Variance Swaps


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Variance Swaps

  1. 1. Yang Han
  2. 2. Outline <ul><li>1. Why and what’s the variance swap. </li></ul><ul><li>2. Replicating variance swap. </li></ul><ul><li>-Hedging a exotic option - log contract provides a payoff equal to the variance of the stock’s return. </li></ul><ul><li>-How to replicate the exotic option by a portfolio of standard options. So the market prices determine the cost of the variance swap. </li></ul>
  3. 3. Outline continue <ul><li>3. The general valuation method. </li></ul><ul><li>4. Effects of Volatility Skew </li></ul><ul><li>5. How to use variance swap: trading strategies. </li></ul>
  4. 4. Part 1 Why variance swap <ul><li>A stock’s volatility is the simplest measure of its riskiness or uncertainty. </li></ul><ul><li>What do you do if you simply want exposure to a stock’s volatility? </li></ul><ul><li>Stock option are impure: they provide exposure to both the direction of the stock price and its volatility. </li></ul><ul><li>Variance swap has more fundamental theoretical significance when compare with volatility swap: from the valuation point of view, it can be replicated most reliably. </li></ul>
  5. 5. What’s Variance Swap <ul><li>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. </li></ul><ul><li>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]. </li></ul>
  6. 6. The payoff of variance swap <ul><li>A variance swap is a forward contract on annualized variance, the square of the realized volatility. Its payoff at expiration is equal to: </li></ul><ul><li>(σ R^2- K var) *N </li></ul><ul><li>Kvar is the delivery price for variance, and the N is the notional amount of the swap in dollars per annualized volatility point squared. </li></ul>
  7. 7. Variance swaps are convex in Volatility
  8. 8. Part 2 Replicating Variance Swaps <ul><li>Creating a portfolio of options whose variance sensitivity is independent of stock price: </li></ul><ul><li>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. </li></ul><ul><li>V: The exposure of an option to a stock’s variance. </li></ul>
  9. 9. Portfolios of call options of different strikes as a function of Stock price
  10. 10. V is virtually independent of Stock price
  11. 11. Portfolio’s payoff <ul><li>At expiration, the payoff of the portfolio is </li></ul>
  12. 12. A long position in a forward contract and a short position in a log contract
  13. 13. Part 3 General Results <ul><li>The fair value of variance is the delivery price that makes the swap have zero value. </li></ul><ul><li>The theoretical definition of realized variance for a given price history: </li></ul>
  14. 14. The math <ul><li>The fair delivery value of future realized variance is the strike Kvar for which the contract has zero present value: </li></ul>
  15. 15. The math <ul><li>By applying Ito’s lemma to logSt: </li></ul><ul><li>Subtracting </li></ul>
  16. 16. The math <ul><li>Therefore: </li></ul>
  17. 17. The math <ul><li>And because the underlyer evolves according to: </li></ul><ul><li>So </li></ul>
  18. 18. The math <ul><li>The log payoff can then be rewritten as: </li></ul><ul><li>And because we already known: </li></ul>
  19. 19. The math <ul><li>The fair value of the swap is: </li></ul>
  20. 20. Part 4 Effects of Volatility Skew <ul><li>Skew linear in Strike: </li></ul>
  21. 21. Skew continue <ul><li>Skew linear in Delta: </li></ul>
  22. 22. Part 5 Hedging Equity Returns <ul><li>We can use variance swaps in portfolio protection. </li></ul><ul><li>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 </li></ul>
  23. 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. 24. Volatility Relative Value and Directional Strategies <ul><li>Variance swaps can also be used in volatility-based relative value trades. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul>
  25. 25. 3M S&P 500 and Nasdaq 100 Index Volatility
  26. 26. Credit and Equity Volatility Relative Value <ul><li>A company’s credit risk is related to its stock price and equity volatility. </li></ul><ul><li>A decline in stock price usually leads to an increase in both equity volatility and credit risk. </li></ul><ul><li>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 </li></ul>
  27. 27. Correlation between Credit Risk and Equity Volatility
  28. 28. Continue <ul><li>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. </li></ul>
  29. 29. Thanks!