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Estimating Volatilities and Correlations Chapter 21PowerPoint Presentation

Estimating Volatilities and Correlations Chapter 21

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Estimating Volatilities and Correlations Chapter 21

Intro

- The methods here are useful for:
- VaR calculations and
- Derivative pricing

- The methods just recognize that volatility changes over time

Standard Approach to Estimating Volatility

- Define sn as the volatility per day between day n-1 and day n, as estimated at end of dayn-1
- Define Si as the value of market variable at end of day i
- Define ui= ln(Si/Si-1)

Simplifications Usually Made

- Defineui as (Si-Si-1)/Si-1
- Assume that the mean value of ui is zero
- Replace m-1 by m
This gives

Weighting Scheme

Instead of assigning equal weights to the observations we can set

ARCH(m) Model

In an ARCH(m) model we also assign some weight to the long-run variance rate, VL:

EWMA Model

- In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time
- This leads to

EWMA Model Example

- Suppose = 0.9, n-1 = 1%, and un-1 = 2%. Then,
- It means that n = 1.14%.
- Also,

Attractiveness of EWMA

- Relatively little data needs to be stored
- We need only remember the current estimate of the variance rate and the most recent observation on the market variable
- Tracks volatility changes with being the sensitivity to current changes in the market variable
- RiskMetrics uses l = 0.94 for daily volatility forecasting

GARCH (1,1)

In GARCH (1,1) we assign some weight to the long-run average variance rate

Since weights must sum to 1

g + a + b =1

Example

- Suppose
- 1- - = 0.01
- The long-run variance rate is 0.0002 so that the long-run volatility per day is 1.4%

Example continued

- Suppose that the current estimate of the volatility is 1.6% per day at t=n-1 and the most recent percentage change in the market variable is 1%.
- The new variance rate is
The new volatility is 1.53% per day

Maximum Likelihood Methods

In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring

Example 1

We observe that a certain event happens one time in ten trials. What is our estimate of the proportion of the time, p, that it happens?

The probability of the event happening on one particular trial and not on the others is

We maximize this to obtain a maximum likelihood estimate. Result: p=0.1

Example 2: Normal with constant variance

Estimate the variance of observations from a normal distribution with mean zero

Application to GARCH (i.e., variance is time-dependent)

We choose parameters that maximize

Excel Application (Table 21.1, page 485)

Start with trial values of w, a, and b

Update variances

Calculate

Use solver to search for values of w, a, and b that maximize this objective function

Important note: set up spreadsheet so that you are searching for three numbers that are the same order of magnitude (See page 486)

Forecasting Future Volatility

A few lines of algebra shows that

The variance rate for an option expiring on day m is

Volatility Term Structures

- The GARCH (1,1) model allows us to predict volatility term structures changes
- It suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities

Impact of Volatility Changes

- Using the GARCH (1,1) model, estimate the volatility to be used for pricing 10-day options on the yen-dollar exchange rate, given
- the current estimate of the variance, (n)2, is 0.00006
- + =0.9602
- VL = 0.00004422

Forecasting Future Volatility continued (equation 21.14, page 489)

Forecasting Future Volatility continued (equation 21.14, page 489)

- In our Japanese Yen example
- The volatility per year for a T-day option:

Volatility Term Structures (Table 21.3)

Yen/Dollar volatility term structure predicted from GARCH(1,1)

Impact of Volatility Changes continued (equation 21.15, page 490)

- When (0) changes by (0), (T) changes by

Volatility Term Structures (Table 21.4)

The GARCH (1,1) suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities

Impact of 1% change in instantaneous volatility for Japanese yen example:

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