Good jump, bad jump, and option valuation

This paper proposes a simple but rich framework to directly price volatility index (VIX) futures by applying the heterogeneous autoregressive structure and asymmetric jumps to the logarithm of the VIX. Compared with other discrete‐time models, our model imposes fewer parameter constraints. The analytical solution is also free from time‐consuming and sometimes unstable numerical integration. Empirical results suggest that our model can significantly reduce pricing errors compared with existing models using realized variance, both in‐ and out‐of‐sample. The improvement indicates that besides looking for a better measure of current volatility, it is also important to utilize information embedded in the VIX itself.

“…

ε_{i} s

are i.i.d. and follow double exponential distribution

K o u (p, q, η_{1}, η_{2})

with the density function:

f_{ε} (x) = p η_{1} e^{- η_{1} x} 1_{MathClass-open{x \geq 0 MathClass-close}} + q η_{2} e^{η_{2} x} 1_{MathClass-open{x < 0 MathClass-close}}

η_{1} > 1, η_{2} > 0 and p + q = 1

1 ∕ η_{1}

and

1 ∕ η_{2}

, respectively.…”

Section: The Modelmentioning

confidence: 99%

“…To keep the model concise, we do not pursue this direction further. For the extended model in the appendix, updating logVIX series requires filtering out the unobservable continuous component and jump component, which can be done using particle filters mentioned in Ornthanalai (2014) and Yang (2018).…”

Section: Data and Estimationmentioning

confidence: 99%

A short cut: Directly pricing VIX futures with discrete‐time long memory model and asymmetric jumps

Yin

Wang

2020

“…Parameters

p_{up}

and

p_{down}

represent the probability of upward and downward jumps.

J_{t}

is widely used in option pricing models such as Kou and Wang (2004) and Yang (2018). The Kou's specification can be reduced to no jumps (

J_{t} = 0

) or symmetric jumps (

η_{up} = η_{down}

p_{up} = p_{down}

), both of them can be tested with real data.…”

Section: The Model Of Vix Dynamicsmentioning

confidence: 99%

Do VIX futures contribute to the valuation of VIX options?

Tong

Huang

Wang

2021

As the volatility index (VIX) is nontradable, most investors use the exchange-traded VIX futures to hedge their exposures in VIX options. However, the information role of VIX futures in pricing VIX options is not fully explored empirically. This paper derives two types of VIX option pricing formula using VIX index and VIX futures as state variables accordingly based on a simple discrete-time VIX dynamics with long memory and asymmetric jumps. Empirical results show that models utilizing VIX futures significantly outperform competing models based on S&P 500 index (SPX) returns, realized volatility, or the VIX index itself. Our findings are robust in out-of-sample.

“…In the literature, there are many papers discussing pricing derivatives (but not yet exchange options) with jump diffusions, for example, Feng and Linetsky (2008), Mina et al (2015), Yang (2018), W. Liu and Zhu (2019) and among many others. There is also empirical evidence in the literature (see, e.g., Eraker, 2004, Chang et al, 2019), showing there are jumps in asset prices.…”

Section: Introductionmentioning

confidence: 99%

Approximate pricing of American exchange options with jumps

Lian

Elliott

Kalev

et al. 2022

This paper presents a numerical method to price American exchange options based on jump‐diffusion processes. We first derive a closed‐form expression for the value of European exchange options, then decompose the value function of an American exchange option into a European counterpart, and an early exercise premium that is derived analytically. The early exercise boundary for an American exchange option approximately satisfies an algebraic equation that can be quickly numerically solved. Consequently, a formula is obtained for efficiently pricing American exchange options. The numerical results reveal that our pricing formula is robust and accurate.