# Modelling the instantaneous funding spread as a log-normal process

Let us consider a stochastic market model with a fixed short (risk-free) rate $$r\in\mathbb{R}$$. A trader can obtain unsecured funding at a rate $$f_t:=r+s_t$$ where $$s_t$$ is its stochastic funding spread. We assume this spread follows geometric Brownian Motion dynamics: $$\label{model:fnd-sprd}\tag{1} \text{d}s_t=\sigma s_t\text{d}B_t, \quad s(0)=s_0$$ for some Brownian Motion $$B$$ and $$\sigma\in\mathbb{R}_+^*$$. Note that this setup naturally enforces the following desirable property: $$\label{lemma}\tag{2} f_t > r$$

This model can be representative of an economy where $$r$$ is the central bank’s main deposit rate, which is fixed over the short to medium term, and the funding cost of a market participant fluctuates according to its perceived credit quality and market conditions.

Is there anything inherently inaccurate in representing the instantaneous funding spread with model \eqref{model:fnd-sprd}?

Note that I am not interested in exponentials involving $$s_t$$, in which case the log-normal assumption can indeed be problematic when defining a funding account, see for example the discussion on the Dothan (1978), Exponential-Vasicek or Black-Karasinski (1991) models in Brigo and Mercurio (2001).

Log-normal models are not normally used for modelling instantaneous risk factors, I was wondering if it is only because of the previous issue, or there are other reasons $$-$$ for example, instantaneous volatility is level-dependent for geometric Brownian motion (due to the $$s_t$$ term) while it is not in a Brownian motion model.

I don't see anything inherently wrong at first sight, although adding a mean-reverting drift might be more realistic, e.g. a CIR process with $$\sqrt{s_t}$$, especially for long maturities? Also I don't know what you mean by instantaneous risk factors exactly but a log-normal process is used for the instantaneous volatility process in the SABR model which is a very popular. Anyways, my opinion is that for short maturities your assumption on the process should be fine, for longer maturities you might want to think about adding a mean-reversion if needed.