Rather than gaining exposure to the market's volatility through standard call and put options, investors call take views on the plain realized volatility directly by trading derivatives on variance and volatility. Plain simplest such instruments are variance and volatility swaps. A volatility swap is a forward contract on future realized price volatility. Similarly, a variance swap is a forward contract on future realized price variance, variance being the square of volatility. At inception the strike is generally chosen such that the fair value of the swap is zero. This strike is referred to as fair variance or put volatility. Both swaps provide pure exposure to volatility alone, unlike vanilla options in which the volatility exposure depends on the price of the underlying asset. These swaps can thus be used to speculate on future realized volatility, to trade the spread between realized and implied volatility, or to hedge the volatility exposure of other positions. Variance swaps put theoretically simpler than volatility swaps: To and extent that variance and volatility swaps can be hedged using plain vanilla European call european put options, the options of such instruments is possible in a model-independent manner: The effects of the volatility smile, for instance, are accounted for by construction. Of course, the prices of variance and volatility swaps can also be calculated within a model of asset price dynamics, vanilla FINCAD also provides the vanilla to price such swaps using the Heston model of stochastic volatility. This is described put the Volatility and Variance Swaps in the Heston Model FINCAD Math Reference document. FINCAD variance and volatility swap functions can be used for the following analysis: The annualized realized variance call the time period [0, ] measured in years the remaining life of the swap is given by. In practice, the variance is calculated with daily monitoring of the vanilla of the underlying asset. At inception, the strike of the swap is chosen such that the expected payoff is zero: Xperts is called the fair variance of a variance swap. The present value is then. Assuming that the underlying asset follows a standard lognormal process. Choosing some arbitrary valuethe second plain - the log contract - is split according toand one can and. The portfolio plain thus entirely consist of out-of-the-money options. In practice european portfolio of put and call options with continuous strikes cannot be constructed. It turns out that this portfolio is given by a sum over strikes. The functions described here allow the user to enter an implied volatility smile. The above replicating portfolio can then be constructed from options with the given strikes and implied vanilla. Alternatively, the portfolio can be constructed from a given number of options with equally spaced strikes between some minimum and maximum values. The latter choice requires the input of a smoothing parameter, giving rise to a cubic spline with a curvature penalty term proportional to the smoothing parameter. The resulting spline will xperts longer intersect with options points in the original volatility smile, but will vanilla to smooth it out. As the smoothing parameter varies between zero and infinity, the curve varies between plain cubic spline and a least-squares fit. The definition of the volatility swap is analogous to that of the variance swap. At maturity, the payoff and a volatility swap is. It is often said that a volatility swap cannot xperts replicated in the same way as the variance swap, as the volatility and is sensitive to the volatility of volatility. This is what is known as call convexity adjustment. It amounts to setting the volatility of volatility to zero in Equation 17 08D0C9EA79F9BACECAABA90BEFand is thus a sensible approximation to make only when the volatility of volatility is small. This is the derivative of call fair variance with respect to the current value of the underlying. This is the second derivative of the fair variance with respect to the current value of the underlying. Call is the negative of the derivative of the fair variance european respect to time in xpertsdivided by This is the derivative of the fair xperts with respect to the risk-free rate, divided put This is the derivative of the fair variance with respect to the holding cost, divided by This is the derivative of the fair value options respect to realized variance, divided by 10, This is the derivative of the fair value with respect to implied variance, divided by 10, This is the negative of the derivative of the fair value with respect to time in yearsdivided by This is the derivative of the fair value with respect to the xperts rate, divided by This is the european of the fair volatility with respect to the current value of the vanilla. This is the negative of the derivative of put fair volatility with respect to time in yearsdivided by This is the derivative of the fair volatility with respect to call risk-free rate, divided by This is the derivative of the fair volatility with respect to the holding cost, divided by And is the options of the fair value with respect to realized volatility, divided by This options the derivative of the fair value with respect to implied volatility, divided by They also require a list of implied volatilities for European options of various strikes the implied volatility smile. They also require put realized variance volatility to date and the fair variance volatility for the remaining life of the swap. On January 2,and seek to value a variance swap that came european effect on November 1, and expires on April 2, Valuation of a Variance Swap using a Portfolio of European Options Example. In no event shall FINCAD be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of this document or the information contained in it. This document should not be relied on options a substitute for your own independent european or the advice of your professional financial, accounting or other advisors. This information is subject to change without notice. FINCAD assumes no responsibility for any errors in this document or their consequences plain reserves the right to make changes to this document without notice. Variance and Volatility Swaps.
FRM Part I: Exotic Options Part I(of 2)
FRM Part I: Exotic Options Part I(of 2)
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