Bounding the values of financial derivatives by the use of the moment problem
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Bounding the values of financial derivatives by the use of the moment problem Mariya Naumova1
· András Prékopa1
Accepted: 20 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Lower and upper bounds are derived on single-period European options under moment information, without assuming that the asset prices follow geometric Brownian motion, which is frequently untrue in practice. Sometimes the entire asset distribution is not completely known, sometimes it is known but the numerical calculation is easier by the use of the moments than the entire probability distribution. As geometric Brownian motion assumption regarding the asset prices is frequently untrue in practice. Some of the bounds are given by formulas, some are obtained by solving special linear programming problems. The bounds can be made close if a sufficiently large number of moments is used, and may serve for approximation of the values of financial derivatives. Keyword Discrete moment problem · European option pricing · Linear programming
1 Introduction Since the publication of the papers by Black and Scholes (1973) and Merton (1973) in 1973 the valuation of financial derivatives is one of the major areas of financial research. The valuation methodology of the above-mentioned authors is based on the assumption that the asset price S(t) follows a multiplicative Brownian motion with drift: S(t) = Seσ B(t)+μt , t ≥ 0,
(1.1)
where B(t), t ≥ 0 is a standard Brownian motion process, volatility σ > 0 and drift μ are constants and S = S(0) is the initial asset price. As it is well-known, S(t) satisfies the stochastic differential equation d S(t) σ2 = σ d B(t) + μ + dt. (1.2) S(t) 2 Based on Eq. (1.2) a parabolic P.D.E. was derived and solved for the price of the derivative that turned out to be independent of the drift parameter μ. In case of the European call and
B 1
Mariya Naumova [email protected] Rutgers The State University of New Jersey, Piscataway, New Jersey, USA
123
Annals of Operations Research
put options the obtained Black–Scholes–Merton (BSM) formulas, designated by c and p, respectively, are the following: c = Se−D(T −t) N (d1 ) − X e−r (T −t) N (d2 ), p = Xe
−r (T −t)
N (−d2 ) − Se
−D(T −t)
(1.3)
N (−d1 ),
(1.4)
where ln XS + (r − D + σ 2 /2)(T − t) , √ σ T −t √ ln XS + (r − D − σ 2 /2)(T − t) = d1 − σ T − t, d2 = √ σ T −t d1 =
(1.5) (1.6)
t is the current time, T the time to maturity, X the striking price, and r , D are the rate of interest and the rate of dividend (if paid continuously at fixed rate), respectively. In what follows we assume D = 0. This does not restrict generality of the method, but some modifications are needed in our formulas if dividend is taken into consideration. The results (1.3)–(1.6) can be obtained in a much simpler way if we assume that r = μ + σ 2 /2, or, what is the same, for the drift parameter we have: μ = r − σ 2 /2. This equation is a consequence of the assumption that the discounted price S(t)e−r t , t ≥ 0 is a martingale. In fact,
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