EQUILIBRIUM VALUATION OF CURRENCY OPTIONS UNDER A DISCONTINUOUS MODEL WITH CO-JUMPS

2020 ◽  
pp. 1-19
Author(s):  
Yu Xing ◽  
Yuhua Xu ◽  
Huawei Niu
Keyword(s):  
Implied Volatility ◽  
Option Price ◽  
Closed Form Solution ◽  
Continuous Model ◽  
Form Solution ◽  
Jump Processes ◽  
Currency Options ◽  
Currency Option ◽  
Jump Intensity ◽  
Discontinuous Model

Abstract In this paper, we study the equilibrium valuation for currency options in a setting of the two-country Lucas-type economy. Different from the continuous model in Bakshi and Chen [1], we propose a discontinuous model with jump processes. Empirical findings reveal that the jump components in each country's money supply can be decomposed into the simultaneous co-jump component and the country-specific jump component. Each of the jump components is modeled with a Poisson process whose jump intensity follows a mean reversion stochastic process. By solving a partial integro-differential equation (PIDE), we get a closed-form solution to the PIDE for a European call currency option. The numerical results show that the derived option pricing formula is efficient for practical use. Importantly, we find that the co-jump has a significant impact on option price and implied volatility.

A closed-form approximation formula for pricing European options under a three-factor model

2021 ◽  
pp. 1-27
Author(s):  
Hye-mee Kil ◽  
Jeong-Hoon Kim
Keyword(s):  
Closed Form ◽  
Implied Volatility ◽  
Factor Model ◽  
Closed Form Solution ◽  
Form Solution ◽  
Approximation Formula ◽  
European Options ◽  
Implied Volatility Surface ◽  
Volatility Surface ◽  
Three Factor

Abstract The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

scholarly journals DECOMPOSITION FORMULA FOR JUMP DIFFUSION MODELS

2018 ◽  
Vol 21 (08) ◽  
pp. 1850052
Author(s):  
R. MERINO ◽  
J. POSPÍŠIL ◽  
T. SOBOTKA ◽  
J. VIVES
Keyword(s):  
Option Pricing ◽  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heston Model ◽  
Approximation Formula ◽  
Option Prices ◽  
Currency Options ◽  
Decomposition Formula

In this paper, we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs [(2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation, Finance and Stochastics 16 (3), 403–422, doi: https://doi.org/10.1007/s00780-012-0177-0 ] for Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ] SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models — models utilizing a variance process postulated by Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ]. In particular, we inspect in detail the approximation formula for the Bates [(1996), Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, The Review of Financial Studies 9 (1), 69–107, doi: https://doi.org/10.1093/rfs/9.1.69 ] model with log-normal jump sizes and we provide a numerical comparison with the industry standard — Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behavior under a specific SVJ model.

scholarly journals Optimal derivative liquidation timing under path-dependent risk penalties

2015 ◽  
Vol 02 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Tim Leung ◽  
Yoshihiro Shirai
Keyword(s):  
Optimal Stopping ◽  
Existence And Uniqueness ◽  
Market Price ◽  
Option Price ◽  
Closed Form Solution ◽  
Form Solution ◽  
Optimal Timing ◽  
Combined Effects ◽  
Optimal Stopping Problem ◽  
Path Dependent

This paper studies the risk-adjusted optimal timing to liquidate an option at the prevailing market price. In addition to maximizing the expected discounted return from option sale, we incorporate a path-dependent risk penalty based on shortfall or quadratic variation of the option price up to the liquidation time. We establish the conditions under which it is optimal to immediately liquidate or hold the option position through expiration. Furthermore, we study the variational inequality associated with the optimal stopping problem, and prove the existence and uniqueness of a strong solution. A series of analytical and numerical results are provided to illustrate the nontrivial optimal liquidation strategies under geometric Brownian motion (GBM) and exponential Ornstein–Uhlenbeck models. We examine the combined effects of price dynamics and risk penalty on the sell and delay regions for various options. In addition, we obtain an explicit closed-form solution for the liquidation of a stock with quadratic penalty under the GBM model.

scholarly journals Fourier Transform of the Continuous Arithmetic Asian Options PDE

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Zieneb Ali Elshegmani ◽  
Rokiah Rozita Ahmad
Keyword(s):  
Fourier Transform ◽  
Direct Method ◽  
Option Price ◽  
Closed Form Solution ◽  
Form Solution ◽  
Three Dimensions ◽  
Asian Option ◽  
Asian Options ◽  
Partial Differential ◽  
Constant Coefficients

Price of the arithmetic Asian options is not known in a closed-form solution, since arithmetic Asian option PDE is a degenerate partial differential equation in three dimensions. In this work we provide a new method for computing the continuous arithmetic Asian option price by means of partial differential equations. Using Fourier transform and changing some variables of the PDE we get a new direct method for solving the governing PDE without reducing the dimensionality of the PDE as most authors have done. We transform the second-order PDE with nonconstant coefficients to the first order with constant coefficients, which can be solved analytically.

scholarly journals Currency Options, Implied Interest Rates and Inflation Targeting

2019 ◽  
Vol 11 (2) ◽  
pp. 119
Author(s):  
Helena G. Keefe ◽  
Erick W. Rengifo
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
Implied Volatility ◽  
Inflation Targeting ◽  
Emerging Market ◽  
Option Price ◽  
Control Group ◽  
Inflation Expectations ◽  
Currency Options ◽  
Market Expectations

The currency option price is a powerful tool used regularly to determine market expectations on volatility in currencies using the implied volatility measure. This research tests and analyzes whether similar inferences can be made regarding interest rate and inflation expectations. Using historical options data, we derive and analyze implied interest rates during non-inflation targeting (non-IT) and inflation targeting (IT) periods for Australia, Canada, and the United Kingdom. We compare the results to a control group of countries that had not yet adopted inflation targeting during the period under study: Germany, Japan and Switzerland. Our results show that options prices can provide insights on market expectations on interest rates, that the adoption of inflation targeting strengthens the relationship between market expectations and inflation, and that shocks in interest rates and inflation lead to higher implied interest rates. In determining the potential uses of implied interest rates derived from currency options prices, our goal is not to replace the Federal Funds futures or equivalent tools in advanced economies, rather to present the usefulness of currency options as a tool to provide information to policymakers in emerging market economies. Central banks, such as the Banco Central de Colombia and Banco de Mexico, have been using currency options as tools for foreign exchange intervention or reserve accumulation/decumulation since the early 2000’s, and options markets in these economies have grown rapidly since then. Therefore, establishing the usefulness of implied interest rate measures derived from currency options prices may provide insights to policymakers and practitioners alike.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

Plane Wave Resonance in the Tire Air Cavity as a Vehicle Interior Noise Source

1995 ◽  
Vol 23 (1) ◽  
pp. 2-10 ◽  
Author(s):  
J. K. Thompson
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Interior Noise ◽  
Cavity Resonance ◽  
Test Results ◽  
Vehicle Interior ◽  
Air Cavity ◽  
Vehicle Interior Noise ◽  
Wave Resonance ◽  
Resonance Frequencies

Abstract Vehicle interior noise is the result of numerous sources of excitation. One source involving tire pavement interaction is the tire air cavity resonance and the forcing it provides to the vehicle spindle: This paper applies fundamental principles combined with experimental verification to describe the tire cavity resonance. A closed form solution is developed to predict the resonance frequencies from geometric data. Tire test results are used to examine the accuracy of predictions of undeflected and deflected tire resonances. Errors in predicted and actual frequencies are shown to be less than 2%. The nature of the forcing this resonance as it applies to the vehicle spindle is also examined.

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