Volume 49 Number 3

Closed-form fair strikes for innovation-based variance-type and volatility-type contracts under a trending Ornstein–Uhlenbeck model

Nontawat Bunchak^a, Udomsak Rakwongwan^a,b,c, Phiraphat Sutthimat^b,c,*

 
ABSTRACT:     Discretesamplingisfundamentalinthevaluationofvariance-typeandvolatility-typecontracts, but explicit formulas are often lost when sampled log returns are serially dependent. This paper studies this issue in a trending Ornstein?Uhlenbeck model for the log price with deterministic coefficients. In this setting, the usual realized variance based on raw log returns is a quadratic form in correlated Gaussian variables and does not admit the standard chi-square reduction. Exact tractability is recovered by working with the exact one-step innovations of the sampled dynamics. These innovations are independent Gaussian variables on disjoint sampling intervals with explicit variances, so the associated standardized quadratic sum has an exact chi-square distribution and its square root has the corresponding chi distribution. This yields explicit formulas for the density, distribution function, Laplace transform, and moments of the standardized innovation statistic. In particular, the fair strike of the innovation-based variance-type contract is explicit for deterministic coefficients and an arbitrary discrete grid, while closed-form fair strikes for both the innovation-based variance-type and innovation-based volatility-type contracts are obtained in the constant-coefficient case on a uniform grid. Numerical results support the theory and illustrate the resulting fair strikes under several trend and coefficient specifications.

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* Corresponding author, E-mail: phiraphat.sut@ku.th

Received 9 Apr 2026, Accepted 17 Jun 2026