3
$\begingroup$

I know following PDE is the continuous payment case, but a caplet pays as rate: $\max(r - r^*,0),$ use the hedge portfolio $\Pi = V- \Delta Z$ $$d\Pi = dV- \Delta dZ +\max(r - r^*,0)dt = r\Pi dt $$ then the result PDE should be $$\dfrac{\partial V}{\partial t} + LV -rV + \max(r - r^*,0) = 0$$ Here $L$ is Black-Scholes operator, and $Z$ is zero=coupon bond respect to interest rate $r.$

But, why in the following book, the constant term is $\min(r,r^*)$ I can't understand that.

enter image description here

$\endgroup$
  • $\begingroup$ The definition of a Caplet given in this book (apparently titled Finite Difference Methods in Financial Engineering, by Duffy) appears to be incorrect. This is not how a Caplet works. It might approximately describe the combination of a Cap and a fixed rate loan. $\endgroup$ – Alex C Mar 15 '17 at 2:58
  • $\begingroup$ @Alex C do you mean the cash flow is impossible to be continuous payed for the such interest swap? $\endgroup$ – A.Oreo Mar 15 '17 at 3:47
3
$\begingroup$

It must be a typo for the equation in the book. That is, the equation for a caplet is of the form \begin{align*} \frac{\partial V}{\partial t} + LV - r_t V +\max(r_t-r^*, 0) = 0, \end{align*} which can also be derived using the martingale approach.

Specifically, note that the accumulated payments from time $t$ up to maturity $T$ is given by \begin{align*} \int_t^T \max(r_s-r^*, 0)e^{\int_s^T r_u du} ds. \end{align*} Let $B_t=e^{\int_0^t r_udu}$ be the money market account value at time $t$. Then, the option value at time $t$ is given by \begin{align*} V_t &= B_tE\left(\frac{\int_t^T \max(r_s-r^*, 0)e^{\int_s^T r_u du} ds}{B_T} \mid \mathcal{F}_t \right)\\ &=B_tE\left(\frac{\int_0^T \max(r_s-r^*, 0)e^{\int_s^T r_u du} ds - \int_0^t \max(r_s-r^*, 0)e^{\int_s^T r_u du} ds}{B_T} \mid \mathcal{F}_t \right)\\ &=B_tE\left(\frac{\int_0^T \max(r_s-r^*, 0)e^{\int_s^T r_u du} ds}{B_T} \mid \mathcal{F}_t\right) \\ &\qquad- B_tE\left(\frac{\int_0^t \max(r_s-r^*, 0)e^{\int_s^T r_u du} ds}{B_T} \mid \mathcal{F}_t \right)\\ &=B_tE\left(\frac{\int_0^T \max(r_s-r^*, 0)e^{\int_s^T r_u du} ds}{B_T} \mid \mathcal{F}_t\right) -B_t\int_0^t \max(r_s-r^*, 0)e^{-\int_0^s r_u du} ds. \end{align*} That is, \begin{align*} M_t \equiv e^{-\int_0^t r_udu} V_t + \int_0^t \max(r_s-r^*, 0)e^{-\int_0^s r_u du} ds \end{align*} is a martingale. We assume that \begin{align*} dr_t = \mu(t, r_t) dt + \sigma(t, r_t) dW_t, \end{align*} where $\{W_t, t \ge 0\}$ is a standard Brownian motion. Then \begin{align*} dM_t &= -r_t e^{-\int_0^t r_udu} V dt + e^{-\int_0^t r_udu}\left(\frac{\partial V}{\partial t} + LV\right)dt\\ &\qquad + e^{-\int_0^t r_udu}\frac{\partial V}{\partial r}\sigma(t, r_t) dW_t + \max(r_t-r^*, 0)e^{-\int_0^t r_u du} dt. \end{align*} Consequently, \begin{align*} \frac{\partial V}{\partial t} + LV - r_t V +\max(r_t-r^*, 0) = 0. \end{align*}

$\endgroup$
0
$\begingroup$

How exactly did you find the first equation:

$$ d\Pi = dV-\Delta dZ + max(r-r^*,0) dt = r \Pi dt\,? $$

I mean how does the term $max(r-r^*,0) dt$ pops up?

Then, it is worth mentioning that:

$$ min(r,r^*) = - max(-r^*,-r) = -(max(r-r^*,0) - r) = r - max(r-r^*,0) $$

Maybe the latter can help.

$\endgroup$
  • $\begingroup$ That is the continuous cashflow, just like dividend paying stock we have $d\Pi = dV - \Delta d S - D\Delta Sd t.$ But one thing I am not sure is that there is a time delay of paying, how to show this in the hedge portfolio? $\endgroup$ – A.Oreo Mar 15 '17 at 1:27
  • $\begingroup$ How can a caplet be considered as a dividend-paying stock??? Besides, a caplet is usually written on a forward rate (e.g. LIBOR). Hence, Black-76 equation should be used instead and therefore the PDE would be different. $\endgroup$ – JejeBelfort Mar 15 '17 at 9:19
  • $\begingroup$ I just make a example of existence of a extra term in Black-Scholes equaiton, and whatever the rate choose, the PDE is only dependent by the dynamic of $r.$ $\endgroup$ – A.Oreo Mar 15 '17 at 15:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.