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Bachelier Caplet Pricing Under the Forward Measure

Article Quant Q&A · Author: Romain
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Summary

The document sets up caplet valuation under the forward measure when the forward rate follows a normal, Bachelier-style dynamic. It expresses the caplet price as a discounted expectation of the positive part of the forward rate less the strike, then attempts to derive the forward-rate process from the difference between two zero-coupon bond prices. A change of measure is used to relate the Brownian motions under the money-market and bond-forward measures.

The author’s calculation reaches an unresolved drift term when applying Itô’s formula, so the document does not provide a completed pricing formula or numerical example. Its useful lesson is the valuation setup and the need to handle the numeraire change consistently when deriving the rate dynamics. The volatility notation and measure-change steps are not fully clarified, and the post does not establish the assumptions required for a closed-form Bachelier price. Readers should treat it as a question motivating a derivation, rather than a validated pricing recipe.

Key ideas

  • A caplet can be valued as a discounted expectation under the measure associated with its payment-date bond numeraire.
  • The payoff is expressed using the positive part of the forward rate minus the strike.
  • The forward rate can be related to a ratio involving two zero-coupon bond prices.
  • Changing to a forward measure changes the Brownian motion through a drift adjustment.
  • The document leaves the Itô derivation unresolved and does not present a completed pricing formula.

Tags

Fixed incomeOptionsDerivatives pricingVolatility
Full text
# Pricing caplet with Bachelier (normal dynamic) using forward measure


# Pricing caplet with Bachelier (normal dynamic) using forward measure












I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution. Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing :

$C_t(T,T+d)$ represents the price of a Caplet at time t, for a Caplet between T and T+d. \

$B(t,T)$ represents the ZCB between at time t, of maturity T. \

$r_t(T,T+d)$ represents the forward rate at time t, between T and T+d. \

$\Big( \dfrac{C_t(T,T+d)}{B(t,T+d)} \Big)_{t \in [0,T]}$ ($\mathbb{F}, \mathbb{P}^{T+d}$)-martingale, and : \begin{align} C_t(T,T+d) &= B(t,T+d) \cdot \mathbb{E}^{T+d} \Big[ \, \dfrac{C_{T+d}(T,T+d)}{B(T+d,T+d)} \; | \; \mathcal{F}_t \Big] \\ &= B(t,T+d) \cdot d \cdot \mathbb{E}^{T+d} \Big[ \, (r_T(T,T+d)-K)^+ \; | \; \mathcal{F}_t \Big] \end{align}

To calculate this, we must find $B^{T+d}$ under $\mathbb{P}^{T+d}$ : \begin{align} \begin{cases} P^{T+d} = L^{T+d}_{T+d} \cdot \mathbb{P}^* \qquad &\Big( L^{T+d}_{T+d} = \dfrac{\mathrm{d}\mathbb{P}^{T+d}}{\mathrm{d}\mathbb{P}^*} \Big) \\ L_t^{T+d} = \dfrac{B(t,T+d)}{P_t \cdot B(0,T+d)} \qquad &L^{T+d}_{T+d} = \dfrac{1}{P_{T+d} \cdot B(0,T+d)} \end{cases} \end{align}

$\sigma(t,T+d) \in \mathcal{F}_t$ represents the local volatility of $B(t,T+d)$, one get : \begin{align} \mathrm{d}B(t,T+d) = r_t.B(t,T+d).\mathrm{d}t + \sigma(t,T+d).\mathrm{d}B^*_t \end{align}

Then, with Girsanov's theorem : \begin{align} B_t^{T+d} = B^*_t - \displaystyle{\int_{0}^{t}} \sigma(s,T+d). \mathstrut{d}s \qquad \mbox{un } (\mathbb{F}, \mathbb{P}^{T+d})\mbox{ - MB} \end{align}

One get : \begin{align*} \begin{cases} X_t &= B(t,T) - B(t,T+d) \\ \mathrm{d}X_t &= r_t.X_t.\mathrm{d}t + \sigma_t^X . \mathrm{d}B_t^* \qquad \mbox{avec } \sigma_t^X = \sigma(t,T) - \sigma(t,T+d) \end{cases} \end{align*}

Then we should apply Ito's formula : \begin{align*} \mathrm{d}r_t(T,T+d) &= \dfrac{1}{d} \cdot \mathrm{d}\Big( \dfrac{X_t}{B(t,T+d)} \Big) \\ &= \rho_t(T,T+d).\mathrm{d}B_t^{T+d} \end{align*}

But here's my problem : when I apply it, I don't find the good Brownian motion, but $\mathrm{d}B_t^{T+d} = dB^*_t-\dfrac{\sigma(t,T+d)}{B(t,T+d)}dt$. \

Has anyone ever done this pricing ? Or do you have any indication to realize it ? I know the derivative of the forward rate, but I can't demonstrate it.

Thank you very much for reading me !

Romain

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.