## Bermudan swaptions: exercise boundaries

The files in this directory plot a two dimensionsal projection of the exercise boundary of a Bermudan swaption in the Jaeckel coordinates

x=L_t(T_t), y=S_{t+1,n}(T_t)

(cf P. Jaeckel, Monte Carlo Methods in Finance, McGrawhill, pp. 171-181). An at the money Bermudan swaption along the tenors

T_p < T_{p+1} < .... < T_n

with quartely accrual T_{t+1}-T_t=0.25, n=60 and p=14 is priced in the two factor lattice for a driftless Libor Market Model with constant volatilities. The T_j denote the Libor reset dates.

We fix t with p < t < n and traverse all nodes in the lattice at time T_t. For each node we compute the statistics x,y and record wether we have exercise or no exercise at the node. This is then plotted in a coordinate system for (x,y) as follows:

The variables x and y are restricted to 0 < x,y < 5k, where k is the strike rate of the swaption. We draw the axes x=k and y=k for orientation and split the coordinate rectangle into 75^2 squares by subdividing each coordinate axis into N=75 intervals of equal length. This gives us a grid size of dx=dy=k/15.

For each square we compute an exercise probability by noting how many nodes with an (x,y) statistic falling into this square exercise and how many do not exercise the swaption. We then color the square according to the following rule:

• p>=0.99: black (always exercised)
• 0.8<=p<0.99: brown
• 0.6<=p<0.8: green
• 0.4<=p<0.6: red (maximum ambiguity)
• 0.2<=p<0.4: blue
• 0.01<=p<0.2: cyan
• p<0.01: yellow (never exercised)

The computation is carried out for t=38 - 49. "Exercise38.ps" is the file with t=38. The coordinate range contains 5625 squares but many squares have no observation (x,y) and remain white. The number of nodes indicates how well the squares are covered:

#### Number of nodes at time T_t

t=38: 36481 nodes.
t=40: 40401 nodes.
t=42: 44521 nodes.
t=44: 48841 nodes.
t=46: 53361 nodes.
t=48: 58081 nodes.