# Parameter Estimation

• Nov 30th 2009, 03:57 AM
Frodo
Parameter Estimation
Dear all!

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

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

where $\displaystyle r$ is a given constant,

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

and

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

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

I then formulate a target function $\displaystyle \Phi$ by

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

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

Thanks for any hint!
Frodo
• Nov 30th 2009, 04:22 AM
CaptainBlack
Quote:

Originally Posted by Frodo
Dear all!

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

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

where $\displaystyle r$ is a given constant,

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

and

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

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

I then formulate a target function $\displaystyle \Phi$ by

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

Does anybody have a clue how to estimate the parameters $\displaystyle a,b$ and $\displaystyle \sigma$ 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
• Dec 1st 2009, 08:35 AM
Frodo
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 $\displaystyle P(t,T)$ for various maturities $\displaystyle (T-t)$ and various points in time $\displaystyle t$.

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

Since there is an explicit formula

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

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

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

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

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!