Asian option pricing via the Karhunen-Loève expansion

ALGORITHM: Asian option pricing via the Karhunen-Loève expansion
PROBLEM: Price discretely monitored Asian options over T monitoring points, where the underlying asset is modeled by a geometric Brownian motion.
IDEA: Prakash, Sun, Chakrabarti, Che, Dandapani, Herman, Kumar, Sureshbabu, Wood, Kerenidis and Pistoia consider the problem of pricing discretely monitored Asian options over T monitoring points, with the underlying asset modeled by a geometric Brownian motion, and give two quantum algorithms for it. Both achieve complexity poly-logarithmic in T and polynomial in 1/ε, where ε is the additive approximation error. One is obtained from an O(log T)-qubit semi-digital quantum encoding of the Brownian motion that allows exponentiation of the stochastic process; the other from analyzing classical Monte Carlo algorithms inspired by that same semi-digital encoding. The better of the two, the abstract states, reaches complexity Õ(1/ε³), where the tilde suppresses factors that are only poly-logarithmic in T and 1/ε. The authors state that their methods generalize to pricing options where the underlying asset price is a smooth function of a sub-Gaussian process and the payoff depends on the weighted time-average of that price. The abstract's title refers to the Karhunen-Loève expansion; it does not use the term Chebyshev polynomials anywhere, so this record makes no claim connecting the two algorithms it describes to any Chebyshev-polynomial construction.
REPORTED COST: Poly-logarithmic in T and polynomial in 1/ε for both quantum algorithms given, where T is the number of monitoring points and ε is the additive approximation error; the better of the two reaches Õ(1/ε³), with the tilde suppressing factors that are poly-logarithmic in T and 1/ε. One algorithm is built from an O(log T)-qubit semi-digital quantum encoding of the Brownian motion. The abstract states no further constant, no explicit qubit or gate count for the Monte-Carlo-derived algorithm, and no total circuit-resource count for either.
BASIS: The abstract of arXiv:2402.10132, the only source read for this record, sets up the task — "We consider the problem of pricing discretely monitored Asian options over $T$ monitoring points where the underlying asset is modeled by a geometric Brownian motion" — and states the headline bound: "We provide two quantum algorithms with complexity poly-logarithmic in $T$ and polynomial in $1/\epsilon$, where $\epsilon$ is the additive approximation error." (TeX rendered into Unicode/plain notation for the reader: $T$ is T and $1/\epsilon$ is 1/ε.) One algorithm comes from "an $O(\log T)$-qubit semi-digital quantum encoding of the Brownian motion that allows for exponentiation of the stochastic process" (rendered O(log T)-qubit), the other "by analyzing classical Monte Carlo algorithms inspired by the semi-digital encodings." The abstract names the better of the two: "The best quantum algorithm obtained using this approach has complexity $\widetilde{O}(1/\epsilon^{3})$ where the $\widetilde{O}$ suppresses factors poly-logarithmic in $T$ and $1/\epsilon$" (rendered Õ(1/ε³), the tilde marking exactly the suppression the abstract states). Beyond these expressions the abstract gives no further constant, no total qubit or gate count for either algorithm, and no cost figure for the Monte-Carlo-derived one beyond its complexity order. The Classiq index entry this record covers, applications/finance/brownian_chebyshev_polynomials, gives a directory path and a file list and states no bound of its own; its name refers to Chebyshev polynomials, a term this abstract never uses, so this field carries no claim about any Chebyshev-polynomial construction.
DEMONSTRATED BY: the Classiq library entry applications/finance/brownian_chebyshev_polynomials
PRIMARY SOURCE: Anupam Prakash, Yue Sun, Shouvanik Chakrabarti, Charlie Che, Aditi Dandapani, Dylan Herman, Niraj Kumar, Shree Hari Sureshbabu, Ben Wood, Iordanis Kerenidis, Marco Pistoia (2024), Quantum option pricing via the Karhunen-Loève expansion — https://arxiv.org/abs/2402.10132

This is a literature reference record, not an executable circuit.