# Short rate models

On the short rate model in Wikipedia

https://en.m.wikipedia.org/wiki/Short-rate_model

Why is the first function, the P(t,T) given? This is not the short rate model this is generating prices for a zero coupon bond. Then a spot curve is taken from the implied rates of that bond

My understanding was that a short rate model projects the instantaneous rate and makes a new forward curve

From that forward curve then you can get the zero curve

What is the relationship between the short rate model and that pricing function?

# The main thing we want is the $$P(t,T)$$ function.

In the short rate model, we model the system as an instantaneous short rate variable which evolves stochastically. Different models assign different dynamics to the short rate (mean reversion, constant or stochastic vol, etc), but they all assume that $$P(t,T)$$ is the expectation of the integral of the instantaneous short rate.

# Because we have to fit it to what we can observe

The model permits you to value things that depend on those paths and derivatives, like path-dependent options or hedges of the greeks. But the things that are observable in the market are largely essentially $$P(t,T)$$ (via cash rates etc), so it is only by relating a model to the things we can observe that we can test it or calibrate its parameters.

# The simplest short rate model is a flat line

Consider the short rate model $$r_t = c$$. We can relate it to observable prices via $$P(t,T) = E_Q(\int_t^T \mathrm{exp}(-r_s))$$, where Q is the risk neutral measure. The shape is fairly boring as an exponential decay, but we could at the very least find the value for $$c$$ which best fits the set of cash rates we can see. We would be able to tell that the model was bad by looking at interest rate options and finding that the price of out of the model options was nonzero.

# The model defines the dynamics - instantaneous changes to the short rate

The model itself defines a set of contributors to the movement of the short rate, rather than defining the level of the short rate itself. For this reason, lognormal models are popular because they prevent the rate evolving below 0% - this turned out to be an artificial barrier in reality. If we can fit the available information both for $$P(t,T)$$ and for option prices, then we probably have a reasonable model to use for pricing things off that standard grid of prices.