Using Geometric Brownian Motion for a Funding Spread
Summary
The document considers modeling an unsecured funding spread as a geometric Brownian motion while treating the short risk-free rate as fixed. Because the spread starts positive and evolves multiplicatively, the model keeps it positive, so the resulting funding rate remains above the risk-free rate. This may represent funding costs that vary with a participant’s perceived credit quality and market conditions.
The response sees no inherent flaw in this setup for short maturities, but suggests a mean-reverting process may be more realistic over longer horizons. It mentions a square-root mean-reverting process as one alternative and notes that log-normal dynamics are also used for instantaneous volatility in SABR. The discussion is qualitative: it supplies no calibration, empirical evidence, or comparison of candidate models. Suitability therefore depends on the intended horizon and how well level-dependent volatility and the lack of mean reversion fit the funding spread being modeled.
Key ideas
- A geometric Brownian motion models the funding spread with volatility proportional to its level.
- A positive initial spread stays positive under the stated dynamics, keeping funding above the fixed short rate.
- The response considers the assumption reasonable for short maturities.
- Mean reversion may be worth considering for longer maturities.
- The answer offers qualitative guidance rather than empirical validation or model calibration.
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Full text
# Modelling the instantaneous funding spread as a log-normal process
# 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.
## Answer by BEQuant (score 2)
https://quant.stackexchange.com/a/71180
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.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.