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BasicConceptsTechniques

BasicConceptsTechniques - IEOR E4602 Quantitative Risk...

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IEOR E4602: Quantitative Risk Management Spring 2010 c 2010 by Martin Haugh Basic Concepts and Techniques of Risk Management We introduce the basic concepts and techniques of risk management in these lecture notes. We will closely follow the content and notation of Chapter 2 of Quantitative Risk Management by McNeil, Frey and Embrechts . This chapter should be consulted if further details are required. 1 Risk Factors and Loss Distributions Let Δ be a fixed period of time such as 1 day or 1 week. This will be the horizon of interest when it comes to measuring risk and calculating loss distributions. Let V t be the value of a portfolio at time t Δ so that the portfolio loss between times t Δ and ( t + 1)Δ is given by L t +1 := - ( V t +1 - V t ) . (1) Note that we treat a loss as a positive quantity so, for example, a negative value of L t +1 denotes a profit. The time 1 t value of the portfolio depends of course on the time t value of the securities in the portfolio. More generally, however, we may wish to define a set of d risk factors , Z t := ( Z t, 1 , . . . , Z t,d ) so that V t is a function of t and Z t . That is V t = f ( t, Z t ) . for some function f : R + × R d R . In a stock portfolio, for example, we might take the stock prices or some function of the stock prices as our risk factors. In an options portfolio, however, Z t might contain stock factors together with implied volatility and interest rate factors. Now let X t := Z t - Z t - 1 denote the change in the values of the risk factors between times t and t + 1 . Then we have L t +1 ( X t + 1 ) = - ( f ( t + 1 , Z t + X t + 1 ) - f ( t, Z t )) (2) and given the value of Z t , the distribution of L t +1 then depends only on the distribution of X t +1 . Linear Approximations to the Loss Function Assuming f ( · , · ) is differentiable, we can use a first order Taylor expansion to approximate L t +1 with b L t +1 ( X t + 1 ) := - ˆ f t ( t, Z t ) + d X i =1 f z i ( t, Z t ) X t +1 ,i ! (3) where the f -subscripts denote partial derivatives and time is measured in Δ units. If time is measured in years then we would replace f t ( t, Z t ) in (3) with f t ( t, Z t ) Δ . The first order approximation is commonly used when X t +1 is likely to be small. This is often the case when Δ is small, e.g. 1 day, and the market is not too volatile. Second and higher order approximations also based on Taylor’s Theorem can also be used. It is important to note, however, that if X t +1 is likely to be very large then Taylor approximations of any order are likely to work poorly. 1 When we say time t we typically have in mind time t Δ.
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Basic Concepts and Techniques of Risk Management 2 Conditional and Unconditional Loss Distributions When we discuss the distribution of b L t +1 it is important to clarify exactly what we mean. In particular, we need to distinguish between the conditional and unconditional loss distributions. Consider the series X t of risk factor changes and assume that they form a stationary 2 time series with stationary distribution F X . We also let F t denote all information available in the system at time t including { X s : s t } in particular. We then have the following two definitions.
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