Pricing Zero-Coupon Bonds as Risk-Neutral Discounted Expectations

Summary

The document asks how to price a zero-coupon bond from a short-rate model, starting with the risk-neutral conditional expectation of the exponential of the negative accumulated short rate. It names several model families and asks how simulated rates, conditioning on current information, the risk-neutral measure, time-dependent rates, and multiple factors fit into the calculation.

The answer emphasizes that without specifying the short-rate dynamics, the conditional expectation is the pricing expression available; a model may also provide a closed-form bond price. This points to the role of model assumptions in turning the expectation into a usable price. The response does not explain how to estimate the expectation from simulated paths, construct risk-neutral dynamics, or handle the time-dependent and multifactor cases. It directs readers toward model references and fixed-income texts, so the numerical implementation questions remain unresolved.

Key ideas

• A zero-coupon bond price is expressed as a risk-neutral conditional expectation of accumulated short-rate discounting.
• The rate dynamics are needed to derive a more explicit price formula.
• Some short-rate models offer closed-form bond pricing expressions.
• The response does not provide a simulation procedure or resolve the time-dependent and multifactor questions.

Tags

# How to price zero coupon bonds with short term rates model?


# How to price zero coupon bonds with short term rates model?












I want to find the price of Zero coupon bond given a short rate model.

I think about Merton, Vasiceck, CIR, Ho & Lee models.

1) Given a simulation of $r_t$ how can I calculate $ P(t,T) = \mathbb{E}^Q\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right] $ ?

Using the simulations i think it would be easy to calculate the integral. But how to calculate the integral knowing $\mathcal{F}_t$ ? Am I supposed to find an expression of $r_s$ depending on $r_t$ ?

2) How to deal with the risk neutral probability here ?

3) Would this approach still be ok with a time dependant model ? (Hull White) Would this approach still be good with multiple factor model ? (Logstaff Schwartz)

## Answer by SRKX (score 3)

https://quant.stackexchange.com/a/11337

If you do not know anything about the dynamics of you short-rate $r_t$, then there is no way to express the price of the zero coupon bond better than what your already have:

$ P(t,T) = \mathbb{E}^Q\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right] $

You can use a model given in this page where you should be able to find close formulas for the zero coupon bond, if available, in their respective wiki pages or in FI books.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.