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Pricing swaps and options on quadratic variation under stochastic time change models
 Columbia University
, 2007
"... We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As ..."
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We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As such a process we consider a class of Lévy models with stochastic time change. Our analysis reveals a natural small parameter of the problem which allows a general asymptotic method to be developed in order to obtain a closedform expression for the fair price of the above products. As examples, we consider the CIR clock change, general affine models of activity rates and the 3/2 power clock change, and give an analytical expression of the swap price. Comparison of the results obtained with a familiar logcontract approach is provided. ∗We thank Arthur Sepp and an anonymous referee for useful comments. We assume full responsibility for any remaining errors.
Hedging variance options on continuous semimartingales. Forthcoming in Finance and Stochastics,
, 2009
"... Abstract We find robust modelfree hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forwardstartin ..."
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Abstract We find robust modelfree hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forwardstarting variance options, by dynamically trading the underlying asset, and statically holding European options. We thereby derive upper and lower bounds on values of variance options, in terms of Europeans.
2012): “Asymptotic and exact pricing of options on variance,”Finance and Stochastics, forthcoming
"... Abstract. We consider the pricing of derivatives written on the discretely sampled realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuoustime limit, the quadratic variation of the underlying logprice. Here, we characterize ..."
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Abstract. We consider the pricing of derivatives written on the discretely sampled realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuoustime limit, the quadratic variation of the underlying logprice. Here, we characterize the smalltime limits of options on both objects. We find that the difference between them strongly depends on whether or not the stock price process has jumps. Subsequently, we propose two new methods to evaluate the price of options on the discretely sampled realized variance. One of the methods is approximative; it is based on correcting prices of options on quadratic variation by our asymptotic results. The other method is exact; it uses a novel randomization approach and applies FourierLaplace techniques. We compare the methods and illustrate our results by some numerical examples. 1.
Option pricing
 in ARCH type models, Mathematical Finance
, 1998
"... options on variance in affine stochastic volatility models ..."
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options on variance in affine stochastic volatility models
Realized Volatility Options
"... Let the underlying process Y be a positive semimartingale, and let X t := log(Y t /Y 0 ). Define realized variance to be [X], where [·] denotes quadratic variation (but see section 5). Define a realized variance option on Y with variance strike Q and expiry T to pay + in the case of a realized vari ..."
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Let the underlying process Y be a positive semimartingale, and let X t := log(Y t /Y 0 ). Define realized variance to be [X], where [·] denotes quadratic variation (but see section 5). Define a realized variance option on Y with variance strike Q and expiry T to pay + in the case of a realized variance call, + in the case of a realized variance put, and define a realized volatility option on Y with volatility strike Q 1/2 and expiry T to pay + in the case of a realized volatility put. We will in some places restrict attention to puts, by putcall parity: for realized variance options, a longcall shortput combination pays [X] T − Q, equal to a Qstrike variance swap; and for realized volatility options, a longcall shortput combination pays [X] 1/2 T − Q 1/2 , equal to a Q 1/2 strike volatility swap. Unlike variance swaps [EQF07/024, EQF07/045], which admit exact modelfree (assuming only continuity of Y ) hedging and pricing in terms of Europeans, variance and volatility options have a range of values consistent with given prices of Europeans. With no further assumptions, there exist sub/superreplication strategies and lower/upper pricing bounds (section 4). Under an independence condition, there exist exact pricing formulas in terms of Europeans (section 2). Under specific models, there exist exact pricing formulas in terms of model parameters (section 1). Unless otherwise noted, all prices are denominated in units of a T maturity discount bond. The results apply to dollardenominated prices, provided that interest rates vary deterministically, because if Y is a dollardenominated share price and Y is that share's bonddenominated price, then log Y − log Y has finite variation, so Expectations E will be with respect to martingale measure P. Transform analysis Some of the methods surveyed here (in particular, sections 1 and 2.1) will price variance/volatility options by pricing a continuum of payoffs of the form e z[X] T . Transform analysis relates the former 1
Hedging of Volatility Hedging of Volatility "An investment in knowledge pays the best interest."
"... Abstract Department of Mathematics Master of Financial Mathematics Hedging of Volatility by Ty Lewis This paper investigates pricing and replication of volatility derivatives; beginning with variance and volatility swaps, moving on to options on those swaps and finally examines a method for pricing ..."
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Abstract Department of Mathematics Master of Financial Mathematics Hedging of Volatility by Ty Lewis This paper investigates pricing and replication of volatility derivatives; beginning with variance and volatility swaps, moving on to options on those swaps and finally examines a method for pricing VIX options. Volatility derivatives are important tools which allow investors to either hedge the volatility risk in a portfolio or speculate on a deviation of for pricing options on variance and volatility swaps as well as the VIX based on the assumption that realized volatility is lognormally distributed as proposed by . We investigate the performance of this method in a small market study and show that the method produces large discrepancies between theoretical and market prices for out of the money calls and discuss the sources of these discrepancies.
DOI 10.1007/s111470099048z
, 2009
"... Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case ..."
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Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case
First Draft: 5/08/2005 – This Draft: 7/01/2006
"... We explore the ability of alternative popular continuoustime diffusion and jump diffusion processes to capture the dynamics of implied volatility over time. The performance of the volatility processes is assessed under both econometric and financial metrics. To this end, data are employed from majo ..."
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We explore the ability of alternative popular continuoustime diffusion and jump diffusion processes to capture the dynamics of implied volatility over time. The performance of the volatility processes is assessed under both econometric and financial metrics. To this end, data are employed from major European and American implied volatility indices and the rapidly growing CBOE volatility futures market. We find that the simplest diffusion/jump diffusion models perform best under both metrics. Mean reversion is of second order importance. The results are consistent across the various markets. JEL Classification: G11, G12, G13.