Continuous vs Discrete monitoring Asian option price

[Antonis]

Hi all,

i am trying to solve Asian option pricing using the PDE of Vecer for both discrete and continuous monitoring. I would like to ask you if my prices from discrete monitoring should be higher or lower than these of the continuous monitoring case?

Thank you very much

 

[Tsotne]

Hi all,

i am trying to solve Asian option pricing using the PDE of Vecer for both discrete and continuous monitoring. I would like to ask you if my prices from discrete monitoring should be higher or lower than these of the continuous monitoring case?

Thank you very much

Hi Antonis! First haven't encountered (or at least have never paid attention on) specifically the standard outcome when you can say this happens always. I mean you cannot say either discrete or continuous monitoring gives ALWAYS higher price. Rather you can derive the relationship between them which is as follows: We know that the discrepancy between option prices under continuous and discrete monitoring can be huge depending on the frequency on monitoring dates. Also the convergence of the discretely monitored option price to the continuous case is known to be extremely slow [order 1/(n^1/2)]. You are observing via the BS right? Since it is the only analytical formula. You have many numerous methods for discrete observations which may or may not give higher price than continuous monitoring.

 

[Antonis]

Thank you for your quick reply. So there is no way to check if my results are accurate or not. I am using numerical methods to find the answer such as Crank-Nicolson.

 

[Mr Doe]

The continuous one should be priced higher, asians are options on a basket, underlyings begin the time. More monitoring, more "underlyings".
Running numerical work on PDE's has nothing to do with the premium of the option, so you should find one of those big books on numerical analysis and find the error.

 

[Tsotne]

The continuous one should be priced higher, asians are options on a basket, underlyings begin the time. More monitoring, more "underlyings".
Running numerical work on PDE's has nothing to do with the premium of the option, so you should find one of those big books on numerical analysis and find the error.

Are you sure it is certain in every situation that continuous monitoring gives higher price? I'm not sure it does.

 

[Mr Doe]

Whats bothering you with my reasoning?

 

[Tsotne]

Whats bothering you with my reasoning?

Just haven't noticed this pattern, and also haven't arrived at the relationship of discrete and continuous monitoring if continuous gives higher price all the time.

 

[Mr Doe]

In an "plain" option on basket you would average the price of the underlyings, here the underlying is the time - that is the price of one single underlying on different dates, so thats what I meant by the basket on time. Furthermore, in asian the "underlyings" are highly (auto)correlated if the monitoring is dense - spot price at current time comands, by (risk neutral) probability, where the next price will be; not very far if the price is driven by geometric brownian motion. In options on basket, if the underlyings are independent and priced far away from each other - they can easily wonder off even further, i.e. first underlying priced at 100, second at 1000, each one of them can end up even further, e.g. first at 50, second at 1100. That isnt the case with asians, if the spot price is 100, where would the next price end up? Even if you cant guess you can use the all mighty standard deviation model, i.e. give a confidence interval.
So, in an dense monitoring you are hedging against a lot of underlyings and a lot of their (auto)correlations.


Edit: Please note that my reasoning is highly intuitive, as it should be when dealing with options.

 

[Tsotne]

Thanks. I believe this is a rational assessment, but also it'd be interesting to see an analytical formula proving it. I'll do it later and let's see what we'll get.

 

[Antonis]

Although it was hard for me to find some intuition, i think i have somethink now at least. Nevertheless my code is giving the opossite results at the moment.