A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

2021 ◽  
pp. 1-10
Author(s):  
Puneet Pasricha ◽  
Anubha Goel
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stochastic Volatility Model ◽  
Strike Price ◽  
Volatility Model ◽  
Exchange Options ◽  
Pricing Formula ◽  
Exchange Option

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

PRICING VARIANCE SWAPS UNDER DOUBLE HESTON STOCHASTIC VOLATILITY MODEL WITH STOCHASTIC INTEREST RATE

2021 ◽  
pp. 1-17
Author(s):  
Huojun Wu ◽  
Zhaoli Jia ◽  
Shuquan Yang ◽  
Ce Liu
Keyword(s):  
Interest Rate ◽  
Closed Form ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Characteristic Functions ◽  
Stochastic Interest Rate ◽  
Modeling Framework ◽  
Variance Swaps ◽  
Volatility Model ◽  
Stochastic Interest

In this paper, we discuss the problem of pricing discretely sampled variance swaps under a hybrid stochastic model. Our modeling framework is a combination with a double Heston stochastic volatility model and a Cox–Ingersoll–Ross stochastic interest rate process. Due to the application of the T-forward measure with the stochastic interest process, we can only obtain an efficient semi-closed form of pricing formula for variance swaps instead of a closed-form solution based on the derivation of characteristic functions. The practicality of this hybrid model is demonstrated by numerical simulations.

A NOVEL ANALYTICAL APPROACH FOR PRICING DISCRETELY SAMPLED GAMMA SWAPS IN THE HESTON MODEL

2016 ◽  
Vol 57 (3) ◽  
pp. 244-268
Author(s):  
SANAE RUJIVAN
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Analytical Approach ◽  
Solution Procedure ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Variance Swaps ◽  
Closed Form Solutions ◽  
Volatility Model ◽  
Closed Form Formula

The main purpose of this paper is to present a novel analytical approach for pricing discretely sampled gamma swaps, defined in terms of weighted variance swaps of the underlying asset, based on Heston’s two-factor stochastic volatility model. The closed-form formula obtained in this paper is in a much simpler form than those proposed in the literature, which substantially reduces the computational burden and can be implemented efficiently. The solution procedure presented in this paper can be adopted to derive closed-form solutions for pricing various types of weighted variance swaps, such as self-quantoed variance and entropy swaps. Most interestingly, we discuss the validity of the current solutions in the parameter space, and provide market practitioners with some remarks for trading these types of weighted variance swaps.

scholarly journals A Closed-Form Pricing Formula for Log-Return Variance Swaps under Stochastic Volatility and Stochastic Interest Rate

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 5
Author(s):  
Mao Chen ◽  
Guanqi Liu ◽  
Yuwen Wang
Keyword(s):  
Interest Rate ◽  
Closed Form ◽  
Hybrid Model ◽  
Approximate Formula ◽  
Closed Form Solution ◽  
Sampling Frequency ◽  
Form Solution ◽  
Variance Swaps ◽  
Return Variance ◽  
Pricing Formula

At present, the study concerning pricing variance swaps under CIR the (Cox–Ingersoll–Ross)–Heston hybrid model has achieved many results ; however, due to the instantaneous interest rate and instantaneous volatility in the model following the Feller square root process, only a semi-closed solution can be obtained by solving PDEs. This paper presents a simplified approach to price log-return variance swaps under the CIR–Heston hybrid model. Compared with Cao’s work, an important feature of our approach is that there is no need to solve complex PDEs; a closed-form solution is obtained by applying the martingale theory and Ito^’s lemma. The closed-form solution is significant because it can achieve accurate pricing and no longer takes time to adjust parameters by numerical method. Another significant feature of this paper is that the impact of sampling frequency on pricing formula is analyzed; then the closed-form solution can be extended to an approximate formula. The price curves of the closed-form solution and the approximate solution are presented by numerical simulation. When the sampling frequency is large enough, the two curves almost coincide, which means that our approximate formula is simple and reliable.

WORST-OF OPTIONS AND CORRELATION SKEW UNDER A STOCHASTIC CORRELATION FRAMEWORK

2012 ◽  
Vol 15 (07) ◽  
pp. 1250051 ◽  
Author(s):  
JACINTO MARABEL ROMO
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Term Structure ◽  
Stochastic Volatility Model ◽  
Closed Form Solutions ◽  
Stochastic Correlation ◽  
Term Structures ◽  
Volatility Model ◽  
Wide Range ◽  
Correlation Term

This article considers a multi-asset model based on Wishart processes that accounts for stochastic volatility and for stochastic correlations between the underlying assets, as well as between their volatilities. The model accounts for the existence of correlation term structure and correlation skew. The article shows that the Wishart specification can generate different patterns corresponding to the correlation skew for a wide range of correlation term structures. Another advantage of the model is that it is analytically tractable and, hence, it is possible to obtain semi-closed-form solutions for the prices of plain vanilla options, as well as for the price of exotic derivatives. In this sense, this article develops semi-closed-form formulas for the price of European worst-of options with barriers and/or forward-start features. To motivate the introduction of the Wishart volatility model, the article compares the prices obtained under this model and under a multi-asset stochastic volatility model with constant instantaneous correlations. The results reveal the existence of a stochastic correlation premium and show that the consideration of stochastic correlation is a key element for the valuation of these structures.

scholarly journals Pricing European options with stochastic volatility under the minimal entropy martingale measure

2015 ◽  
Vol 27 (2) ◽  
pp. 233-247 ◽  
Author(s):  
XIN-JIANG HE ◽  
SONG-PING ZHU
Keyword(s):  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Martingale Measure ◽  
Minimal Entropy Martingale Measure ◽  
Volatility Model ◽  
In Series ◽  
Expected Utility Maximization ◽  
Equivalent Martingale ◽  
The One ◽  
Pricing Formula

In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance10(1), 39–52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method.

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