Pricing American Options with Multiple Noise Sources
Summary
The document considers numerical valuation of American options when the underlying model has multiple sources of randomness, such as stochastic volatility. One response argues that lattice methods can remain preferable to least-squares Monte Carlo (LSMC) with a small number of noise sources, though it gives no benchmark data to support that general comparison.
It also describes estimating the American exercise premium as the difference between American and European values. If a complex model supports reliable European pricing and a simpler model can approximate the premium, their estimates can be combined. This approximation is less suitable when the premium is large or early exercise is likely, because errors in the premium or reliance on long-horizon European dynamics can matter. For validation without a closed-form price, the discussion suggests simplifying or adjusting parameters until results can be compared with an alternative valuation method, then checking that values behave sensibly.
Key ideas
- Lattice methods may remain practical with a small number of stochastic drivers.
- The American exercise premium is the difference between American and European option values.
- A European price from a complex model can be combined with a premium estimate from a simpler model.
- The premium approximation is less reliable when early exercise is likely or the premium is large.
- Numerical methods can be checked by reducing model parameters to cases with an alternative valuation method.
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# Efficient numerical approaches for pricing American Options with multiple sources of noise
# Efficient numerical approaches for pricing American Options with multiple sources of noise
I am looking for efficient numerical approaches for pricing American options when two or more sources of noise are involved (the simplest case coming to mind would be the Heston Model)
Eventhough I am familiar with lattice methods I don't see how this could work in a "poly-noise" setting. A solution might present itself in the form of MCLSQ (Monte Carlo Least Squares). To my knowldge this method however produces somewhat large deviations ?
Questions:
- What are the prevaliant approaches to dealing with this type of situation numerically ?
- How does one benchmark these approximating algorithms if no close form solution for the option-price exists ?
## Answer by Brian B (score 2, accepted)
https://quant.stackexchange.com/a/10821
Generally speaking, if you have two or three sources of noise, you are still going to be much better off pricing American options on a lattice than via LSMC. Too often, LSMC becomes the refuge of academics lacking patience to learn proper lattice techniques.
Now, you can frequently reduce the difficulty of pricing American options by considering the american exercise premium $P$, defined as the difference in value between an american-exercise option and its european-exercise equivalent
$$ P = A - E $$
If you have some complicated stochastic model, but enjoy a technique $f(\cdot)$ for pricing european-exercise options
$$ \tilde{E} = f(x_E;\vec\mu) $$
and you can define some much simpler model $g(\cdot)$ that is good enough for estimating the premium
$$ \tilde{P} \approx g(x_A; \vec\nu) - g(x_E; \vec\nu) $$
then your american option price can be estimated as
$$ \tilde{A} \approx \tilde{E} + \tilde{P} $$
If the american exercise premium is large then relative error in $\tilde{P}$ will be important and this trick will not work as well.
Also, if exercise probability is large, or exercise is likely to happen long before the option tenor, then the trick will fail, since we have introduced a dependency on $\vec\mu$ at (european) timescales well past the relevant timescales for the actual american option.
## Answer by Oblomov (score 1)
https://quant.stackexchange.com/a/10816
Regarding your second question: one possible approach is to reduce the instrument you are trying to value to something simpler, for which an analytical solution are an alternative methodology does exist. You can then vary parameters and check that the valuation is behaving as expected.
If you are using simulations because your price process is more complicated, you can generally tweak the parameters so that it reduces to some other process, which allows an alternative valuation method.
The answer is a bit generic, but your question does not have a lot of detail either.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.