A chaos expansion approach for the pricing of contingent claims

Abstract:In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H > 1/2, (ii) is proven to be satisfied by a rough volatility model with H < 1/2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist.

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“…For the proof of Lemma 1, we apply the chaos expansion approach proposed by [12]. Namely, consider the solution (6), i.e.,…”

Section: Appendix a Proof Of Lemmamentioning

confidence: 99%

“…If the volatilities σ n (t) are deterministic functions and σ t = max n σ n t ∈ L 2 ([0, t]) is sufficiently small, then Proposition 2.2 of [12] assures that the sum of the iterated integrals converges very quickly.…”

Section: Appendix a Proof Of Lemmamentioning

confidence: 99%

A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

Funahashi

Kijima

2017

Fractal Fract

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“…In this paper, we apply the chaos expansion approach recently developed by Funahashi and Kijima (2013) to approximate the random variable V T by a truncated sum of iterated Ito stochastic integrals. We then derive the probability density function (PDF for short) of the approximated random variable.…”

Section: The Setupmentioning

confidence: 99%

“…In this paper, we propose an approximation method based on the chaos expansion approach, recently proposed by Funahashi and Kijima (2013), for the pricing of Asian basket options. Through ample numerical examples, we show that the accuracy of our approximation remains quite high even for a complex basket option with long maturity and high volatility under various diffusion models.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is organized as follows. After explaining our problem concisely in the next section, we extend the chaos expansion approach of Funahashi and Kijima (2013) to the multi-asset case in Section 3. Each asset price is approximated by a truncated sum of iterated Ito stochastic integrals, and the Asian basket variable is also described, after the change of order of integration, as a sum of iterated Ito stochastic integrals.…”

Section: Introductionmentioning

confidence: 99%

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An Extension of the Chaos Expansion Approximation for the Pricing of Exotic Basket Options

Funahashi

Kijima

2013

Applied Mathematical Finance

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Funahashi and Kijima (2013) have proposed an approximation method based on the Wiener-Ito chaos expansion for the pricing of European-style contingent claims. In this paper, we extend the method to the multi-asset case with general local volatility structure for the pricing of exotic basket options such as Asian basket options. Through ample numerical experiments, we show that the accuracy of our approximation remains quite high even for a complex basket option with long maturity and high volatility.

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Does the Hurst index matter for option prices under fractional volatility?

Funahashi

Kijima

2016

Ann Finance

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A chaos expansion approach for the pricing of contingent claims

Cited by 17 publications

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A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

An Extension of the Chaos Expansion Approximation for the Pricing of Exotic Basket Options

Does the Hurst index matter for option prices under fractional volatility?

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