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Applied Business Statistics Case studies Market risk management. Mauro Bufano Risk Management – Banca Mediolanum Spa. Market risk.
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Risk Management – Banca Mediolanum Spa
Market risk is not unique, but there are different sources of risk:
In the chart above we can see different yield curves at different rates. This could impact dramatically in the price of interest rate related assets (e.g. bonds)
Illiquid bond: the bid-ask spread is high and there are few counterparties. Prices are very different among them
Liquid bond: the bid-ask spread is low and there are many counterparties. Prices are very close to each others
Volatility can be measured in different ways:
Even if it’s not a direct observable variable and it’s derived under strong assumptions, implied volatility is used (especially for pricing purposes) because:
In some cases, it’s necessary to simulate volatility paths in order to get a distribution of future assets’ or portfolios’ values.
Different models are available:
GARCH models are widely used for many applications (especially for VaR or scenario analysis), but is generally not a market model for derivative pricing
Its calibration is obtained via maximum likelihood estimation
A common GARCH(1,1) model is the Risk Metrics one, where:
ω = 0
α = 1- β
β = 0.94 for daily data, β = 0.97 for monthly data
In this way the volatility depends on the path of the underlying and on the time. For its calibration it’s necessary to know the first derivative of the option price on time and the derivatives (first and second order) of the option price on the strike. It’s therefore necessary to have a volatility surface
where WtSand WtV are correlated Brownian motions
Heston model has to be calibrated on the quotations of the options listed on the market. It’s generally used for complex derivatives’ pricing
Example of bond pricing:
As we have already seen, risk measurement can be divided into two different families:
We will focus on the 2), that in market risk is quantified with the Value at Risk (VaR)
The VaR is therefore a quantile in the loss distribution
For each asset we need to know the sensitivities to risk factors:
Then, we need to know the variance-covariance matrix of the risk factors (to determine volatilities and correlations)
There are many databases (e.g. Risk Metrics) that provide on a daily basis the variance-covariance matrix of risk factors
Let’s consider a USD dollar bond with a market value (MV) of 1,000,000 €. We have (at least) 2 risk factors:
The VaR of the asset p will result:
i.e. approximately 688,000 €
Volatility measure are generally annualized, if we want the daily VaR we can use the following transformation:
If we consider 252 working days in a year, the daily VaR will be:
N.B. The previous transformation, even if it’s widely used, is given under the assumption that changes in market factors are independent through time (which has been generally denied by empirical studies!)
In historical simulation VaR, risk factors changes are assumed to be represented by their historical empirical distribution:
it’s a mean reverting process: the interest rate r tends to its long term average b with speed a
Therefore VaR cannot be considered a coherent measure of risk
where MV is the market value of the portfolio