What is the Role of Martingale Measures in Pricing Options?

[Focus]

Hey,

I was wondering if anyone knows a bit about financial probability theory. I was wondering when you are pricing options, you take the expectation under Q (a martingale measure). In the case of Black and Scholes, this seems simple as it is unique. When the martingale measure is not unique (in the case of Levy processes for example), does it matter which measure you choose to take E under? If so which one do you choose when you wish to price options?

Thanks in advance

 

[gel]

Yes it does matter which one you take, and there is no single correct answer. If there is more than one equivalent martingale measure, and the expectation of the contingent claim is different under different measures, then it can't be hedged exactly and there is not a unique arbitrage price.

However, there are some things you can do (by no means a complete list)
1) find upper and lower bounds on the value. If you are long the claim, to be conservative you should look at the lower bound on its value.
2) look at other related tradable market intruments. Is there a freely traded option market on the underlyer? If there is, these can also be used for hedging and you can restrict yourself to equivalent martingale measures which also correctly price these options.
3) define a utility function. You should hedge so as to maximise your total utility. The price at which you should value the claim is the maximum price which you can pay and still achieve non-negative utility.

Other things such as the so-called minimal martingale measure can be used, although I don't really know if this is used much.

 

[BWV]

If there is not a unique risk neutral price, then there are limits on the possibility of arbitrage

In practice Monte Carlo methods would be most commonly used rather than BSM

 

[Focus]

If there is not a unique risk neutral price, then there are limits on the possibility of arbitrage

In practice Monte Carlo methods would be most commonly used rather than BSM

If the martingale exists then surely there is no arbitrage in the model?!

Yes it does matter which one you take, and there is no single correct answer. If there is more than one equivalent martingale measure, and the expectation of the contingent claim is different under different measures, then it can't be hedged exactly and there is not a unique arbitrage price.

However, there are some things you can do (by no means a complete list)
1) find upper and lower bounds on the value. If you are long the claim, to be conservative you should look at the lower bound on its value.
2) look at other related tradable market intruments. Is there a freely traded option market on the underlyer? If there is, these can also be used for hedging and you can restrict yourself to equivalent martingale measures which also correctly price these options.
3) define a utility function. You should hedge so as to maximise your total utility. The price at which you should value the claim is the maximum price which you can pay and still achieve non-negative utility.

Other things such as the so-called minimal martingale measure can be used, although I don't really know if this is used much.

Sorry I am a bit confused. I have a model on the underlying asset of the option and I wish to price the option to see how well this pricing fits the market. I've read various papers on this by Carr and Madan and such. I ideally want to produce the circles and crosses plots they have. I also have the book Financial Modeling with Jumps by Cont and Tankov that has a section on hedging strategies. The problem is I have no idea which one to use to price options. I really just want to check the model fits against option prices. Sorry I am a bit impaired when it comes to applying stuff.

Thanks for the replies

 

[gel]

yes, it's true that if you have a martingale measure then there is no arbitrage. However, under the Black-Scholes model there is also a unique price at which you can buy/sell the option without introducing any arbitrage (which is what I mean by the arbitrage price). This is also the unique price from which you can exactly replicate the payoff of the option by continuous trading.
If you have more than one equivalent martingale measure then there can be many different prices at which it could trade without introducing arbitrage, and you can't replicate the option payoff. So the maths does not help as much here. In practise, if you were buying or selling an option then the price you would be prepared to pay will also depend on your attitude to the unhedgeable risk that you would be taking on.

Hoever, if you just want to replicate some graphs and generate a consistent set of theoretical option prices, then you need to choose one equivalent martingale measure and use that. The papers you're reading should say which measure is being used to do this, and you should use the same.

 

[Focus]

Thanks for the reply, I will go over the papers again, they seem to cross referencing a lot so it looks like a long road.

Sorry to be such a pain but do you have any idea where I can get some option data from (I have about £120 which is about $200 i think + a library :D)?

 

[gel]

The papers should say which measure they are using to generate the graphs -- i.e. if they specify a SDE for the process which is already in martingale form, and it is not explicitly mentioned which measure they use, then I think you should just use the measure under which the SDE takes that form.

Sorry, don't know where's the best place to get option data from. I could look around, but I'd only be searching with google, which you can do yourself just as easy.

 

[Focus]

My bad, they use Esscher Transform. I will read up on it. Don't worry about searching google, I tried already. Thanks a lot for your help. Nice to see some experts in this field :D. It is certainly interesting.