Abstract
Using the Donsker–Prokhorov invariance principle, we extend the Kim–Stoyanov–Rachev–Fabozzi option pricing model to allow for variably-spaced trading instances, an important consideration for short-sellers of options. Applying the Cherny–Shiryaev–Yor invariance principles, we formulate a new binomial path-dependent pricing model for discrete- and continuous-time complete markets where the stock price dynamics depends on the log-return dynamics of a market influencing factor. In the discrete case, we extend the results of this new approach to a financial market with informed traders employing a statistical arbitrage strategy involving trading of forward contracts. Our findings are illustrated with numerical examples employing US financial market data. Our work provides further support for the conclusion that any option pricing model must preserve valuable information on the instantaneous mean log-return, the probability of the stock’s upturn movement (per trading interval), and other market microstructure features.
Highlights
The Donsker–Prokhorov invariance principle (DPIP), known as the Functional Limit Theorem, is a fundamental result in the theory of stochastic processes and a limit theorem for sequences of random variables.1 Cox et al (1979) were the first to use DPIP in their seminal Cox–Ross–Rubinstein (CRR)-binomial option pricing model.2 there are several extensions of the CRR model3, rigorous proofs of the corresponding limit results leading to continuous-time option pricing formula are often not provided.4 In this paper, we provide the proofs for various extensions of DPIP to obtain a variety of new binomial option pricing models
Using the Donsker–Prokhorov invariance principle, we extend the Kim–Stoyanov–Rachev –Fabozzi option pricing model to allow for variably-spaced trading instances, an important consideration for short-sellers of options
Inclusion of more information on market microstructure will inevitably lead to better option pricing models
Summary
The Donsker–Prokhorov invariance principle (DPIP), known as the Functional Limit Theorem, is a fundamental result in the theory of stochastic processes and a limit theorem for sequences of random variables.1 Cox et al (1979) were the first to use DPIP in their seminal Cox–Ross–Rubinstein (CRR)-binomial option pricing model. there are several extensions of the CRR model, rigorous proofs of the corresponding limit results leading to continuous-time option pricing formula are often not provided. In this paper, we provide the proofs for various extensions of DPIP to obtain a variety of new binomial option pricing models. Step 1: Introduce a continuous-time arbitrage-free model for the underlying stock price, for example, a geometric Brownian motion with instantaneous mean log-return μ > r and volatility σ > 0, where r is the risk-free rate. Addressing the misconception about binomial option pricing being independent of μ and p∆t is the main motivation for the results in Sections 2 and 3 in this paper In these two sections, general binomial option pricing formulas are derived and the corresponding continuous-time limits are shown by applying DPIP. Billingsley (1962); Silvestrov (2004), Chapter 4; Gut (2009), Chapter 5; Tanaka (2017), Chapter 2; Cherny et al (2003); and Crimaldi et al (2019). 14 The options data are from CBOE options on MSFT, see https://datashop.cboe.com/option-trades
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