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tibbar
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Expected European option price

January 3rd, 2018, 2:44 pm

Hi,

If I want to estimate the expected price of a vanilla call or put option with strike K, tenor 3 months, in three months time what's the recommended (practical) approach?

If we assume stock price grows at expected rate x%, then a simple approximation might be to put the expected stock price in three month into Black-Scholes using the forward volatility derived from the three and six month implied vols.

Is this a reasonable approximation? Alternatively you could compute this expectation numerically by integrating the B-S formula wrt stock price or go down a stochastic vol approach etc. But it feels intuitively the approx should be reasonable.

Any thoughts appreciated.
 
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outrun
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Re: Expected European option price

January 3rd, 2018, 4:54 pm

This is called a forward start option, it has a simple analytical solution (under B&S conditions), but I can't remember. It'll be easy to find, I think you can even reason you way towards the formula using hedging ideas.
 
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Paul
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Re: Expected European option price

January 3rd, 2018, 5:02 pm

What is practical to you depends on your abilities.

The problem you want to solve has the real expected growth for the first period (and no discounting?) and then risk neutral for the second. That’s easy in the basic Black Scholes model/formulae.

Whether your approximation is any good depends on vols and how far in or out of the money. But generally no it’s not good. If it were good then it would also be good for option valuation which it’s not.
 
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tibbar
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Re: Expected European option price

January 3rd, 2018, 6:13 pm

Thanks. This isn't a forward starting option since no premium is paid up front, it is about estimating the premium that is expected when the roll over rate comes for a hedging strategy where 3 month puts are rolled.

Paul is correct on the approach. So i can simulate geometric Brownian motion for the stock's real world trajectory from time 0 to time 0.25, value the option using black scholes at each path using an appropriate implied vol (risk neutral), and take the median of the simulation.

What would the right implied vol be? Is it the forward implied vol observed at time zero?

I don't see an obvious closed form solution to this, hence my question of whether the above approximation would be reasonable.

Many thanks
 
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outrun
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Re: Expected European option price

January 3rd, 2018, 6:44 pm

If you use GBM in both your simulation and pricing -and two different constant vols- then I think you can get a closed form solutions.

Today you can (continiously) hedge the premium you will pay 3 months from now, it will be a function of just the underlying 3 months from now. Are you going to buy it no matter what?

I think Paul was saying that with GBM you can find closed form, but it's not going to be practical because the market isn't gbm and you'll have smiles in the implied vol etc
 
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outrun
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Re: Expected European option price

January 3rd, 2018, 6:47 pm

It's indeed not a forward starting option since K is fixed. I think the expectation it's just the normal call with an interest rate correction for the 3 months?
 
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Paul
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Re: Expected European option price

January 3rd, 2018, 6:55 pm

On the downside you have:

1. You don't know what the real drift is
2. You don't know what implied vols will be in three months
3. And usual disclaimers about the actual model

On the upside:

1. you have talked about approximations and you don't need  to do this to meet any requirements set by regulators or bosses (correct me if I'm wrong)
2. The fact that there are so many unknowns might give you some flexibility

Since interest rates are so low (am assuming you are talking about typical low-rate economy) then the present valuing doesn't make much difference. And since the time periods are short the growth doesn't matter much anyway. The conclusion is that a simple approximation is just the option value today!!!
 
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Alan
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Re: Expected European option price

January 3rd, 2018, 7:24 pm

What's the underlying?
 
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Paul
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Re: Expected European option price

January 3rd, 2018, 7:30 pm

What's the underlying?
Genius!
 
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tibbar
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Re: Expected European option price

January 3rd, 2018, 8:24 pm

Thanks. My intuition is that the current option price but calculated using the forward volatility implied by the term structure of implied volatility would be a better approximation?
 
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outrun
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Re: Expected European option price

January 3rd, 2018, 9:42 pm

Thanks. My intuition is that the current option price but calculated using the forward volatility implied by the term structure of implied volatility would be a better approximation?
I you *always* but the option 3 months from now, then you might as well buy it now, the only difference is that you can't keep the money in the bank for the next 3 months when you do. 

You would also have to start hedging the option today (or put the delta in your books). ..if you have an agreement -with yourself- to always buy that option 3 months from now, then it means you *already* have exposure. It's not in your trade account yet, but if the stock goes down they you will have to pay more for your put at the time you plan to buy it.
 
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tibbar
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Re: Expected European option price

January 3rd, 2018, 9:51 pm

The background here is analysing the merits of hedging equity downside risk by rolling either short term options or longer tenors - e.g. comparing expected return and expected costs of a rolling 3 month hedging programme versus simply going for a 1 year tenor hedge.

In comparing the price of a 1 year option to the expected costs of rolling 3 month options I need to account for the shape of implied vols.  So there will be an option in place on day 1 and when that expires in 3 months it will be replaced with a new option to provide the same level of protection.
 
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tibbar
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Re: Expected European option price

January 3rd, 2018, 9:55 pm

What's the underlying?
SPX
 
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outrun
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Re: Expected European option price

January 3rd, 2018, 10:11 pm

Thanks, that makes it clear.

One important aspect is: when you replace it, do you buy the same strike as the current option, or do you buy a strike that's a function of the stock at the time of the roll (e.g. 80% of the stock price) ?
 
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tibbar
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Re: Expected European option price

January 3rd, 2018, 10:23 pm

It would be a function of the stock price - e.g. 90% of stock price (it would actually be a collar to make the strategy self financing).