Challenges in approximating the Black and Scholes call formula with hyperbolic tangents
- PDF / 622,676 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 17 Downloads / 155 Views
Challenges in approximating the Black and Scholes call formula with hyperbolic tangents Michele Mininni1 · Giuseppe Orlando1,2
· Giovanni Taglialatela1
Received: 21 November 2019 / Accepted: 13 August 2020 © The Author(s) 2020
Abstract In this paper, we introduce the concept of standardized call function and we obtain a new approximating formula for the Black and Scholes call function through the hyperbolic tangent. Differently from other solutions proposed in the literature, this formula is invertible; hence, it is useful for pricing and risk management as well as for extracting the implied volatility from quoted options. The latter is of particular importance since it indicates the risk of the underlying and it is the main component of the option’s price. That is what trading desks focus on. Further we estimate numerically the approximating error of the suggested solution and, by comparing our results in computing the implied volatility with the most common methods available in the literature, we discuss the challenges of this approach. Keywords Black and Scholes model · Hyperbolic tangent · Implied volatility JEL Classification G10 · C02 · C88 Mathematics Subject Classification 65-02 · 91G20 · 91G60
B
Giuseppe Orlando [email protected] Michele Mininni [email protected] Giovanni Taglialatela [email protected]
1
Department of Economics and Finance, Università degli Studi di Bari “Aldo Moro”, Largo Abbazia S. Scolastica, 70124 Bari, Italy
2
School of Science and Technologies, Università degli Studi di Camerino, Via M. delle Carceri 9, 62032 Camerino, Italy
123
M. Mininni et al.
1 Introduction For investors and traders, a key component in their decision making is to assess the risk they run. A common way to do so is to focus on the dispersion of returns. However, this measure has the problem that is computed on past performance and may have little to do with the current level of risk. Within the Black–Scholes framework (Black and Scholes 1973), later extended by Merton (1973), it is possible to identify a relation between the value of an asset and the option written on it. Though such a formula is subjected to strong criticism (Derman and Taleb 2005), and notwithstanding its pitfalls, the formula and its related approximations (Taylor or delta-gamma) are widely used in risk management (Estrella 1995) and “in many respects the story of the establishment of the Black–Scholes–Merton model (BSM) simply marks the emergence of contemporary financial risk management” (Millo and MacKenzie 2009). Many other models have been developed to address some of the BSM weaknesses, such as those proposed by Engle and Mustafa (1992), Heston (1993), Duan (1995) Ritchken and Trevor (1999), Christoffersen and Jacobs (2004) and Duan et al. (2006) as well as models based on neural networks (for a review see Mostafa et al. 2017) but those are beyond the scope of the present paper. This is for the simple reason that we share the same experience of most quants such as Paul Wilmott: when it comes to pric
Data Loading...
