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Total Posts: 21
Joined: Mar 2005
Posted: 2017-04-27 18:05
Years ago we used this as the starting point for an oversimplified yield curve model, which resulted in zero coupon bond yield of maturity T having a formula like

y = r + lam * T - sig^2 * T^2

which we then fit to a bunch of zero coupon bond yields to find the market implied lambda and sigma. r was the risk free rate, lambda was the "market price of risk" and sig was the market implied volatility.

While this model doesn't fit well and has major shortcomings, can someone tighten up my fuzzy recollection of this, and show the correct result of dr = sig*dW. And is there a similar sort of parabolic expression for zero-coupon yields when using

dr_t = a (mu - r_t)*dt + sig*dW_t

I'd like to have a simple model for zero coupon bonds that gives me a market price of risk and vol both implied by a fit to the Treasury zero curve


Total Posts: 5
Joined: Jun 2016
Posted: 2017-04-28 04:09
I barely understand your formulae and desired output but I have worked hard last months into a (brute force) technique that I want to put on test

Given a dataset of n inputs at time t and a $result(t) then; for known n(t+1) I would try to predict $result (t+1) with a high degree of accuracy between constraints.

However not all is easy data liquefaction. The shortcomings may be real for sure because the bounds shall (or may be) too wide to be of practical use.

But if you got an old spreadsheet I would like to experiment with it because the bond market is unexplored to me and would return to you my findings and the outputs, and may be we may work further to optimize the calculations


Total Posts: 5
Joined: Dec 2015
Posted: 2019-04-21 23:31
The short rate model

has ZCB price to be

The yield

is thus

Hold t fixed, or for simplicity set t=0, then interpreting y as a function of T, we will see that y will go to -inf as T goes to inf, i.e., the function y is proportional to the function T |-> -T^2.

To stop the yield having this behaviour, you need to have mean reversion in the short rate model (i.e., a non-zero drift) so that the yield approaches a constant level as T approaches inf. There is no such parabolic expression in the Vasicek model or the Hull White model for this reason.

The model you are looking for is the Hull White 1 factor model dr(t)=[theta(t)-a r(t)]dt+sig*dw(t), where theta is chosen so as to exactly fit the term structure of interest rates observed in the market. You can show that the function theta is given by

... maybe one day ...
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