We are going to deal with a simple but common situation in finance - if I have a contract where I am going to receive $100 dollars in 3 years time what is that 'contract' worth to me now. How much would I pay to obtain that contract? In order to calculate the worth we need to consider what else I would do with the money and the most obvious action is to deposit it into a bank account that attracts interest.
The question is reposed then as: if I put x into a bank account then what is x if the final amount after 3 years is $100. This is easy if the interest is fixed, not so easy if it varies.
This blogpiece will provide a fragment of the implementation of HWFC that answers the above.
As this is literate Haskell some preliminaries:
> module Main where
>
> import Probability
HWFC introduces the concept of a value process which is a function from time to a random variable. We shall equate a random variable with a probability distribution and a definition of a value process is:
> type PR a = Int -> Dist a
For our interest rate model let us say that from year to the next the interest rate can either stay the same, increase by 1% or decrease by 1% all with equal likelihood. We can express this as:
> interest :: Floating a => a -> PR a
> interest i n = (n *. one) i where one start = uniform [start+1/100,start,start-1/100]
The *. function allows us to repeat a random process n times. The process here is to start with an interest rate and to move to the next years rate.
If this year the rate is 10%, after a couple of years the distribution looks like:
interest 10 2
10.0 33.3%
9.99 22.2%
10.01 22.2%
9.98 11.1%
10.02 11.1%
Let us put that to one side and look at the contracts side of things. I will short circuit the approach in the paper and dive directly into the valuation
> data Obs a = O { evalObs :: PR a }
>
> konst k = O (\t -> certainly k)
> lift f (O pr) = O (\t -> fmap f (pr t))
> lift2 f (O pr1) (O pr2) = O (\t -> joinWith f (pr1 t) (pr2 t))
> date = O (\t -> certainly t)
>
> data Contract = C { evalContract :: PR Float }
> cconst k = C $ \ _ -> certainly k
> when o c = C $ disc (evalObs o) (evalContract c)
>
>
> at t = lift2 (==) date (konst t)
> zcb t x = when (at t) x
>
> whenFirstTrue :: PR Bool -> Int
> whenFirstTrue prb = f 0 where f i = if prb i == certainly True then i else f (i+1)
>
> baseRate = 10
This is a value process such that if when the first argument is true, return the second otherwise calculated the discounted value of the first argument.
> disc :: PR Bool -> PR Float -> PR Float
> disc prb prd t = if prb t == certainly True then prd t else let s = prd t
> t' = whenFirstTrue prb
> in discount baseRate s (t'-t)
>
> discount :: Floating a => a -> Dist a -> PR a
> discount int final time = let intspread = interest int time
> in joinWith (\i s -> s / (1+i/100)) intspread final
>
Lets start with a trivial example to make sure that things are working as planned
> ex1 = cconst 100
The value of this contract, as a random variable, is:
evalContract ex1 0
100.0 100.0
> ex2 = zcb 3 (cconst 100)
The value of this contact is:
evalContract ex2 0
90.90909 25.9%
90.900826 22.2%
90.91736 22.2%
90.89256 11.1%
90.92562 11.1%
90.88431 3.7%
90.93389 3.7%
The PFP library has a function to provide the expected value which can be ask of a distribution. The expected value of our contract is:
expected $ evalContract ex2 0
90.9091
