Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching
133
Citation
45
Reference
10
Related Paper
Citation Trend
Abstract:
A model is developed for pricing volatility derivatives, such as variance swaps and volatility swaps under a continuous‐time Markov‐modulated version of the stochastic volatility (SV) model developed by Heston. In particular, it is supposed that the parameters of this version of Heston's SV model depend on the states of a continuous‐time observable Markov chain process, which can be interpreted as the states of an observable macroeconomic factor. The market considered is incomplete in general, and hence, there is more than one equivalent martingale pricing measure. The regime switching Esscher transform used by Elliott et al. is adopted to determine a martingale pricing measure for the valuation of variance and volatility swaps in this incomplete market. Both probabilistic and partial differential equation (PDE) approaches are considered for the valuation of volatility derivatives.Keywords:
Heston model
Variance swap
We consider the pricing of discretely sampled volatility swaps under a modified Heston model, whose risk-neutralized volatility process contains a stochastic long-run variance level. We derive an analytical forward characteristic function under this model, which has never been presented in the literature before. Based on this, we further obtain an analytical pricing formula for volatility swaps which can guarantee the computational accuracy and efficiency. We also demonstrate the significant impact of the introduced stochastic long-run variance level on volatility swap prices with synthetic as well as calibrated parameters. doi: 10.1017/S144618112200013X
Variance swap
Heston model
Interest rate swap
Cite
Citations (0)
Variance swap
Realized variance
Local volatility
Cite
Citations (0)
Trading in derivatives has caused investors, and especially market makers, to be concerned with the volatility of asset returns along with their direction. Uncertain and time-varying volatility imparts risk to an otherwise hedged position, and volatility risk is not easy to manage with ordinary instruments. Volatility swaps are a new class of derivative, for which an asset9s volatility itself is the underlying. This article describes how volatility swaps work, and derives pricing and hedging equations for them. Interestingly, the natural derivative instrument in this family would be based on variance, rather than volatility, since a variance swap can be replicated (pretty well) by a static portfolio of ordinary European calls and puts on the price of the underlying asset. The authors also show how to set up a volatility hedge when the available traded options exhibit a smile or skew pattern.
Variance swap
Volatility risk
Cite
Citations (474)
Variance swap
Heston model
Volatility risk
Realized variance
Cite
Citations (3)
In this paper, Malliavin calculus is applied to arrive at exact formulas for the difference between the volatility swap strike and the zero vanna implied volatility for volatilities driven by fractional noise. To the best of our knowledge, our estimate is the first to derive the rigorous relationship between the zero vanna implied volatility and the volatility swap strike. In particular, we will see that the zero vanna implied volatility is a better approximation for the volatility swap strike than the ATMI.
Variance swap
Local volatility
Heston model
Cite
Citations (0)
The scope of this diploma thesis is to examine the four generations of asset pricing models and the corresponding volatility dynamics which have been devepoled so far. We proceed as follows: In chapter 1 we give a short repetition of the Black-Scholes first generation model which assumes a constant volatility and we show that volatility should not be modeled as constant by examining statistical data and introducing the notion of implied volatility. In chapter 2, we examine the simplest models that are able to produce smiles or skews - local volatility models. These are called second generation models. Local volatility models model the volatility as a function of the stock price and time. We start with the work of Dupire, show how local volatility models can be calibrated and end with a detailed discussion of the constant elasticity of volatility model. Chapter 3 focuses on the Heston model which represents the class of the stochastic volatility models, which assume that the volatility itself is driven by a stochastic process. These are called third generation models. We introduce the model structure, derive a partial differential pricing equation, give a closed-form solution for European calls by solving this equation and explain how the model is calibrated. The last part of chapter 3 then deals with the limits and the mis-specifications of the Heston model, in particular for recent exotic options like reverse cliquets, Accumulators or Napoleons. In chapter 4 we then introduce the Bergomi forward variance model which is called fourth generation model as a consequence of the limits of the Heston model explained in chapter 3. The Bergomi model is a stochastic local volatility model - the spot price is modeled as a constant elasticity of volatility diffusion and its volatility parameters are functions of the so called forward variances which are specified as stochastic processes. We start with the model specification, derive a partial differential pricing equation, show how the model has to be calibrated and end with pricing examples and a concluding discussion.
Heston model
Variance swap
Local volatility
Cite
Citations (0)
In this paper, Malliavin calculus is applied to arrive at exact formulas for the difference between the volatility swap strike and the zero vanna implied volatility for volatilities driven by fractional noise. To the best of our knowledge, our result is the first to derive the rigorous relationship between the zero vanna implied volatility and the volatility swap strike. In particular, we will see that the zero vanna implied volatility is a more accurate approximation for the volatility swap strike than the at-the-money implied volatility.
Variance swap
Heston model
Local volatility
Cite
Citations (4)
In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is the multi-factor Heston stochastic volatility model. We obtain pricing formulas for the weighted variance swap and approximate expressions for the weighted volatility swap. The bounds of the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to compute in real time and can be applied efficiently for practical applications. We study the problem of hedging volatility swap with variance swap. We also determined the optimal amount of the underlying asset that has to be held for minimizing the hedging error by taking positions in options and weighted variance swap. From the numerical analysis, a couple of important features of the usefulness of the multi-factor Heston stochastic volatility model are discussed.
Variance swap
Heston model
Cite
Citations (7)
In this paper, Malliavin calculus is applied to arrive at exact formulas for the difference between the volatility swap strike and the zero vanna implied volatility for volatilities driven by fractional noise. To the best of our knowledge, our estimate is the first to derive the rigorous relationship between the zero vanna implied volatility and the volatility swap strike. In particular, we will see that the zero vanna implied volatility is a better approximation for the volatility swap strike than the ATMI.
Variance swap
Local volatility
Cite
Citations (0)
Abstract We consider the pricing of discretely sampled volatility swaps under a modified Heston model, whose risk-neutralized volatility process contains a stochastic long-run variance level. We derive an analytical forward characteristic function under this model, which has never been presented in the literature before. Based on this, we further obtain an analytical pricing formula for volatility swaps which can guarantee the computational accuracy and efficiency. We also demonstrate the significant impact of the introduced stochastic long-run variance level on volatility swap prices with synthetic as well as calibrated parameters.
Variance swap
Heston model
Interest rate swap
Cite
Citations (0)