Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

Mininni, Michele; Orlando, Giuseppe; Taglialatela, Giovanni

doi:10.1007/s10203-020-00305-8

Cited by 9 publications

(15 citation statements)

References 42 publications

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“…Using the results of Appendix A, we can easily compute the expressions of the kernels for the differential operators (25), resulting in:…”

Section: Slv Model Local Volatility (Lv) Models First Proposed Bymentioning

confidence: 99%

“…Volatility "is of particular importance since it is the main component of the option's price and because, among traders, options are quoted in terms of volatility rather than price" [27,5]. Volatility may be obtained by inverting the Black-Scholes-Merton (BSM) model for stock option pricing [25], [28]. However the BSM model, while providing a simple formula, relies on the assumption that the volatility is constant and that the stock return is normally distributed.…”

mentioning

confidence: 99%

“…In the new variables, the differential operators (10) have the same expressions as in (25), except for the case of K 1 , which now is written as:…”

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confidence: 99%

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Stochastic local volatility models and the Wei-Norman factorization method

Guerrero¹,

Orlando²

2022

DCDS-S

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<p style='text-indent:20px;'>In this paper, we show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Then, we compare the results of traditional Monte Carlo simulations with the explicit solutions obtained by said techniques. This approach is new in the literature and, in addition to reducing a non-autonomous problem into an autonomous one, allows for shorter time in numerical computations.</p>

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“…Using the results of Appendix A, we can easily compute the expressions of the kernels for the differential operators (25), resulting in:…”

Section: Slv Model Local Volatility (Lv) Models First Proposed Bymentioning

confidence: 99%

mentioning

confidence: 99%

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Stochastic local volatility models and the Wei-Norman factorization method

Guerrero¹,

Orlando²

2022

DCDS-S

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“…Its relatively simplicity and analytical tractability allowed the CIR model to become one of the most widely used short‐term structure models in financial institutions. Additional applications are stochastic volatility modeling in option pricing (Heston, 1993; Mininni et al, 2020) or default intensities in credit risk (Duffie, 2005).…”

Section: Literature Reviewmentioning

confidence: 99%

Interest rates forecasting: Between Hull and White and the CIR#—How to make a single‐factor model work

Orlando

Bufalo

2021

Journal of Forecasting

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In this work, we present our findings of the so‐called CIR#, which is a modified version of the Cox, Ingersoll, and Ross (CIR) model, turned into a forecasting tool for any term structure. The main feature of the CIR# model is its ability to cope with negative interest rates, cluster volatility, and jumps. By considering a dataset composed of money market interest rates during turmoil and calmer periods, we show how the CIR# performs in terms of directionality of rates and forecasting error. Comparison is carried out with a revamped version of the CIR model (denoted CIRadj), the Hull and White model, and the exponentially weighted moving average (EWMA) which is often adopted whenever no structure in data is assumed. To confirm the analysis, testing and validation is performed on both historical and ad hoc data with different metrics and clustering criteria.

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“…In this paper the authors extended some considerations and analyses regarding a closed-form formula to approximate the BS formula by means of an appropriate parameterization of the hyperbolic tangent, as shown by Mininni et al (2021). In particular, this paper presents further literature, with the purpose of calculating implied volatility and its use in decision making, and provides further graphical and numerical evidence.…”

Section: Introductionmentioning

confidence: 99%

A generalized derivation of the Black-Scholes implied volatility through hyperbolic tangents

Mininni¹,

Orlando²,

Tagliatela³

2022

Argumenta Oeconomica

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This article extends the previous research on the notion of a standardized call function and how to obtain an approximate model of the Black-Scholes formula via the hyperbolic tangent. Although the Black-Scholes approach is outdated and suffers from many limitations, it is still widely used to derive the implied volatility of options. This is particularly important for traders because it represents the risk of the underlying, and is the main factor in the option price. The approximation error of the suggested solution was estimated and the results compared with the most common methods available in the literature. A new formula was provided to correct some cases of underestimation of implied volatility. Graphic evidence, stress tests and Monte Carlo analysis confirm the quality of the results obtained. Finally, further literature is provided as to why implied volatility is used in decision making.

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Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

Cited by 9 publications

References 42 publications

Stochastic local volatility models and the Wei-Norman factorization method

Stochastic local volatility models and the Wei-Norman factorization method

Interest rates forecasting: Between Hull and White and the CIR#—How to make a single‐factor model work

A generalized derivation of the Black-Scholes implied volatility through hyperbolic tangents

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