Hull-White and the LMM Framework

Hard·25 min read

Derivatives PricingInterest Rate ModelsHull-WhiteLibor Market ModelTerm Structure

Quick Quiz

1. In the Hull-White one-factor model, the time-dependent drift $\theta(t)$ is set by requiring exact calibration to the initial term structure. If the initial term structure is flat at rate $r_0$ , what is $\theta(t)$ ?

\theta(t) = 0

\theta(t) = a r_0

\theta(t) = a r_0 + \frac{\sigma^2}{2a}(1 - e^{-2at})

\theta(t) = r_0

2. A caplet paying $\delta\max(L(T_0;T_0,T_1) - K, 0)$ at time $T_1$ can be rewritten as a put on a zero-coupon bond. What is the effective bond strike $X$ ?

X = K

X = \frac{1}{1 + \delta K}

X = e^{-KT_1}

X = \delta K

3. In the Libor Market Model under the spot Libor measure, the drift of $L_i(t)$ is a sum over forward rates $L_j(t)$ for $j = \beta(t), \ldots, i$ . Which of the following correctly identifies the source of this drift?

The drift arises from the Itô correction when changing from log to level representationThe drift arises from a change-of-numeraire from the

T_{i+1}

-forward measure (where

L_i

is a martingale) to the spot Libor measureThe drift represents the risk premium demanded by investors for holding the Libor rateThe drift is a numerical artefact of the discretisation scheme

4. The Libor Market Model recovers Black's caplet formula exactly for each individual caplet, without approximation.

truefalse

5. The HJM (Heath-Jarrow-Morton) no-arbitrage drift condition states that, under the risk-neutral measure, the drift of $f(t,T)$ is:

Zero for all

T

r_t \cdot f(t,T)

\boldsymbol{\sigma}(t,T)^\top \int_t^T \boldsymbol{\sigma}(t,u)\,du

Equal to the risk-neutral short rate

r_t

6. Why is the Libor Market Model generally not amenable to PDE-based pricing methods, unlike Hull-White?

LMM does not satisfy the Markov property in a finite-dimensional senseLMM requires negative interest rates, which PDE solvers cannot handleLMM has no closed-form characteristic functionLMM uses discrete tenors, which PDEs cannot represent

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