Consistency of risk-neutral valuation

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Consistency of risk-neutral valuation

Postby jsg » Mon Nov 25, 2013 5:34 am

Hi Mark. In page 181, you defined the price of an option O such that Ot/Bt is martingale under the risk-neutral measure (i.e. where the drift of St is equal to the risk-free rate). But how can you make sure that this pricing is "consistent"?

I don't know how to formalise what I mean by "consistent", so let me illustrate with an example. Suppose that, using Girsanov's theorem, we change the drift of St to be 0. In this case, St is a martingale. Hence I could define the price of an option O such that Ot is martingale under this new measure. However I realized that this pricing is inconsistent. For example, when O = B is a bond, E(Bt) = B0 * exp(rt), hence it is not martingale.

What I did above is like taking 1 as numeraire. It seems that this causes a problem because there is no asset whose value is constantly 1. However I'm not convinced that taking Bt as numeraire will give a consistent pricing.
jsg
 
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Re: Consistency of risk-neutral valuation

Postby mj » Thu Nov 28, 2013 3:50 am

I am not sure what page you mean.

If you make the price of everything a martingale there can be no arbitrage. Generally this is too hard since the bond is deterministically growing.
But we can make the discounted price of everything a martingale and that is enough.
mj
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