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FINCAD offers the most transparent solutions in the industry, providing extensive documentation with every product. This is complemented by an extensive library of white papers, articles and case studies. The classic derivatives that allow investors to take a view on volatility are straddles or strangles. A long position in a straddle, for example, will generate a profit if the underlying asset price moves up or down, or if the implied volatility rises.
However, these options are also sensitive to the underlying asset price, as the delta of a straddle or a strangle is zero only when the option is at-the-money. Unlike these options, variance and volatility swaps provide pure exposure to volatility. A volatility swap is essentially a forward contract on future realized price volatility.
At expiry the holder of a long position in a volatility swap receives or owes if negative the difference between the realized volatility and the initially chosen volatility strike, multiplied by a notional principal amount.
A variance swap is analogously a forward contract on future realized price variance, which is the square of future realized volatility. In both cases, at inception of the swap the strike is chosen such that the fair value of the swap is zero. This strike is then referred to as fair volatility and fair variance, respectively. Variance and volatility swaps can 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.
For example, variance swaps can effectively protect against drops in the underlying price: Variance swaps can be replicated and valued in a model-independent manner, using a static portfolio of European vanilla options [ 1 ].
For this reason, variance swaps are more popular than volatility swaps - for which there exist only approximate static replication strategies. On the other hand, both types of swaps may be valued within a calibrated Heston model [ 2 ].
The variance swap replication is accomplished using a portfolio of options with different strikes. The construction of this portfolio can be understood intuitively in the Black Scholes model.
This "variance vega" 2 is largest when the underlying price is closest to the strike of the option, and is also an increasing function of the strike. Both these dependencies are illustrated in the graph below, which shows the variance vega as a function of the underlying asset price for a range of strikes. The variance vega of a portfolio of options that replicates the variance swap payoff must be independent of the underlying price. To achieve this, each option has to be weighted by the inverse of the strike squared.
The following graph shows the variance vega of such a portfolio: As can be seen, the variance exposure of the portfolio is largely independent of the underlying asset price, as long as the price lies within the range of option strikes. In fact, as the spacing between strikes is decreased and the range of strikes in the options portfolio increases, the variance exposure becomes entirely independent of the underlying stock price.
The rigorous derivation given in [ 1 ] shows that such a portfolio indeed replicates the payoff of a variance swap. From an input volatility smile table that lists the available options, the function computes the value of each option and the number of options at each strike required in the portfolio.
The replication of a volatility swap is not as simple as that of a variance swap. Volatility on the other hand should be viewed as a derivative of variance. As volatility is the square root of variance, the relation is non-linear, which does not allow replication by a static options portfolio.
K vol , where K vol is the volatility strike of the volatility swap. The replication strategies discussed above are largely model-independent, as they only assume that the underlying asset price process be continuous. However, given some model of the underlying asset price dynamics, variance and volatility swaps can also be priced.
A natural model to use is the Heston model of stochastic volatility. In this model the underlying asset price S follows a standard lognormal process, and the variance V follows a mean-reverting square root process:.
The five Heston model parameters are: In the Heston model this expectation value follows from the differential equation for the variance process given above [ 2 ]:.
Similarly, the expected total volatility required for the valuation of a volatility swap can be computed in the Heston model. However, the expression is more complicated than the variance expectation and its calculation involves a numerical integration [ 2 ].
In this example we use the FINCAD Analytics Suite workbook "Variance or Volatility Swap" to calculate the replicating portfolio for a variance swap and its fair variance which would result in a zero fair value of the swap at inception. We also calculate the fair value of the swap some time after inception. For the replication we have ten options that we enter in the worksheet "Smile Table":.
The annually compounded risk free rate is 3. Using the input volatility smile table the replicating options portfolio is calculated on the worksheet "Portfolio":. The first column in the "Hedging Portfolio" table lists the type of options required at the strike price given in the second column.
The implied volatility in the third column follows directly from the given smile table. The fourth column is the option's value as computed in the Black Scholes model.
The replicating portfolio is given in the fifth column, which lists the number of put and call options that are required at each strike.
The final column is the price of the position in each option. Clearly, the positions of the at-the-money options are the most expensive. We now calculate the fair variance and the associated risk statistics for the variance swap using the worksheet "Fair Var".
The fair variance is close to the variance strike. The delta of the variance swap is very small; it is different from zero, because the replication with just ten options is not perfect.
After two weeks we compute the present value of the variance swap. For simplicity we assume that all inputs including the volatility smile are the same, except for the value date which is now March On the worksheet "Realized Var" we enter the realized prices for the two week period and calculate the realized variance, which is 0.
The realized variance 0. The implied variance on the value date that follows from the options portfolio is 0.
The present value of the swap is therefore slightly negative:. We enter the variance swap details on the worksheet "Main" as shown in the following screenshot. Note that the valuation of a variance swap in the Heston model does not depend on the volatility of volatility or on the correlation.
The fair variance of the variance swap evaluates to 0. This value is close to the value we calculated with the options portfolio replication strategy in the first example, as should be expected.
One would find exact agreement only for a perfectly calibrated model and an options portfolio with continuous strikes. FINCAD Analytics Suite provides functions to value variance and volatility swaps using model-independent replication strategies as well as using the Heston model.
The former functions allow the user to compute portfolios that replicate the floating leg of a variance swap. Based on this replication strategy, the fair variance and the fair value including all risk statistics can be calculated. If the user has calibrated the Heston model to European options or to variance or volatility swaps, the Heston model functions can be used to compute the fair variance and the fair value of a variance or a volatility swap.
A Practitioner's Guide', Wiley Finance. Your use of the information in this article is at your own risk. The information in this article is provided on an "as is" basis and without any representation, obligation, or warranty from FINCAD of any kind, whether express or implied.
We hope that such information will assist you, but it should not be used or relied upon as a substitute for your own independent research. Valuation of Variance and Volatility Swaps. Overview The classic derivatives that allow investors to take a view on volatility are straddles or strangles.
Replication with Vanilla Options The variance swap replication is accomplished using a portfolio of options with different strikes. Variance vega for individual vanilla call option The variance vega of a portfolio of options that replicates the variance swap payoff must be independent of the underlying price. Variance vega for a portfolio of vanilla options In fact, as the spacing between strikes is decreased and the range of strikes in the options portfolio increases, the variance exposure becomes entirely independent of the underlying stock price.
Valuation in the Heston Model The replication strategies discussed above are largely model-independent, as they only assume that the underlying asset price process be continuous. In this model the underlying asset price S follows a standard lognormal process, and the variance V follows a mean-reverting square root process: In the Heston model this expectation value follows from the differential equation for the variance process given above [ 2 ]: Model-independent Valuation with a Replicating Portfolio In this example we use the FINCAD Analytics Suite workbook "Variance or Volatility Swap" to calculate the replicating portfolio for a variance swap and its fair variance which would result in a zero fair value of the swap at inception.
For the replication we have ten options that we enter in the worksheet "Smile Table":