolio, but instead demanding that the market risk aversion is such that Risks 2023, 11, 24 5 of 5 assets with a drift are martingales when discounted properly. The well known discrepancy between option market data and the values predicted by the Black-Scholes model has been empirically veriﬁed many times. The practical beneﬁt of the present model for the investor is that the model could give more accurate fair values for ﬁnancial derivatives. It should also be noted that if the present approach performs better than the original Black-Scholes model, implied volatilities must change as well. The present model includes the drift of the underlying asset, which can be estimated from data, as has been the case for the risk-free interest rate in the original model. The present model suggests that when pricing ﬁnancial derivatives with an underlying following geometric Brownian motion, one should use the pricing PDE (5) instead of the original Black-Scholes PDE. In similar fashion, for European call options, one should use pricing Formulas (6)–(8). Finally, it is indeed interesting to note that the pricing formula in the present approach is similar to the pricing formula of (Boness 1964), see also the discussion in (O’Brien and Selby 1986). Funding: This research received no external funding. Data Availability Statement: Not applicable. Conﬂicts of Interest: The author declares no conﬂict of interest. References Atmaz, Adem, and Suleyman Basak. 2019. Option prices and costly short-selling. Journal of Financial Economics 134: 1–28. [CrossRef] Bana, Gergei. 2007. A note on portfolios with risk-free internal gains. Expositiones Mathematicae 25: 83–93. [CrossRef] Bartels, Hans-Jochen. 1995. The hypotheses underlying the pricing of options. Presented at the 5th International AFIR Colloquium, Brussels, Belgium, September 7–8; vol. 1, pp. 3–15. Bergman, Yaacov Zvi. 1982. Pricing of Contingent Claims in Perfect and Imperfect Markets. Ph.D. thesis, University of California, Berkeley, CA, USA. Black, Fischer, and Myron Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–59. [CrossRef] Boness, J. James. 1964. Elements of a Theory of Stock-Option Value. Journal of Political Economy 72: 163–175. [CrossRef] Carbonneau, Alexandre, and Frédéric Godin. 2021. Equal risk pricing of derivatives with deep hedging. Quantitative Finance 21: 593–608. [CrossRef] Derman, Emanuel, and Michael B. Miller. 2016. The Volatility Smile. Hoboken: John Wiley & Sons. Fama, Eugene F. 1965. The Behavior of Stock-Market Prices. The Journal of Business 38: 34–105. [CrossRef] Fama, Eugene. 1970. Efﬁcient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance 25: 383–417. [CrossRef] Guo, Ivan, and Song-Ping Zhu. 2017. Equal risk pricing under convex trading constraints. Journal of Economic Dynamics and Control 76: 136–51. [CrossRef] Hao, Tao. 2008. Option pricing and hedging bounds in incomplete markets. Journal of Derivatives and Hedge Funds 14: 78–89. [CrossRef] LeRoy, Stephen F. 1989. Eﬁicient Capital Markets and Martingales. Journal of Economic Literature 27: 1583–621. Lindgren, Jussi. 2020. Efﬁcient Markets and Contingent Claims Valuation: An Information Theoretic Approach. Entropy 22: 1283. [CrossRef] [PubMed] Marzban, Saeed, Erick Delage, and Jonathan Yu-Meng Li. 2022. Equal risk pricing and hedging of ﬁnancial derivatives with convex risk measures. Quantitative Finance 22: 47–73. [CrossRef] Musiela, Marek, and Marek Rutkowski. 2005. Martingale Methods in Financial Modelling. Milan: Springer. O’Brien, Thomas J., and Michael J. P. Selby. 1986. Option pricing theory and asset expectations: A review and discussion in tribute to James Boness. The Financial Review 21: 399–418. [CrossRef] Pavliotis, Grigorios A. 2014. Stochastic Processes and Applications. New York: Springer Science+Business Media. Samuelson, Paul A. 1965. Proof That Properly Anticipated Prices Fluctuate Randomly. Industrial Management Review 6: 41–49. Samuelson, Paul A. 1973. Proof That Properly Discounted Present Values of Assets Vibrate Randomly. Bell Journal of Economics 4: 369–74. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Risks – Multidisciplinary Digital Publishing Institute

**Published: ** Jan 17, 2023

**Keywords: **options pricing; financial derivatives; efficient market hypothesis; martingale; Feynman-Kac; Black-Scholes

Loading...

You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!

Read and print from thousands of top scholarly journals.

System error. Please try again!

Already have an account? Log in

Bookmark this article. You can see your Bookmarks on your DeepDyve Library.

To save an article, **log in** first, or **sign up** for a DeepDyve account if you don’t already have one.

Copy and paste the desired citation format or use the link below to download a file formatted for EndNote

Access the full text.

Sign up today, get DeepDyve free for 14 days.

All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.