Pricing Swaption Analytically using Libor Market Model

Question

I was asked the following question in a recent interview: "(i) Express a forward swap rate in terms of forward Libor rates. (ii) Apply Ito's lemma to this expression to derive the process for the forward swap rate. (iii) Finally, use this expression to price a swaption". I couldn't quite figure out the full question on the spot, and wanted to work through it here to see how to solve it properly.

(i) Forward Swap Rate: this is straight forward, nothing complicated here (underlying swap has n-years maturity, is fixed vs. 6-m float)

$$r_T(t)= \frac{\sum_{j=0}^{2n-1}\tau_j L_j\left(t\right) Df(T+h_{j/2})}{\sum_{i=1}^{n}*Df(T+h_i)}$$

Above, $r_T(t)$ is the forward swap rate as of time $t$, where the corresponding swap sets at time $T\geq t$. $L_j(t)$ is the "j-th" forward Libor at time "$t$", that sets at time $T+h_{j/2}$ and expires six month thereafter. $\tau_j$ is the annual fraction. $DF(T+h_i)$ is the discount factor at a particular point in time (i.e. $DF(T+h_i)=P(t,T+h_i)$, with $P(t,T+h_i)$ being a zero-coupon bond expiring at $T+h_i$). Each forward Libor rate follows log-normal diffusion: $$dL_j=\mu_j L_j dt+\sigma_j L_j dW_j$$

(ii) Ito's Lemma: We need to take the first and second derivatives of $r_T(t)$ with respect to each forward Libor $L_j(t)$, and also with respect to time. Here goes: $$\frac{\partial r}{\partial L_j}=\frac{\sum_{j=0}^{2n-1}\tau_j Df(t_{j/2})}{\sum_{i=1}^{10}*Df(t_i)}, \frac{\partial^2 r}{\partial L_j^2}=0, \frac{\partial r}{\partial t} = 0$$

Great news, the first order derivatives are a constant, the second order derivative is zero and the time derivative is also zero, so that:

$$ r(L_1, ..., L_j, ...,L_n)=r_0+\int_{s=0}^{s=t} \left( \sum_{j=0}^{2n-1} \frac{\partial r}{\partial L_j} * L_j(s) \mu_j \right) dt+\\+\sum_{j=0}^{2n-1}\left(\int_{s=0}^{s=t} \left(\frac{\partial r}{\partial L_j}L_j(s) \sigma_j \right) dW_j(s) \right) $$

Could anyone double check if I applied Ito's Lemma correctly above pls?

(iii) Swaption Pay-off: we're now interested in valuing the Swaption denoted $C$ (where $N_j(t)$ is a Numeraire of our choice & $C(r_T(t_0),T_1)$ is the value as of time $t_0$ of a Swaption expiring at time $T_1\leq T$):

$$ \frac{C(r_T(t_0),T_1)}{N_j(t_0)}=E^{N_j}\left[\frac{\left(r_T(T_1)-K,0 \right)^{+}}{N_j(T_1)} \right] $$

Question 1: Is the application of Ito's Lemma in part (ii) correct in yielding the right equation for $r(t)$?

Question 2: I suppose the Swaption formula, with $r(t)$ as derived in part (ii), cannot be solved analytically - is this right?

Answer 1

The present value of a Vanilla Swap (the word Vanilla is used since I am considering the simplest swap, i.e., notional equal to one, contiguous time intervals, constant rate, etc) is given by:

\begin{align} V_s(t) &= \mathbb{E}_t^Q \left[ \sum_{i=1}^N D(t, T_{i+1}) \cdot \tau_i \cdot (L(T_i, T_i, T_{i+1}) - k) \right] \end{align}

where $T$ describes the tenor structure of the fixings and payments, i.e. $0 \leq T_1 \leq T_2, \dots, T_{N+1}$, $\tau_i = T_{i+1} - T_i$, $D(t, T)$ is the discount factor and $L$ is the Libor spot rate.

Let's recall that the forward Libor rate is a martingale under a specific measure:

$$ L(t, T, T + \tau) = \mathbb{E}_t^{T + \tau} \left[ L(T, T, T + \tau) \right] \quad \text{with } t \leq T. $$

Now, performing a change of measure in the swap valuation and using the result given above, we get:

$$ V_s(t) = \sum_{i=1}^N P(t, T_{i+1}) \cdot \tau_i \cdot (L(t, T_i, T_{i+1}) - k). $$

The forward swap rate is defined such the swap value can be computed as:

$$ V_s(t) = A(t) \cdot ( S(t) - k) $$

where $A(t)$ represents the annuity and $S(t)$ the forward swap rate. After some algebra, you get that:

$$ S(t) = \frac{P(t, T_1) - P(t, T_N)}{\sum_{n=1}^{N} \tau_n \cdot P(t, T_{n+1})} = \frac{P(t, T_1) - P(t, T_N)}{A(t)} \quad \text{with } t < T_1, $$

or, equivalently:

$$ S(t) = \frac{\sum_{n=1}^N \tau_n \cdot P(t, T_{n+1}) \cdot L(t, T_n, T_{n+1})}{\sum_{n=1}^{N} \tau_n \cdot P(t, T_{n+1})} \quad \text{with } t < T_1, $$

Now, knowing the dynamics of the Libors $dL(t, T_n, T_{n+1})$ given by the Libor market model, you can apply Ito's Lemma and find the dynamics for $dS(t)$.

Now, suppose that in the European Swaption the holder has the right to enter the previous Swap in $T_1$. Its value at time $t = T_1$ is given by:

$$ V_{es}(T_1) = \max(V_s(T_1), 0) = \left( V_s(T_1) \right)^+. $$

Then, its value at time $t < T_1$ is given by:

\begin{align} V_{es}(t) &= \mathbb{E}_t^Q \left[ D(t, T_1) \cdot V_{es}(T_1) \right]\\ V_{es}(t) &= \mathbb{E}_t^Q \left[ D(t, T_1) \cdot \left( V_s(T_1) \right)^+ \right]\\ V_{es}(t) &= \mathbb{E}_t^Q \left[ D(t, T_1) \cdot \left( A(T_1) \cdot ( S(T_1) - k) \right)^+ \right]\\ V_{es}(t) &= \mathbb{E}_t^Q \left[ D(t, T_1) \cdot A(T_1) \cdot \left( S(T_1) - k \right)^+ \right]\\ \end{align}

Now, switching to the annuity measure (also known as Swap measure $Q^A$), the swaption value is given by:

$$ V_{es}(t) = A(t) \cdot \mathbb{E}_t^A \left[ \left( S(T_1) - k \right)^+ \right]\\ $$

This last expectation can be solved since it is a call option with the Swap Forward Rate $S$ as underlying (using the Black model for example). The only thing remaining is the Swap rate dynamics under the Swap measure $Q^A$. The swap rate is a martingale under this measure since it is given by the subtraction of two numeraire deflated assets, namely $P(t, T_1)/A(t)$ and $P(t, T_N)/A(t)$. The dynamics of $S(t)$ under under the $Q^A$ measure and using the Libor market model are presented in equation (14.30) of the Andersen and Piterbarg Interest Rate Modeling book. Since you are not considering a stochastic volatility model for the Libor market model, it can be simplified a lot. I will do that ASAP and edit my answer.