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- 1. Greeks and volatiltiy By A V Vedpuriswar September 21, 2009
- 2. Volatility
- 3. Basics of volatility <ul><li>Volatility is a huge issue in risk management. </li></ul><ul><li>The science of volatility measurement has advanced a lot in recent years. </li></ul><ul><li>Here we look at some basic concepts and tools. </li></ul>
- 4. Estimating Volatility <ul><li>Calculate daily return u 1 = ln S i / S i-1 </li></ul><ul><li>Variance rate per day </li></ul><ul><li>We can simplify this formula by making the following simplifications. </li></ul><ul><li>u i = (S i – S i-1) / S i-1 </li></ul><ul><li>ū = 0 m-1 = m </li></ul><ul><li>If we want to weight </li></ul>
- 5. Estimating Volatility <ul><ul><li>Exponentially weighted moving average model means weights decrease exponentially as we go back in time. </li></ul></ul><ul><ul><li> n 2 = 2 n-1 + (1 - ) u 2 n-1 </li></ul></ul><ul><ul><li>= [ n-2 2 + (1- )u n-2 2 ] + (1- )u n-1 2 </li></ul></ul><ul><ul><li> = (1- )[u n-1 2 + u n-2 2 ] + 2 n-2 2 </li></ul></ul><ul><ul><li> = (1- ) [u n-1 2 + u 2 n-2 + 2 u n-3 2 ] + 3 2 n-3 </li></ul></ul><ul><ul><li>If we apply GARCH model, </li></ul></ul><ul><ul><li> n 2 = Y V L + u n-1 2 + 2 n-1 </li></ul></ul><ul><ul><li>V L = Long run average variance rate </li></ul></ul><ul><ul><li>Y + + = 1. If Y = 0, = 1- , = , it becomes exponentially weighted model. </li></ul></ul><ul><ul><li>GARCH incorporates the property of mean reversion. </li></ul></ul>
- 6. Problem <ul><li>The current estimate of daily volatility is 1.5%. The closing price of an asset yesterday was $30. The closing price of the asset today is $30.50. Using the EWMA model, with λ = 0.94, calculate the updated estimate of volatility . </li></ul>
- 7. Solution <ul><li>h t = λ σ 2 t-1 + ( 1 – λ ) r t-1 2 </li></ul><ul><li>λ = .94 </li></ul><ul><li>r t-1 = ln[(30.50 )/ 30] </li></ul><ul><li>= .0165 </li></ul><ul><li>h t = (.94) (.015) 2 + (1-.94) (.0165) 2 </li></ul><ul><li>Volatility = .01509 = 1.509 % </li></ul>
- 8. Greeks
- 9. Delta <ul><li>Delta is the rate of change in option price with respect to the price of the underlying asset. </li></ul><ul><li>It is the slope of the curve that relates the option price to the underlying asset price. </li></ul><ul><li>A position with delta of zero is called delta neutral. </li></ul><ul><li>Because delta keeps changing, the investor’s position may remain delta neutral for only a relatively short period of time. </li></ul><ul><li>The hedge has to be adjusted periodically. </li></ul><ul><li>This is known as rebalancing. </li></ul>
- 10. Theta <ul><li>Theta of a portfolio is the rate of change of value of the portfolio with respect to change of time, with all else remaining same. </li></ul><ul><li>Theta is also called the time decay of the portfolio. </li></ul><ul><li>T heta is usually negative for an option. </li></ul><ul><li>As time to maturity decreases with all else remaining the same, the option tends to become less valuable. </li></ul>
- 11. Gamma <ul><li>The gamma is the rate of change of the portfolio’s delta with respect to the price of the underlying asset. </li></ul><ul><li>It is the second partial derivative of the portfolio price with respect to the asset price. </li></ul><ul><li>If gamma is small, it means delta is changing slowly. </li></ul><ul><li>In such a case, adjustments to keep a portfolio delta neutral can be made only relatively infrequently. </li></ul><ul><li>However, if gamma is large, it means the delta is highly sensitive to the price of the underlying asst. </li></ul><ul><li>It is then quite risky to leave a delta neutral portfolio unchanged for any length of time. </li></ul>
- 12. Vega <ul><li>The Vega of a portfolio of derivatives is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset. </li></ul><ul><li>High Vega means high sensitivity to small changes in volatility. </li></ul><ul><li>A position in the underlying asset has zero Vega. </li></ul><ul><li>The Vega can be changed by adding options. </li></ul><ul><li>If V is Vega of the portfolio and V T is the Vega of the traded option, a position of –V/ V T in the traded option makes the portfolio Vega neutral. </li></ul><ul><li>If a hedger requires the portfolio to be both gamma and Vega neutral, at least two traded derivatives dependent on the underlying asset must usually be used. </li></ul>
- 13. Rho <ul><li>Rho of a portfolio of options is the rate of change of value of the portfolio with respect to the interest rate. </li></ul>
- 14. Problem <ul><li>Suppose an existing short option position is delta neutral and has a gamma of － 6000. Here, gamma is negative because we have sold options. Assume there exists a traded option with a delta of 0.6 and gamma of 1.25. Create a gamma neutral position. </li></ul>
- 15. Solution <ul><li>To gamma hedge, we must buy 6000/1.25 = 4800 options. </li></ul><ul><li>Then we must sell (4800) (.6) = 2880 shares to maintain a gamma neutral and original delta neutral position. </li></ul>
- 16. Problem <ul><li>A delta neutral position has a gamma of － 3200. There is an option trading with a delta of 0.5 and gamma of 1.5. How can we generate a gamma neutral position for existing portfolio while maintaining delta neutral hedge? </li></ul>
- 17. Solution <ul><li>Buy 3200/1.5 = 2133 options </li></ul><ul><li>Sell (2133) (.5) = 1067 shares </li></ul>
- 18. <ul><li>Suppose a portfolio is delta neutral, with gamma = - 5000 and vega = - 8000. A traded option has gamma = .5, vega = 2.0 and delta = 0.6. How do we achieve vega neutrality? </li></ul>Problem
- 19. <ul><li>To achieve Vega neutrality add 4000 options. Delta increases by (.6) (4000) = 2400 </li></ul><ul><li>So sell 2400 units of asset to maintain delta neutrality. </li></ul><ul><li>As the same time, Gamma changes from – 5000 to ((.5) (4000) – 5000 = - 3000. </li></ul>
- 20. <ul><li>Suppose there is a second traded option with gamma = 0.8, vega = 1.2 and delta = 0.5. </li></ul><ul><li>if w 1 and w 2 are the weights in the portfolio, </li></ul><ul><li> - 5000 + .5w 1 + .8w 2 = 0 - 8000 + 2.0w 1 + 1.2w 2 = 0 </li></ul><ul><li>w 1 = 400 w 2 = 6000. </li></ul><ul><li>This makes the portfolio gamma and vega neutral. </li></ul><ul><li>Now let us examine delta neutrality. </li></ul><ul><li>Delta = (400) (.6) + (6000) (.5) = 3240 </li></ul><ul><li>3240 units of the underlying asset will have to be sold to maintain delta neutrality. </li></ul>

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