3. Volatility models (a) Current market volatility is higher than the historical average of market volatil-...

 

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3. Volatility models (a) Current market volatility is higher than the historical average of market volatil- ity from a long sample. Given this information and the stylized facts about market return volatility, what can you say about expected future market volatility? (b) Explain what 'realized variance' is. Use both a mathematical expression and an intuitive description. (c) A Variance Swap is a very popular over-the-counter derivative contract. In the contract, the fixed leg pays a fixed dollar amount every month, while the floating leg pays an amount proportional to the realized variance in each month based on daily data. Assume any variance risk premium and the risk-free rate are both zero. In particular, the payoff to a long variance position in a 2-month swap is: Payofft+2 RVt+1+RVt+2 - 2 × Ft, where RV, is the realized variance in months and F is the fixed payment decided at the start of the swap (at time t). Recall that swaps have zero value at inception, thus Ft is set so that Et (Payofft+2) 0. Assume RV follows an AR(1) process. Derive a formula for the fair time t swap rate, Ft. = t+1 (d) Consider the GARCH(1,1) process o7+₁ = 0.1+0.07² +0.92o². Using the fact that o² = Et-1 [7], write this GARCH(1,1) as an ARMA process in & includ- ing the values of the ARMA coefficients. Why is it not technically appropriate to assume the residuals in this ARMA process are Normally distributed? 3. Volatility models (a) Current market volatility is higher than the historical average of market volatil- ity from a long sample. Given this information and the stylized facts about market return volatility, what can you say about expected future market volatility? (b) Explain what 'realized variance' is. Use both a mathematical expression and an intuitive description. (c) A Variance Swap is a very popular over-the-counter derivative contract. In the contract, the fixed leg pays a fixed dollar amount every month, while the floating leg pays an amount proportional to the realized variance in each month based on daily data. Assume any variance risk premium and the risk-free rate are both zero. In particular, the payoff to a long variance position in a 2-month swap is: Payofft+2 RVt+1+RVt+2 - 2 × Ft, where RV, is the realized variance in months and F is the fixed payment decided at the start of the swap (at time t). Recall that swaps have zero value at inception, thus Ft is set so that Et (Payofft+2) 0. Assume RV follows an AR(1) process. Derive a formula for the fair time t swap rate, Ft. = t+1 (d) Consider the GARCH(1,1) process o7+₁ = 0.1+0.07² +0.92o². Using the fact that o² = Et-1 [7], write this GARCH(1,1) as an ARMA process in & includ- ing the values of the ARMA coefficients. Why is it not technically appropriate to assume the residuals in this ARMA process are Normally distributed?

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a If current market volatility is higher than the historical average of market volatility it suggests that there is an expectation of higher future ma... View the full answer