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Variance Swaps
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Variance Swaps



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  • 1. Yang Han
  • 2. Outline
    • 1. Why and what’s the variance swap.
    • 2. Replicating variance swap.
    • -Hedging a exotic option - log contract provides a payoff equal to the variance of the stock’s return.
    • -How to replicate the exotic option by a portfolio of standard options. So the market prices determine the cost of the variance swap.
  • 3. Outline continue
    • 3. The general valuation method.
    • 4. Effects of Volatility Skew
    • 5. How to use variance swap: trading strategies.
  • 4. Part 1 Why variance swap
    • A stock’s volatility is the simplest measure of its riskiness or uncertainty.
    • What do you do if you simply want exposure to a stock’s volatility?
    • Stock option are impure: they provide exposure to both the direction of the stock price and its volatility.
    • Variance swap has more fundamental theoretical significance when compare with volatility swap: from the valuation point of view, it can be replicated most reliably.
  • 5. What’s Variance Swap
    • A variance swap is an over-the-counter derivative contract in which two parties agree to buy or sell the realized volatility of an index or single stock on a future date—the swap expiration date—for a pre-determined price, the swap-strike.
    • Variance swap vega is the dollar-change in swap value for each percentage-point movement in volatility. The vega of the swap equals twice the volatility times the notional amount of the swap: [2 * Volatility * $ Notional].
  • 6. The payoff of variance swap
    • A variance swap is a forward contract on annualized variance, the square of the realized volatility. Its payoff at expiration is equal to:
    • (σ R^2- K var) *N
    • Kvar is the delivery price for variance, and the N is the notional amount of the swap in dollars per annualized volatility point squared.
  • 7. Variance swaps are convex in Volatility
  • 8. Part 2 Replicating Variance Swaps
    • Creating a portfolio of options whose variance sensitivity is independent of stock price:
    • The portfolio with weights inversely proportional to K^2 produces a V that is virtually independent of stock price S, as long as S lies inside the range of available strikes and far from the edge of the range.
    • V: The exposure of an option to a stock’s variance.
  • 9. Portfolios of call options of different strikes as a function of Stock price
  • 10. V is virtually independent of Stock price
  • 11. Portfolio’s payoff
    • At expiration, the payoff of the portfolio is
  • 12. A long position in a forward contract and a short position in a log contract
  • 13. Part 3 General Results
    • The fair value of variance is the delivery price that makes the swap have zero value.
    • The theoretical definition of realized variance for a given price history:
  • 14. The math
    • The fair delivery value of future realized variance is the strike Kvar for which the contract has zero present value:
  • 15. The math
    • By applying Ito’s lemma to logSt:
    • Subtracting
  • 16. The math
    • Therefore:
  • 17. The math
    • And because the underlyer evolves according to:
    • So
  • 18. The math
    • The log payoff can then be rewritten as:
    • And because we already known:
  • 19. The math
    • The fair value of the swap is:
  • 20. Part 4 Effects of Volatility Skew
    • Skew linear in Strike:
  • 21. Skew continue
    • Skew linear in Delta:
  • 22. Part 5 Hedging Equity Returns
    • We can use variance swaps in portfolio protection.
    • Index implied volatility (therefore variance) is negatively correlated to equity returns. In addition, the payoff of a long variance swap increases more rapidly than equity returns decrease. Therefore, we can use variance swaps as crash-protection, insulating a portfolio from very negative equity market performance
  • 23. Relationship between monthly changes in 30-day S&P 500 variance (as measured by the VIX Index) and monthly changes in the S&P 500 Index. Vix: A high value corresponds to a more volatile market and therefore more costly options, which can be used to defray risk from this volatility by selling options
  • 24. Volatility Relative Value and Directional Strategies
    • Variance swaps can also be used in volatility-based relative value trades.
    • For example, an investor can buy S&P 500 variance and sell a similar amount of Nasdaq 100 variance. The trade will be profitable if NDX volatility decreases relative S&P 500 volatility, or if S&P 500 volatility increases more than Nasdaq 100 volatility.
    • This trade was very profitable in the months following the tech bubble burst in 2000. During this period, the weight of technology stocks in the Nasdaq 100 decreased significantly; this caused Nasdaq 100 Index volatility to decrease significantly as well, especially relative to S&P 500 Index volatility.
  • 25. 3M S&P 500 and Nasdaq 100 Index Volatility
  • 26. Credit and Equity Volatility Relative Value
    • A company’s credit risk is related to its stock price and equity volatility.
    • A decline in stock price usually leads to an increase in both equity volatility and credit risk.
    • The charts on the next page illustrate the correlation between credit risk (as measured by the spread between BAA-rated US corporate 30-year bonds and the equivalent treasury rate) and volatility in the S&P 500 Index (as measured by the VIX Index) since April 2000
  • 27. Correlation between Credit Risk and Equity Volatility
  • 28. Continue
    • An investor could use a variance swap to hedge the credit risk of a basket of corporate bonds because of this relationship. A variance swap could also be used in a credit versus equity volatility relative value trade.
  • 29. Thanks!