# Pricing bonds of float coupon rate by stochastic interest rate

So I am not sure whether the following pricing of the bond is possible. Given the stochastic interest rate, one wants to price the bond with the floating coupon rate or the coupon rate being unknown. How should one price this bond if both forward interest rate and coupon rate are not known in the sense that both are random.

• Is something special in this setting that does not allow you to compute the Expected Value under the martingale measure? I do not see where a problem creeps in. – muffin1974 Jan 17 '16 at 13:05
• @muffin1974 I would expect that the expected value valuation is possible under martingale measure. However, the troublesome part is that given interest rates being stochastic and coupon rate unknown. How do you even solve the equation with more unknowns including coupon rates than equations? Maybe the first proper question should be whether there is a lower bound for the coupon rates or individual coupon rate given the profile of interest rate? Then we might be able to ask how much the bond worth. – user45765 Jan 17 '16 at 16:27
• So you are interested in an analytical solution rather than in a numerical one? Numerically it should be fine to simulate paths and to compute the expected value even if you have many stochastic parameters. Unfortunately I am not familiar with analytical solutions. – muffin1974 Jan 17 '16 at 16:48
• @muffin1974 Do you have any recommendation on readings of analytical solutions on this subject? I have only seen Jensen inequality applied to obtain the bound for the expectation value in a textbook so far. And I also have another unrelated question. Was there a well-defined (either accounting or economical) costs associated with the floating coupon rates(i.e. you will be paid different amount at a different time instead of the flat rate.)? My first guess is that there will definitely something related to shoe leather cost from the seller side and similarly on the buyer side. – user45765 Jan 17 '16 at 16:58
• I am sorry, I am not familiar with that field of research, but I am interested in the answers you'll get on that question. However, I suggest to edit it a little bit as your interest seems broader than you indicate with your question: Maybe you could try to state your question as a mathematical model, this could look familiar to experts in the field. – muffin1974 Jan 17 '16 at 17:02

Consider the calculation period $[T_1, T_2]$ and the floating coupon rate \begin{align*} L(T_1; T_1, T_2) = \frac{1}{T_2-T_1}\left(\frac{1}{P(T_1, T_2)} -1 \right) \end{align*} set at $T_1$ and paid at $T_2$, where $P(t, u)$ is the price at time $t$ of a zero-coupon bond with maturity $u$ and unit face amount.
Let $B_t= \exp\left(\int_0^t r_s ds \right)$ the money market account value to time $t$. Moreover, let $Q$ be the risk-neutral measure and $Q_{T_2}$ be the $T_2$-forward measure. Then \begin{align*} \frac{dQ}{dQ_{T_2}}\big|_{\mathcal{F}_t} = \frac{P(0, T_2)B_{t}}{P(t, T_2)}. \end{align*} Moreover, the value of the floating coupon payment is given by \begin{align*} E_Q\left(\frac{L(T_1; T_1, T_2) \times (T_2-T_1)}{B_{T_2}} \right)&=E_{Q_{T_2}}\left( \frac{dQ}{dQ_{T_2}}\big|_{\mathcal{F}_{T_2}}\frac{L(T_1; T_1, T_2) \times (T_2-T_1)}{B_{T_2}}\right)\\ &=P(0, T_2) E_{Q_{T_2}}\left( L(T_1; T_1, T_2) \times (T_2-T_1)\right)\\ &=P(0, T_2) E_{Q_{T_2}}\left( \frac{1}{P(T_1, T_2)} -1 \right)\\ &=P(0, T_2) E_{Q_{T_2}}\left( \frac{P(T_1, T_1)}{P(T_1, T_2)} -1 \right)\\ &=P(0, T_2)\left( \frac{P(0, T_1)}{P(0, T_2)} -1 \right)\\ &=P(0, T_1) - P(0, T_2). \end{align*} For a floating rate note, or bond, the value is the sum of the values of all coupon payments and the value of the notional payment at maturity. Here the valuation is model-free, no matter the interest rate is deterministic or stochastic.
• So if I consider a FRN that pays coupon one at $T_1$ set at $T_0:=0$, pays coupon two at $T_2$ set at $T_1$, and so on until it pays the last coupon (set at $T_{n-1}$) together with the principal (assumed $1$) at $T_n$. Then its initial value should be $$V=\sum_{i=1}^nV(\text{coupon}_i) + P(0, T_n) = \sum_{i=1}^n(P(0, T_{i-1})-P(0, T_i)) + P(0, T_n) = P(0, T_1).$$ Now I am interested in calculating the duration of this FRN, but am stuck to even what definition of "duration" for a stochastic rate model should be used. Would you care to enlighten me? Thanks! – Vim Oct 27 '18 at 14:08