Value the variance swap in Example 26.4 of Section 26.16...

Value the variance swap in Example 26.4 of Section 26.16 assuming that the implied volatilities for options with strike prices $800,850,900,950,1,000,1,050,1,100,1,150$, 1,200 are $20 \%, 20.5 \%, 21 \%, 21.5 \%, 22 \%, 22.5 \%, 23 \%, 23.5 \%, 24 \%$, respectively.

Key Concepts

Variance Swap

A variance swap is a financial derivative contract that allows investors to trade future realized variance against a pre-agreed strike variance. The payoff of a variance swap depends on the difference between the realized variance of an underlying asset, typically computed over a specified period, and the fixed variance strike. This contract provides exposure to the volatility of the asset without direct exposure to its price movements.

Implied Volatility

Implied volatility is the market’s expectation of an asset’s future volatility, derived from the prices of options on that asset. It reflects the consensus forecast of the asset’s price fluctuations over the life of the options. In the context of variance swap valuation, the term structure of implied volatilities across different strikes or maturities is essential, as it informs the replication strategies and the fair value of the contract.

Static Replication

Static replication is a method used to price complex derivatives by constructing a portfolio of standard options whose payoffs mimic the payoff of the derivative. For variance swaps, a continuum of out?of?the-money calls and puts can be combined in such a way that their weighted payoff replicates the realized variance of the underlying asset. This technique provides a way to price variance swaps using observable market prices of vanilla options.

Risk-Neutral Valuation

Risk-neutral valuation is a cornerstone concept in derivative pricing, whereby expected future payoffs are computed under a probability measure in which all investors are indifferent to risk. Under this measure, the expected returns on all assets equal the risk-free rate, and derivatives are priced by discounting the expected payoff at the risk-free rate. This framework is integral to the valuation of variance swaps, as it enables the linkage between market-observed option prices and the expected future realized variance.

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