The future CFs of a Bermudan swaption trade are determined by 1) the T&Cs contained in the trade’s term sheet and 2) by the path taken by the underlying fwd rate. The T&Cs include the early exercise dates and maturity date. The SABR model is used to simulate the fwd rate paths. Its three parameters – alpha, beta & rho – control the width and variability of the simulation. The paths determine possible future CFs as they hit the exercise dates. The price of the Bermudan is the probability-weighted avg of the discounted future expected CFs.
The left-to-right channels A, B, & C in the diagram below illustrate three different ways of generating the Bermudan price. Channel B uses the SABR model to simulate the fwd rate paths. The model is deemed to be arbitrage-free and risk-neutral because its parameters and the yc are calibrated to market-quoted rates.
Channel A, on the other hand, is a non-parametric, data-driven approach that does not use market quotes. It is therefore not risk-neutral. Instead, a neural network architecture learns the relationships btw Bermudan trade characteristics and the resulting Bermudan prices. The architecture includes a T&C encoder which encodes the trade contract structure and a market encoder which uses patterns in the historical Bermudan prices to create latent factors.
The NN in each encoder learns by taking data from its input layer and passing it through the nodes of its hidden layers. The NN learning happens using forward propagation and backpropagation. With forward propagation, each neuron in the hidden layer receives inputs from the previous layer. The neuron computes a weighted sum of its inputs, and this weighted sum is passed through an activation function to produce the neuron’s output. The layer’s output is a vector consisting of the individual neuron outputs.
Backpropagation uses a loss fn to measure the diff btw the NN-derived Bermudan price the actual price taken from the input time series of Bermudan prices. The loss fn minimizes this diff or error by propagating backward through the network, updating the weights and biases of the neurons. The network’s accuracy is improved as each Bermudan trade + price passes thru it. An epoch = one full pass of all the Bermudan trades + prices through the NN. The training loop for the NN stops when pre-defined stopping criteria (e.g., train for 200 epochs) are satisfied.
The last layer in the multi-hidden layer of the market encoder represents the latent factors or latent vectors. The latent vector is a compressed view of the actual physical risk factors – the yc and SABR params– from Channel B.
Channel C is a hybrid approach where only the SABR parameters are learned using an NN. The final Bermudan price is not learned. Once learned by the NN, the SABR params are used in the Bermudan valuation process in the same way as they are in the risk-neutral Channel B approach.
