The_Role_of_Volatility

The Key Role of the Volatility for
Financial Engineering
Dr. Zbigniew Krysiak [e-mail] Associate Professor of Finance - Warsaw School of Economics
Visiting Professor at Northeastern Illinois University, Chicago

Financial Mathematics
Mathematics Department at Northeastern Illinois University, Chicago
Wednesday, October 3rd, 2012
1

Agenda
• Approaches to Modeling Volatilities
• Volatility Models in Capital Allocation - VaR
• Application of GARCH to Modeling Volatility
• Volatility Impacts Option Value
• Portfolio Hedging Against Volatility Change

2

Standard Approach to Estimating
Volatility
• Define sn as the volatility per day between day n-1 and day n, as estimated at end of day n-1
• Define Si as the value of market variable at end of day i
1 m
• Define ui= ln(Si/Si-1)
2
sn 

( un  i  u ) 2


m  1 i 1

1 m u   un i m i 1
3

Simplifications Usually Made in
Risk Management
• Define ui as (Si −Si-1)/Si-1
• Assume that the mean value of ui is zero • Replace m−1 by m
This gives

1 m 2 s  i 1 un i m 2 n 4

Weighting Scheme
Instead of assigning equal weights to the observations we can set

s  i 1  i u m 2 n 2 n i

where m  i 1

i

1
5

ARCH(m) Model
AutoRegressive Conditional Heteroskedasticity

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

s  VL  i 1  i u m 2 n 2 n i

where m    i  1 i 1

6

ARCH(m) Model
AutoRegressive Conditional
Heteroskedasticity
Robert Fry Engle is an American economist and the winner of the 2003 Nobel Memorial
Prize in Economic Sciences, sharing the award with Clive Granger, "for methods of analyzing economic time series with time-varying volatility (ARCH)".

7

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