Cap Pricing - 2023.2 English

A cap is a basket of caplets, where all caplets have the same strike (caprate). Each caplet will have a payoff at time \(T_1, T_2, ..., T_n\). The price of the cap will be the sum of all the caplets. The pricing of caps with the LMM framework is interesting because we can use it to validate the model and the calibrations by comparing the output of the MonteCarlo simulation with the output from the analytical Black76 model. Once we are satisfied with the results from the model, we can use the same parameters to compute the pricing of other options that don’t have analytical formulas.

The general formula for the price of a cap with notional \(N\) and caprate \(K\) is given by:

\[Cap = \sum_{i=1}^n Caplet(T_i)\]

Analytically, we can use the Black76 formula to calculate the price of a caplet with:

\[Caplet_{Black76}(t) = P(t, T_{i+1})\tau_iN(L_i(t)\phi(d_1) - K\phi(d_2))\]

\[d_1 = \frac{log(\frac{L_i(t)}{K})+\frac{1}{2}\sigma^2t}{\sigma\sqrt{t}}, d_2 = d_1 - \sigma\sqrt{t}\]

\[\phi(t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{t}e^{-\frac{1}{2}x^2}\]

\[P(t, T_i) = e^{-rT_i}, r = \frac{1}{\tau_i}log(1+\frac{1}{i}\sum_{k=1}^{i}(L_k(t)\tau_k))\]

With the LIBOR market model, we can calculate the price of a caplet with the following formula:

\[Caplet_{LMM}(t)=\tau_{t-1}N(L_t(t)-K)^+\frac{B(0)}{B(t+1)}\]

\[B(t) = [\prod_{k=t}^{n}(1+\tau_kL_k(t))]^{-1}\]

After generating enough paths, the average of all Cap prices with the LMM will converge to the value from the Black76 formula provided the model is correctly calibrated.