Model-free stochastic collocation for an arbitrage-free implied volatility: Part I

Floc'h, Fabien Le; Oosterlee, Cornelis W.

doi:10.1007/s10203-019-00238-x

Cited by 4 publications

(5 citation statements)

References 38 publications

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“…While most-if not all-models used in finance satisfy it, it is possible to construct degenerate models violating it. Such an atypical behavior may also occur when interpolating option prices instead of implied volatilities (Le Floc'h and Oosterlee, 2019). Theorem 3.12.…”

Section: Pricing Formulae For European Optionsmentioning

confidence: 98%

“…While most—if not all—models used in finance satisfy it, it is possible to construct degenerate models violating it. Such an atypical behavior may also occur when interpolating option prices instead of implied volatilities (Le Floc'h and Oosterlee, 2019). Theorem For

\mathfrak {q}\in [1, \infty )

, let

\Psi \in \mathcal {P}_{q}^{-}

with

q \in [0, \mathfrak {q})

such that

\Psi ^{\prime } \in \mathcal {P}^{+}\cap \mathcal {P}_{q^{\prime }}^{-}

with

q^{\prime } \in [0, \mathfrak {q}-\frac{1}{2})

.…”

Section: Variance Swaps and The Log‐moment Formulamentioning

confidence: 99%

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The log‐moment formula for implied volatility

Raval

Jacquier

2023

Mathematical Finance

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We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that in the absence of arbitrage, if the underlying stock price at time T admits finite log‐moments for some positive q, the arbitrage‐free growth in the left wing of the implied volatility smile for T is less constrained than Lee's bound. The result is rationalized by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log‐returns, and requires no assumption for the underlying price to admit any negative moment. In this respect, the result can be derived from a model‐independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral–Fukasawa formula expressing variance swaps in terms of the implied volatility.

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Section: Pricing Formulae For European Optionsmentioning

confidence: 98%

\mathfrak {q}\in [1, \infty )

, let

\Psi \in \mathcal {P}_{q}^{-}

with

q \in [0, \mathfrak {q})

such that

\Psi ^{\prime } \in \mathcal {P}^{+}\cap \mathcal {P}_{q^{\prime }}^{-}

with

q^{\prime } \in [0, \mathfrak {q}-\frac{1}{2})

.…”

Section: Variance Swaps and The Log‐moment Formulamentioning

confidence: 99%

The log‐moment formula for implied volatility

Raval

Jacquier

2023

Mathematical Finance

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“…In many other methods this is solved by filtering the input prices or applying some other form of de-arbitraging. (Jäckel 2014;Kahale 2004;Le Floc'h and Osterlee 2019a).…”

Section: Optimization Problem For Finding the Densitymentioning

confidence: 99%

“…It has been mentioned in the literature (Le Floc'h and Osterlee 2019a) that short-term SPX500 options pose a challenge, particularly to stochastic volatility models and the similar SVI smile model (Gatheral and Jacquier 2014), since their volatility smiles are quite steep. We imported the market data from Table 11 in Le Floc'h and Osterlee (2019a), which corresponded to SPX500 1M (one month) options on 5 February 2018. We calculated Call and Put option prices from these market data for 75 strikes in the range between 1900 and 2900, using the Black (1976) model.…”

Section: Density Implied From Sandp 500 Option Pricesmentioning

confidence: 99%

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Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities

Guterding

2023

Risks

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We present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations as a system of linear equations. We show that the kernel matrix of this system is, in general, ill-conditioned, so that it cannot be solved for the discretized density using a naive approach. Instead, we construct a sparse model for the kernel matrix using singular value decomposition (SVD), which allows us not only to systematically improve the condition number of the kernel matrix, but also determines the computational effort and accuracy of our method. In order to allow for the treatment of realistic inputs that may contain arbitrage, we reformulate the system of linear equations as an optimization problem, in which the SVD-transformed density minimizes the error between the input prices and the arbitrage-free prices generated by our method. To further stabilize the method in the presence of noisy input prices or arbitrage, we apply an L1-regularization to the SVD-transformed density. Our approach, which is inspired by recent progress in theoretical physics, offers a flexible and efficient framework for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, without the need to explicitly specify a stochastic process, expansion basis functions or any other kind of model. We demonstrate the capabilities of our method in a number of artificial and realistic test cases.

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