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.


1 Answer 1


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.

  • $\begingroup$ Thank you @BEQuant. The SABR model is usually used as an interpolation method for implied vols directly, so market-observable rates such as USD 3M Libor as opposed to modelled instantaneous rates such as the short rate in a Hull-White model. Noted the mean-reversion, actually my intention is to use an exponential Vasicek model which includes mean-reversion. $\endgroup$ Jun 8 at 12:26

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