PDE Discretization for American Swaption Pricing

Summary

The document explores adapting a short-rate pricing PDE to an American swaption. It gives a PDE with rate diffusion, drift, and discounting terms, along with a terminal condition based on the positive part of swap value minus an exercise amount. It also raises questions about how accrual payments enter the swap value, how to define the exercise value, and how the rate, zero-coupon bond price, and swap rate notation relate.

The proposed model example is Vasicek, which gives simple forms for the diffusion variance and drift. The author asks whether this setup is appropriate for simulating bond and swap values from the short rate and for calibrating the model, then suggests discretizing the PDE. No answer or numerical evidence is included, and the American exercise treatment, cash-flow schedule, calibration data, and boundary conditions remain unresolved. The document is therefore a problem formulation and tentative modeling plan rather than a validated pricing method.

Key ideas

• The proposed swaption PDE includes short-rate diffusion, drift, discounting, and an exercise-value terminal condition.
• The document asks how accrual payments and the exercise amount should enter the swap value.
• It proposes a Vasicek short-rate model to simplify the PDE coefficients.
• The relationship among the short rate, zero-coupon bond prices, and swap rates needs clarification.
• No pricing results or validation are supplied, and model calibration and American exercise details remain open.

Tags

# American Swaption Pricing with PDE discretization


# American Swaption Pricing with PDE discretization












So I am still trying to price an american swaption. (MC approach here: American Swaption Pricing with Monte-Carlo method)

I've found in Paul Wilmott, The mathematics of financial derivatives, a PDE for swaption:

$$ \partial_tV + \frac{1}{2}w^2\partial_{rr}V+(u-\lambda w) \partial_rV-rV = 0 $$

with terminal condition:

$$ V(r,T)= max(V_S(r,T)-E,0) $$

where $V_S(r,t)$ is the price of a swap of maturity $T_S$. I think I can use the same formula:

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

Source

1) Is it ok to work with that ? Where is handled the fact that we have accrual payments ? what would be the expression of E ? $V_S(\hat{r},T)$ ? I think in my case that $T=T_S$ is this possible ? this would mean that the last value is 0, no ?

The code seems simple to change between European and american option.

2) The notations are not consistants, what would be the link between r,Z,R ?

Then I think we have to choose a model for r (such that $w², u-\lambda w$ are simple).Vasicek for exemple will give:

$$ w^2 = \alpha_0 $$ $$ u - \lambda w = -\gamma_0 r(t) + \eta_0 $$

3) Is this a good approach ? In term of simulating Z,R after r ? In term of calibrating the model ? (I understand that it depends on what data I will have access to, but for the moment I don't know that, I am not working with swaption for financial market).

I think after the choice of the model it won't be a problem to discretize my PDE.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.