Value-at-risk and trading impact
Suppose we consider a trading impact model where the price reduction factor is of the familiar form 1 − αkπ, for selling k shares, except that … π is random. To make matters simple, suppose there are 100 different possible values for π, all equally likely. For example let these be
π0 = .6,π1 = .601,π2 = .602,…,π99 = .699.
Each time we sell some number of shares, one of these π values is chosen with equal probability.
Here we discuss how to compute a selling strategy to sell N shares over T time periods, so as to minimize the Var (value at risk) of the loss at any given target level.
Here, loss is the amount that we don’t get, as a result of the trading impact. That is to say, if we assume that we begin with unit price, then the revenues we would get, if there were no price impact, would be exactly N. Loss is the difference between N and what we actually get.
And what is VaR? Historically, there have been many outlooks on what VaR is, precisely. Con- sider a probability level, 0 < δ ≤ 1. The VaR of a loss function is the maximum number L such that the probability that the loss is more than L is at least δ. To be more precise, we should say “sup” rather than maximum, above.
Exercise. Consider a loss function under a discrete probability distribution, i.e. you have a finite set of possible values for the loss, each with a probability. How do you compute the VaR at any given level?
A related concept is what we will call risk. Consider a set of potential losses under a probability distribution (mass function of pdf). The risk associated with a value x is the probability that the loss is at least x.
Exercise. How do we compute the risk associated with any value?
Exercise. How are VaR and risk related to one another? Suppose you have the VaR for every
probability level – how do you get the risk function?
How do we apply these concepts to our problem? Let S be any selling strategy that sells the N shares over T days. Let δ be one of the values 1/100, 2/100, . . . , 99/100, 1. Then define V (S, δ) to be the Var of the loss attained by strategy S, at level δ.
OK, now we can define the appropriate dynamic programming recursion. Let t be any time period (so 1 ≤ t ≤ T). Let 0 ≤ k ≤ N be a number of shares. Finally let δ be one of the values 1/100, 2/100, . . . , 99/100.
Then, define Ft(k, δ) as the
• minimum value V (S, δ), over all strategies S that • sell k shares between days t and T,
• assuming we start day t at price 1.
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Exercise. How do we get FT (k, δ)?
Now assume we have computed (or rather as we will see, closely estimated) for some time period
t, all values Ft(k,δ) (i.e. over all k and δ). How do we get the Ft−1(k,δ)?
Suppose it is period t − 1 and we choose to sell h shares today. So 0 ≤ h ≤ k. What happens?
• With probability 1/100, our loss today is αhπj +1 for 0 ≤ j ≤ 99, and price changes to 1−αhπj . To apply a Dynamic Programming procedure, we need to identify a quantity as follows, for any
probability level δ.
“The minimum VaR, at level δ over all strategies that sell k shares between day t − 1 and day T ,
assuming we sell h today.”
Let us call this quantity: Wt−1(k, h, δ).
Then: Ft−1(k, δ) = min{Wt−1(k, h, δ) : 0 ≤ h ≤ k}
So let us focus on computing Wt−1(k,h,δ). How do we do it? Let us assume that we sell h shares today (i.e. on day t − 1). Then
• Wt−1(k, h, δ), as defined, is a VaR. Instead of computing this VaR, let us compute the related risk function.
• In other words, for any value x we want to compute the probability that the total loss between now and day T will be x or higher, given that (i.e, conditional on the event that ) we sell h shares today.
• And let us additionally condition on getting some particular exponent πj today. Then today’s loss is αhπj+1. And as a result the probability that the loss between today and time T is at least x equals
Prob(loss from selling k − h shares between t and T is at least x − αhπj +1).
• This computation was based on conditioning on getting a particular exponent πj. So, decon-
ditioning, we get that the probability that the loss is x or higher equals
1 100
Prob(loss from selling k − h shares between t and T is at least x − αhπj +1).
100 j=1
• This formula shows the following fact. The formula shows that if today we want to minimize the VaR at any given risk level, then tomorrow we will ideally minimize each of the terms in the sum. This means that tomorrow we will also seek to minimize VaR.
• Careful! In the above sum, when we compute the loss, we should account for the fact that we do not begin day t at unit price.
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Exercise. Convince yourself that this is the case.
Having computed the risk function, we “invert” it to get the VaR value Wt−1(k,h,δ). Question: Are we approximating any quantity?
Mon.May..6.224754.2019@littleboy
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