Vasicek Bond Pricing, the Maturity Payoff, and Market Price of Risk

Summary

The document addresses three conceptual questions about the Vasicek and extended Vasicek term-structure models: why a zero-coupon bond is worth one unit at maturity, why bond prices may be written in an exponential affine form, and what the market price of risk represents. A zero-coupon bond is defined here as paying one unit at its maturity, so its value at that same time is one. If the payoff convention is changed to another principal, the maturity value changes accordingly; the normalization reflects the contract’s units rather than a requirement that every bond pay one.

The exponential affine pricing form is described as an assumption or guess used in early model work, which proves useful in this model and other affine term-structure models. The answer does not derive the form or explain the market price of risk, referring readers elsewhere for that topic. It offers intuition about payoff normalization and model structure, but not a full derivation, calibration method, or empirical assessment of Vasicek model performance.

Key ideas

• A zero-coupon bond’s value at maturity equals its specified principal payment.
• The maturity value of one reflects a one-unit payoff convention and can be rescaled with the principal.
• An exponential affine bond-pricing form is presented as a useful model guess.
• The answer does not derive the affine form or explain the market price of risk.

Tags

# Vasicek and Extended Vasicek Model


# Vasicek and Extended Vasicek Model












I want to ask about basic reasoning in Vasicek and Extended Vasicek model.

- Why $P(T,T) = 1$ for non arbitrage model? Can we place $P(T,T) = 10$ or other numbers? Is it correlated with The Law of Single Price?

- How can you have an idea to write $P(t,T,r) = A e^{-Br}$ or $e^{A - Br}$. Why not other function?

- What is actually market price of risk?

## Answer by AdB (score 1, accepted)

https://quant.stackexchange.com/a/45759

- In general, for $t<T$, $P(t, T)$ is the price at time $t$ of a $T$-maturity zero coupon bond with a principal of $1$. It is commonly called the discount factor between time $t$ and $T$, since it is the value at time $t$ of receiving $1$ unit at time $T$. Using this idea, for $t=T$, it is easy to see why $P(T, T)=1$. What is the value at time $T$ of receiving $1$ unit at time $T$? It is simply $1$! Of course, you can change your notation and let $P(t, T)$ denote a zero coupon bond with a principal of e.g. $10$, thereby changing your numeraire, which would result in the no-arbitrage condition $P(T, T) = 10$.

- This was simply presented as a "guess" in the initial papers. This guess in turn seems to work out very nicely (in this model and other affine term structure models).

- The market price of risk is extensively covered many places on this site and elsewhere.

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.