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Basic question

DoubleTrouble

DoubleTrouble

Joined
7/19/11
Messages
38
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18
Hello!

I want to discuss what is the most reasonable way to model historical option prices when you only have daily prices of the underlying and the implied vols. It is the first time I'm doing something applied in mathematical finance, so please bear with me :)

Let's make an example out of it. Consider a 1 week call option on a stock index. The option is initialized on a Monday and expires on Friday. Just to be clear about the setting, we have the index closing price and the implied vols for Monday, Tuesday, Wednesday, Thursday and Friday. I let Monday's closing price represent the price on Monday. If we think about the theory and a continuum of prices it's like a piecewise constant price process.

So when a contract with moneyness 105% is initialized on Monday, the strike price will be 1.05*[closing price on Monday]. And on Tuesday one should use the closing price on Tuesday etc. So there is no change in the prices during the days, but rather between the days, which is kind of confusing theoretically but (at least I think so) the most reasonable way to use discrete data.

On the Friday we get a situation where we first use the closing price of the index to get the option price with 1 day left to maturity and then again for the settlement price which would be min(closing price minus the strike, 0).

An alternative way, that I also taught of was to let one day's closing price represent the next day. So the closing price on Thursday will be used to represent the price on Friday and the closing price on Friday will be used to calculate the settlement price. But then we get problem when is comes to initializing the option. Should one then use the closing price from the previous Friday? Which way is most reasonable?

I'd appreciate your input

Thanks in advance!

/DT
 
the option value is replicated by a self-financing portfolio, which is set up prior to price changes and can still hedge small random shift of the underlying price in a short period. Both the value and the component of the portfolio is "previsible" about the underlying price change. Therefore you should always evaluate an option based on all the information you have right now, rather than unveiled information sometime later.

Hull's book have a chapter on how to hedge an option by replication on a daily basis, you can check that out.
 
quotes said:
the option value is replicated by a self-financing portfolio, which is set up prior to price changes and can still hedge small random shift of the underlying price in a short period. Both the value and the component of the portfolio is "previsible" about the underlying price change. Therefore you should always evaluate an option based on all the information you have right now, rather than unveiled information sometime later.

Hull's book have a chapter on how to hedge an option by replication on a daily basis, you can check that out.
Click to expand...

Thank you for your answer, but you misunderstood my problem. I know how to delta hedge. The problem is how I should get the historical option prices. I know how to do it using historical implied vol and the underlying index, just unsure about which values to use when I have discrete data!
 
You can change sample time so long as your option price and stock price are observed at the same time. So it is okay to use Monday opening for both prices.
 
1) it seems wrong to use the closing price to represent that day's price.. it seems more correct to use the day prior's closing price as a proxy for that day's open price.. but both are a bit wrong obviously
2) a 1 week option that is 5% out of the money will have some skew on it, so using the at the money implied vol is wrong (though i guess you've got to weigh in the added pain of figuring out an adjustment with how important the added accuracy is to your problem)
3) if the option expires on friday, you need to settle against friday's price, there's no fudge or workaround for that - use common sense here rather than worrying about consistency in your rules, that's wrong
you can either use open prices (or some volume weighted price) as the days price and then ultimately settle against friday's close, or you can use the day before's close prices as proxies for the day's open prices, and then still settle against friday's close... if you always use every day's close prices to backtest this, you're in effect removing 1 day's worth of time value of the option every single day
 
Thank you very much for your answer financeguy!

financeguy said:
you can either use open prices (or some volume weighted price) as the days price and then ultimately settle against friday's close, or you can use the day before's close prices as proxies for the day's open prices, and then still settle against friday's close... if you always use every day's close prices to backtest this, you're in effect removing 1 day's worth of time value of the option every single day
Click to expand...

But if I use the previous day's closing price as the next day's opening price then I will get a problem on Monday, right? Should I then use the previous Friday's price? That seems odd to me! When I simulate these historical prices, should I even care about opening and closing prices? I think of the trading day as an instant where there exist only one price and for example: The option price on Friday will therefore be 1 day left to maturity and similarily the option price on Monday will be 5 days left to maturity. I don't understand what you mean by "removing 1 day's worth of time value"

I have in fact both opening and closing prices but my advisor suggested that I only use closing prices. Maybe he meant for hedging, where of course the closing prices are the only thing of interest.

Again, thank you for taking your time to answer!
 
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