Pricing Swaption Analytically using Libor Market Model

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?

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.

• Thank you. Two questions: (i) My equation for swap rate is the same as yours, we just use different notation (and my summation for the floating forward rates is over six-month intervals, rather than annual). So applying Itol's lemma to the equation for S(t) (in my case (r(t)), do you get the same result as me? (iii) Piterbarg wrote three books, Volume 1 through to 3: which book specifically do you mean when you say Proposition 14.4.4? Sep 8, 2020 at 16:08
• Hi Jan, (i) My equation for the swap rate does not assume annual intervals, they can be anything. Actually, they can refer to non-constant intervals. I have not derived the dynamics of $S(t)$ but it is presented, in a general manner, in proposition 14.4.2 (under the annuity measure). (ii) Yes, they have 3 books but they share chapter index. So chapter 14 refers to the II volume. Sorry for not clarifying this! I would like to point out that the dynamics of S(t) might be presented in another chapter as well, I will try to find them! Sep 8, 2020 at 16:32
• Andersen expression should be simplified a lot to get to your answer but, my intuition believes that your result is correct! I will try to give it a try as soon as I can. Sep 8, 2020 at 16:47
• Thank you for the clarifications, I much appreciate these. Btw, I think in your equation for S(t), the sum in the numerator and the sum in the denominator cannot be over the same index: because the swap rate S(t) is annual (I assume) but the Libors are most probably not annual. If you assume semi-annual Libors, then the sum should be 2n, if quarterly, then 4n: do you agree? Sep 8, 2020 at 17:14
• Just a quick reply: The swap rate $S(t)$ is multiplied by the annuity $A(t)$ and the denominator for $S(t)$ is equal to the annuity, so it vanishes. We have a difference in the Swap valuation since you used $r(t) - K$ and I used $A(t) \cdot (S(t) - k)$. I will take a look at it once I finish working! Sorry for the delay! Sep 8, 2020 at 17:46