Monte Carlo Methods for Pricing American Swaptions
Summary
The discussion considers how to price an American swaption, whose holder can exercise before expiry. It distinguishes the backward exercise decision—comparing immediate payoff with the discounted conditional value of continuing—from Black’s formula, which is for a European swaption and does not itself provide the early-exercise rule. A model for interest rates or swap rates is needed to generate paths and value future cash flows under an appropriate pricing measure.
The central Monte Carlo difficulty is estimating continuation value using only information available at each exercise date. Longstaff–Schwartz addresses this by regressing later realized values on current state variables; simply taking the larger of current payoff and the next realized path value would use future information and would not produce a valid exercise policy. The answer notes that trees or lattice methods can suit non-path-dependent models, while Monte Carlo is useful for path dependence. It gives no model specification, calibration procedure, convergence evidence, or implementation details, so method choice remains model dependent.
Key ideas
- An American swaption’s value compares immediate exercise payoff with discounted continuation value.
- Black’s formula prices a European swaption and does not resolve early exercise.
- Monte Carlo pricing requires a model for the relevant rates and an appropriate pricing measure.
- Regression methods such as Longstaff–Schwartz estimate continuation value from simulated paths.
- Using future realized path values directly at an exercise date introduces look-ahead bias.
- Trees and lattice methods may be preferable when the model is not path dependent.
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# American Swaption Pricing with Monte-Carlo method
# 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 $V_t = max(\Phi(S_t),V_{t + \delta t})$ ?
## Answer by athos (score 2)
https://quant.stackexchange.com/a/10970
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.