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Orthogonal Polynomials, Concentration Principle, and the Black-Scholes Formula

Kronick, Zachary J
http://rave.ohiolink.edu/etdc/view?acc_num=osu1618322171292041

Abstract Details

2021, Master of Mathematical Sciences, Ohio State University, Mathematical Sciences.
The derivation of the Black-Scholes formula as a continuous extension of the discrete binomial model invokes two fundamental questions. The first of these questions deals with how we are to choose the upper and lower values over one time step of the binomial model. After choosing these values, we are then faced with the question of how to determine the probability p* that the stock moves to the upper value in the binomial model. In the case of the classic Black-Scholes formula, the upper and lower values are chosen symmetrically about the expected return of the stock. The probability p* is then determined using a replicating portfolio, or the mind of a risk-neutral investor. In this paper, we approach the first question by formulating a k-value Concentration Principle, which uses the roots of orthogonal polynomials to generate k values on which to concentrate a set of data. We then defer the thinking of a risk-neutral investor until the end of the derivation of the stock process, and answer the second question by utilizing the first moment of the return rate. In this way, we derive a more general Black-Scholes formula, one that considers skewness of the return rate random variable. The Black-Scholes formula thus becomes a marginal (limiting) formula corresponding to the very particular scenario when the return rate of the stock possesses third order symmetry with respect to its mean. Finally, we use the k-value Concentration Principle to construct an alternative option pricing model, derived from a symmetric trinomial model, which utilizes information about higher order moments.
Aurel Stan (Advisor)
Rodica Costin (Committee Member)
53 p.
Mathematics
Black-Scholes; binomial model; orthogonal polynomials; concentration principle; option pricing

Recommended Citations

Citations

  • Kronick, Z. J. (2021). Orthogonal Polynomials, Concentration Principle, and the Black-Scholes Formula [Master's thesis, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618322171292041

    APA Style (7th edition)

  • Kronick, Zachary. Orthogonal Polynomials, Concentration Principle, and the Black-Scholes Formula. 2021. Ohio State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1618322171292041.

    MLA Style (8th edition)

  • Kronick, Zachary. "Orthogonal Polynomials, Concentration Principle, and the Black-Scholes Formula." Master's thesis, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618322171292041

    Chicago Manual of Style (17th edition)

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This open access ETD is published by The Ohio State University and OhioLINK.