# Can you model the LIBOR rate as a geometric Brownian motion?

i.e. The LIBOR rate is driven in the same way as a stock price in the Black Scholes model.

For example let $$R_t$$ denote the LIBOR rate at time t. the stochastic differential equation (sde) would take the following form:

$$\frac{dR_t}{R_t}=\mu dt + \sigma dW_t$$

Where $$\mu$$ is the drift parameter (mean increment of the LIBOR rate). $$\sigma$$ the (constant) volatility of the LIBOR increment. $$dW_t$$ is the increment of $$W_t$$, a standard Brownian motion under P.

Then you apply Girsanov's theorem for switching to the risk neutral measure Q. Under Q the sde becomes:

$$\frac{dR_t}{R_t}=r dt + \sigma d \tilde{W_t}$$

where $$\tilde{W_t}$$ is a Q-Brownian motion $$r$$ is the risk free rate, (i.e. a risk-free interest rate other than the libor) which is assumed to be constant, as in the Black Scholes model.

Is this approach reasonable?

It is not reasonable because rates display a stationarity but brownian motion is not stationary.

The variance of libor at a future time $$t>0$$ conditional on the value at time $$t=0$$ does not scale as $$\sqrt{t}$$

the equation should reflect part of the true dynamics of the libor. do you really think libor will continue drift in one way? mean reversion is much better. r is not risk free rate by your definition

• as I said r is not the libor. r is another interest rate which is assumed to be constant, and it is used for discounting future (libor based) cash flows. Jan 9, 2019 at 23:24

Your idea is somewhat related to the Black's 76 formula. First of all, one should bear in mind the pricing axioms:

1) Positivity of the LIBOR rates: $$\mathcal{L}(t,T_{i-1},T_i)\geq 0$$.

2) Martingale property under the corresponding forward measure.

3) Analytical tractability. Our goal is to model the LIBOR market: $$\mathcal{L}(t_s,T_{i-1},T_i)=\frac{1}{\delta}\left(\frac{P(t,T_{i-1})}{P(t,T_i)}-1\right)$$

In the Black's 76 model, given the assumption $$\log F(t, T_{i-1}, T_i)\sim\mathcal{N}(\mu,\varrho^2)$$, where $$\mu=\log F(0,T_{i-1}, T_i)-\frac{\sigma^2(0)}{2}T_{i-1}$$ and $$\varrho^2=\sigma^2(0)T_{i-1}$$, the cash flow of the $$i^{th}$$ caplet at time $$T_{i-1}$$ is equivalent to the cash flow of a put option on a $$T_i$$-bond at maturity $$T_{i-1}$$, i.e. $$\text{Cpl}(t,K;T_{i-1},T_i)=\delta P(0,T_i){(F(0,T_{i-1},T_i)\Phi(d_1(i;0))-K\Phi(d_2(i;0)))}$$ with $$d_{1,2}(i;0)=\frac{\log(\frac{F(0,T_{i-1},T-i)}{K})\pm \frac{1}{2}\sigma^2(0)T_{i-1}}{\sigma(0)\sqrt{T_{i-1}}}$$.

Hence, the main assumption is that the forward rate is an exponential Brownian motion under $$\mathbb{Q}_{T_i}$$. This does not mean that all rates are lognormally distributed; only one LIBOR rate for a certain tenor is lognormal. This makes it an unsuitable model for consistent pricing on a term structure.