Limits of Geometric Brownian Motion for Modeling LIBOR

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


# 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?

## Answer by Ezy (score 6)

https://quant.stackexchange.com/a/43403

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 $\sqrt{t}$

## Answer by XYQ (score 0)

https://quant.stackexchange.com/a/43402

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

## Answer by FunnyBuzer (score 0)

https://quant.stackexchange.com/a/43438

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.