Efficient Methods for Pricing Long-Dated American Options

# How to price long dated options most efficiently?


# How to price long dated options most efficiently?












hi question is how to price a long dated option most computationally efficiently? With European, you use Black Shoals (yes assumption constant vol/rates...etc) but it's a simple algebraic formula.

With American, then what? If you use binomial tree, each step one day, with 20 days you'd have a million nodes...

How do you efficiently for example price an American option with say 10 years tenure?

Anyone implemented with variable "Delta T"? where prices relatively stable coarser "Delta Time" and where prices fluctuates more uses a finer "Delta Time"? Any good paper (Url please)?

Thanks

## Answer by Brian B (score 3, accepted)

https://quant.stackexchange.com/a/11259

Briefly: instead of using trees you should be using implicit (or Crank-Nicholson) PDE schemes. They allow the timesteps to be much larger for a given equity price grid, and allow for boundary conditions to limit the range of equity prices to a realistic regime.

There are (at least) two major markets that have a lot of long-dated american-exercise options: bermudan interest-rate swaptions and convertible bonds. Though I generally agree with Matt that there is good reason to use stochastic vol in these markets, they do not traditionally do so, leaving stochastic vol modeling mainly to exotics desks. Bermudan swaptions, for example, are usually handled in multifactor interest rate models and don't provide a close analogue for your question.

In convertible bonds, the embedded conversion option is exercised at the discretion of the bondholder and typically lasts for many years. This is much closer to what you are asking about. You can therefore get some good inspiration by looking into that literature.

One trick that works (surprisingly?) well is to include random volatility without specifying an extra stochastic factor for it. This is done by linking volatility to the stock price, as in Andersen's paper.

The SDE changes from $$ \frac{dS}S = r(t) dt + \sigma(t) dW $$ to $$ \frac{dS}S = r(t) dt + \sigma(S,t) dW $$ where we can take a variety of forms for $\sigma$, such as $$ \sigma(S,t) ={ \sigma(t) \over S^{2}} $$

The discretization for an implicit PDE solver is then almost exactly as for Black-Scholes.

## Answer by Matt Wolf (score 3)

https://quant.stackexchange.com/a/11255

The standard Black Scholes pricing framework (and its required inputs) is not an optimal model for long-dated European options. Among others put delta is severely understated. Also, you want to keep in mind that implied volatility for long term options exhibits strong auto-correlation with time and reflects a geometric decay pattern. This should lead to the next question then whether the volatility process modeled should not be given much more importance for long-term options over short-term options. Keyword: Stochastic volatility.

Regarding the modeling of rates that are input to your pricing model, it has been shown that stochastic rate processes do not really improve the model for long-dated options).

American options of such long-dated nature can be conveniently modeled using monte-carlo simulations. The seminal paper by Longstaff and Schwartz show how this is done but there are a number other papers that target American option pricing through monte-carlo as well.