# American Swaption Pricing with Monte-Carlo method

I want to price an American swaption but I am not sure about what I am doing.

Tree methods and PDE discretization seem difficult to adapt to a swaption. I am trying a Monte-Carlo approach. (in another subject I am trying a PDE approach.

First I have american option retrograde equations (timestep $\delta$t):

$$V_t = max(\phi(S_t), E(e^{-r \delta t} V_{t+\delta t} | F_t )$$ $$V_T = \phi(S_T)$$

(source: my old courses)

And Black's formula for an European call swaption:

$$C_t = (\delta \sum_{j=n+1}^{M+1} Z_t^{T_j})[R(t,T_n,T_m) \Phi(d_1) - \hat{R} \Phi(d_2)]$$

(source)

Here are my questions:

1) Is it possible to mix american option retrograde equation with the Black's formula ? What do I need to use for the payoff $\phi$ ? for the expectation (under probability ?) ?

2) What do I need then ? I think the next step is to introduce a model for r, Z or R, calibrate it and then I can simulate it and go for the classical monte carlo method for american option. What are my options now ?

3) Is there any better MC method (QMC or Longshaft-Schwartz) wich would be more adapted ?

I have asked another question to the community about PDE Pricing for American swaption: American Swaption Pricing with PDE discretization

Edit: I think my main question is in fact really simple. If I want to work with known simulated paths ($S_t$).

Can I calculate the $V_t$ backwards simply using $V_t = max(\Phi(S_t),V_{t + \delta t})$ ?

-
American exercise is precisely when you want to avoid using Monte Carlo. – Brian B Apr 16 '14 at 16:20
Why ? what should I use then ? – Were_cat Apr 16 '14 at 18:23
Use trees or other PDE discretization. Though you say they are "difficult" they are the right approach. – Brian B Apr 16 '14 at 18:40
Ok, it is what I have done today. I will write another question. I am still interessed for an answer. – Were_cat Apr 16 '14 at 18:50
Is there any interesting results for swaption with Malliavin calculus ? – Were_cat Apr 16 '14 at 18:50

## 1 Answer

American options pricing (swaption is just a kind of option) is a bit tricky due to the early exercise. Here is a page listing possible approaches, including some numeric methods, and some close form approximation formula.

As I understand, lattice methods (tree, PDE discretization such as forward shooting) are fine to price American options. There're complains on the converge speed but I'm not sure how serious it is.

When it comes to path dependent options, Monte Carlo probably is the most popular method, as now the tree cannot recombine. The problem is now in each path we cannot look ahead into the future to compute $\mathbb{E}\{V(t_{i+1},W_{t_{i+1} } ) | W_{t_i} \}$ , as that information we will not have in real life. Binomial tree won’t have such a trouble, but Monte Carlo world is no longer a filtration. Longstaff (2001) proposed a regression estimation. L.C.G.Rogers improved it a bit afterwards.

So if your model is not path dependent, I'd second Brian on adopting trees or other lattice methods.

-
I begin to understand your answer. If I have multiple single path for $St$. Could I use it backward to find $V_{t+\delta t}$ as a max then $V_t$ and so on ? – Were_cat Apr 28 '14 at 22:56
I don't understand "as a max". maybe you could try read Longstaff first? – athos Apr 29 '14 at 0:05