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 $V_t = max(\Phi(S_t),V_{t + \delta t})$ ?