# 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

-
BS is not a good 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. (contrary, it has been shown that stochastic rate processes do not really improve the model for long-dated options). My 2 cents... –  Matt Wolf May 12 '14 at 5:59
Yes thanks Matt I understand BS is not good for American especially long dated - problem is, with tree implementation (lattice), price tree grows to very big very quickly. How do you deal with this...? (Of course, you can have a coarse "Delta T" for example, one year per step as supposed to one day per step) –  Swab.Jat May 12 '14 at 6:04
I said European. I do not have much experience pricing long-dated American options. Maybe someone else can chime in. –  Matt Wolf May 12 '14 at 6:39
@MattWolf You may want to add that comment as an answer. –  John May 12 '14 at 14:24

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.

-
Thanks Brian ... –  Swab.Jat May 13 '14 at 1:10

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.

-
Sorry I'm not asking European - the question is American perhaps I didn't state this clearly. Monte Carlo less computationally intensive but you get different numbers (Valuation, thus unrealised pnl as well which our risk managers don't appreciate,,) each trial, we only use for Asian and look back options, path dependent structures only. –  Swab.Jat May 12 '14 at 23:20
I do not find your criticism towards Monte Carlo simulations valid - if you use the same underlying model, for example a specific discretization, you get the exact same value (option price or greek or what have you), regardless of whether you derived the value through BS or MC. The difference originates from the underlying pricing model or model that you use to drive the asset price evolution. –  Matt Wolf May 13 '14 at 1:02
No that's incorrect. It's description of fact, not criticism. And further, Monte Carlo is RAND by nature. excelvbaexplained.wordpress.com/2012/09/09/… –  Swab.Jat May 13 '14 at 1:05
You should read up on the underlying methodologies. If you make the time spans between discretizations small enough the solution converges to exactly the same as if you used a closed form solution (given same underlying model). There is a reason monte carlo approaches are perused at most every exotic rates and equity desk. By the way a lot of banks locate the pricing of long-dated option contracts with the exotic guys because those are the ones that are better at stripping out dividend curves, and anything else that affects long-date contract pricing. –  Matt Wolf May 13 '14 at 1:22
You are welcome, and the difference in final price are negligible, given you compare apples with apples, model-wise. Unless I know exactly what alternative pricing approach we are talking about I would not know which one is computationally more intensive. –  Matt Wolf May 13 '14 at 5:13