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Math Help - Parameter Estimation

  1. #1
    Newbie
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    Parameter Estimation

    Dear all!

    I have some difficulties with estimating parameters. Given a function P(T) with

    P(T) = A\cdot\exp(-B \cdot r),

    where r is a given constant,

    B = B(T,a) = \frac{1-\exp(-a \cdot T)}{a}

    and

    A = A(T,a,b,\sigma) = \exp\left(\frac{(B-T)\cdot(a^2b - \sigma^2/2)}{a^2} - \frac{\sigma^2 B^2}{4a}\right)<br />

    Let P^{\mbox{\tiny obs.}}(T) be some observed datapoints for (e.g.) T \in [1,2,3,...,30] .

    I then formulate a target function \Phi by

    \Phi = \min_{a,b,\sigma}\sum\limits_T \left( P(0,T) - P^{\mbox{\tiny obs.}}(0,T)\right)^2 <br />

    Does anybody have a clue how to estimate the parameters a,b and \sigma to solve the problem above?

    Thanks for any hint!
    Frodo
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Frodo View Post
    Dear all!

    I have some difficulties with estimating parameters. Given a function P(T) with

    P(T) = A\cdot\exp(-B \cdot r),

    where r is a given constant,

    B = B(T,a) = \frac{1-\exp(-a \cdot T)}{a}

    and

    A = A(T,a,b,\sigma) = \exp\left(\frac{(B-T)\cdot(a^2b - \sigma^2/2)}{a^2} - \frac{\sigma^2 B^2}{4a}\right)<br />

    Let P^{\mbox{\tiny obs.}}(T) be some observed datapoints for (e.g.) T \in [1,2,3,...,30] .

    I then formulate a target function \Phi by

    \Phi = \min_{a,b,\sigma}\sum\limits_T \left( P(0,T) - P^{\mbox{\tiny obs.}}(0,T)\right)^2 <br />

    Does anybody have a clue how to estimate the parameters a,b and \sigma to solve the problem above?

    Thanks for any hint!
    Frodo

    Use a non-linear optimisation (Non-Linear Least Squares) tool, the "solver" in Excel should (may) work, as should (may) the Nelder-Mead algorithm (or any other of the multitude of methods available)

    CB
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  3. #3
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    Nov 2009
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    CB, thanks for your hint. I've found a pretty nice simplex-solver in R to solve the minimizing problem mentioned above.

    However, there is another thing I'm thinking about... In the end I want to generate stochastic zero-coupon bond prices P(t,T) for various maturities (T-t) and various points in time t.

    Using the risk-neutral process for the spot-rate suggested by Vasicek (1977) dr = a(b-r)dt + \sigma dz I can simulate the (instantaneous) spot rate for all t and derive all ZCB-prices I need.

    Since there is an explicit formula

    P^{\mbox{\small calc.}}(t,T) = A\cdot \exp(-B \cdot r(t))

    to solve the differential equation dr = ..., one only needs to determine the parameters a,b and \sigma to solve the whole thing.

    Given some today's ZCB-Prices P^{\mbox{\small obs.}}(0,T), T \in \{1,2,3,5,10,15,20\}, does it make sense to estimate a,b and \sigma according to

    \min_{a,b,\sigma} \sum_T \left(P^{\mbox{\small calc.}}(0,T) - P^{\mbox{\small obs.}}(0,T)\right)^2?

    Actually I now know how to solve the minimizing problem, but I'm not sure if this approach makes sense in general.

    Thanks for any hints!
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