From Short-Rate Models to Bond Yields and Forward Rates

Summary

A short-rate model specifies the evolution of the instantaneous rate over time. The response explains that this modeled rate can be used to derive zero-coupon bond prices by taking the expected discount factor over the life of each bond. The bond price is then converted into a continuously compounded yield to maturity, which contributes to the spot term structure.

Forward rates can subsequently be calculated from the spot curve using standard term-structure relationships. Thus, the model does not directly choose between producing a discount rate or a forward rate: it models the short rate, from which bond prices and yield curves are derived. The document gives the conceptual chain but does not specify a particular model, pricing measure, or the assumptions needed for a concrete implementation.

Key ideas

• A short-rate model describes the instantaneous rate through time.
• Zero-coupon bond prices are derived from expected discounting at the modeled short rate.
• Bond prices imply yields to maturity and a spot term structure.
• Forward rates can be calculated from the resulting spot curve.

Tags

# Concept Question Regarding Short Rate Model


# Concept Question Regarding Short Rate Model












I have a conceptual question that needs help. Does anyone know whether the short rate model generate discount rate or forward rate?

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/24861

Which is the paper/book you are reading? It should be noted there. But basically in a short-rate model you have

- a model for the short rate $r_t$

- you can calculate zero-coupon bond prices from it by $P_T = E[\exp(-\int_{0}^T r_u du)]$

- from these prices you can calculate the yield-to-maturity $Y_T$ which fulfills $$ P_T =\exp( - Y_T T) $$ thus $Y_T = - \log(P_T)/T$.

From the yield-to-maturity of a zero bond you get the spot term structure and then by the usual calculations forward rates.

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