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Options on Realized Variance by Transform Methods: A Non-Affine Stochastic Volatility Model
Gabriel G Drimus University of Copenhagen - Institute for Mathematical Sciences
September 19, 2009
Abstract: In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model, by developing efficient transform based pricing methods. This non-affine model gives prices of options on realized variance which allow upward sloping implied volatility of variance smiles. Heston's (1993) model, the benchmark affine stochastic volatility model, leads to downward sloping volatility of variance smiles - in disagreement with variance markets in practice. We show a robust method, using control variates, to express the Laplace transform of the variance call function in terms of the Laplace transform of realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm which gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps. Discuss this paper
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by Peter Carr, Roger Lee, Liuren Wu Under purely continuous price dynamics, the risk-neutral expected value of the return variance is equal to the negative of twice the risk-neutral expected value of the log return over the same horizon. The former represents the variance swap rate, and the latter is often referred to as the log profile, and can be synthesized by a particular portfolio of European options across a continuum of strikes and at the same maturity. When the price process contains discontinuous movements, variance swap rates and the log profile are no longer equal to each other. This paper derives the relation between the two quantities under different return dynamics specifications. Using quotes on both variance swaps and European options on the S&P 500 index, the paper tests alternative classes of index dynamics specifications, infers the return innovation structures, and extracts the stochastic variance from different return innovations.
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by DANIEL EGLOFF, MARKUS LEIPPOLD,AND LIUREN WU With increasing appreciation of the fact that stock return variance is stochastic and variance risk is heavily priced, the industry has created a series of variance derivative products to span variance risk. The variance swap contract is the most actively traded of these products. It pays at expiry the difference between the realized return variance and a fixed rate, called the variance swap rate, determined at the inception of the contract. We obtain a decade worth of variance swap rate quotes at five maturities. With the data, we first exploit the information in both the time series and the term structure of the variance swap rates to analyze the return variance rate dynamics and market pricing of variance risk. We then study both theoretically and empirically how investors can use variance swap contracts across different maturities to span the variance risk and to revise their dynamic asset allocation decisions. We find that with the swap contract to span the variance risk, an investor increases her investment in the underlying stock. In addition, the investor Discuss this paper
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Conditional variance swaps are claims on realised variance that is accumulated when the underlying asset price stays within a certain range. Being highly sensitive to movements in both asset price and its variance, they require a very reliable model for pricing and risk-managing. We apply the Heston stochastic volatility model to derive closed-form solutions for pricing and risk-managing of such swaps Discuss this paper
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Peter Carr and Roger Lee Abstract Variance swaps, which pay the realized variance of [the returns on] an underlying price process, have become a leading tool for managing exposure to volatility risk. Variance options Discuss this paper
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