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APPENDIX: Example calibration to market implied volatilities using the analytical weighted Monte Carlo approach

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APPENDIX: Example calibration to market implied volatilities using the analytical weighted Monte Carlo approach

 

As at some date in the past a leading commercial vendor’s risk system included the following standard deviations and correlations within its covariance matrix risk model for the following four UK equities:

 

Security

Standard Deviation (%pa)

Correlation with

1

0.29

0.33

0.30

AL

16.92

0.29

1

0.35

0.29

BLT

28.56

0.33

0.35

1

0.45

AVZ

36.64

0.30

0.29

0.45

1

BAY

32.85

1

0.29

0.33

0.30

 

The predicted tracking error of a model portfolio consisting of 35% Alliance and Leicester, 35% BHP Billiton, 15% Amvescap and 15% British Airways versus an equally weighted benchmark of these four stocks using this risk model was 5.35%pa.

 

As at approximately the same date the (at-the-money) implied volatilities for (call) options on these equities were approximately as follows:

 

Security

Implied Volatility (%pa)

AL

22

BLT

31

AVZ

30

BAY

27

 

Calibrating simultaneously to these four pieces of market information using the analytical weighted Monte Carlo approach gives a calibrated covariance matrix as shown below.  The calibrated volatility of each individual security now matches its implied volatility.  There are also some changes to individual correlations. The tracking error of the model portfolio using this calibrated covariance matrix is 4.77%pa.

 

Security

Standard Deviation (%pa)

Correlation with

AL

BLT

AVZ

BAY

AL

22

1

0.35

0.33

0.30

BLT

31

0.35

1

0.33

0.26

AVZ

30

0.33

0.33

1

0.36

BAY

27

0.30

0.26

0.36

1

 

If we were also to calibrate to an ‘index’ implied volatility, then the individual security implied volatilities would remain the same, since they are already calibrated to the market. What instead would happen is that correlations between securities would change. There is no listed option relating to a basket of these four stocks.  If the above model fully matched market implieds then the ‘index’ implied volatility would be 19.3%pa. But suppose that there was an observable ‘index’ implied volatility and that it was 15%pa. The covariance matrix would then be as follows. The tracking error of the model portfolio would increase to 5.73%pa.

 

Security

Standard Deviation (%pa)

Correlation with

AL

BLT

AVZ

BAY

AL

22

1

0.13

0.09

0.06

BLT

31

0.13

1

0.03

-0.05

AVZ

30

0.09

0.03

1

0.12

BAY

27

0.06

-0.05

0.12

1

 

Conversely, if our ‘index’ basket actually had an implied volatility of 21%pa then the covariance matrix would become:

 

Security

Standard Deviation (%pa)

Correlation with

AL

BLT

AVZ

BAY

AL

22

1

0.46

0.45

0.41

BLT

31

0.46

1

0.46

0.40

AVZ

30

0.45

0.46

1

0.47

BAY

27

0.41

0.40

0.47

1

 

The average correlation between individual securities is quite sensitive to divergent movements between index implied volatility and average single security implied volatility, as is the tracking error of the model portfolio, which would now be 4.30%pa.

 

As we might also expect, each individual security calibration point disproportionately affects the volatility of that particular security. For example, suppose we only calibrated the original risk model to one implied volatility, namely the one for Amvescap. The calibrated volatilities would then be as follows:

 

 

Volatility prior to calibration (%pa)

Volatility post calibration (%pa)

Change

AL

16.92

16.62

-2%

BLT

28.56

27.97

-2%

AVZ

36.64

30.00

-18%

BAY

32.85

31.75

-3%

 


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