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Extra info for An Explicit Approach To Elementary Number Theory
In this new “currency” the claim pays , for t ∈ [0, T] and at maturity; its price is at t ∈ [0, T]. 17) where E t is the expectation under Qπσ. In other words the process , expressing prices and payments in the new numeraire, is a Qmartingale. The answer to our first question is indeed affirmative. But it is important to realize that the equivalent martingale measure had to be constructed in a very specific way in order to preserve the martingale property in the new currency system. In this operation changes in units of account are intimately related to changes of measure (and conversely).
8. OPTION AND FORWARD CONTRACTS 23 for t ∈ [0, T]. 23) The key idea behind put-call symmetry is a change of measure converting a put option in the original economy into a call option with symmetric characteristics in the auxiliary economy. In effect this change of measure amounts to a change of numeraire, as outlined in the previous section, in which the dividend-adjusted underlying asset is taken as the new unit of account (the numeraire is and . Expressing the payoff in this new numeraire and correcting for dividends where .
7. APPENDIX: PROOFS 53 Substituting in the prior expression and eliminating the Q-martingale component gives for all t ∈ [0, T]. 10) follows Proof of Theorem 24: Using the definition of the Snell envelope we can write, for any , that . 11) is an upper bound for the claim’s price. 11) is also a lower bound for the claim’s price. Recall the definition of the Snell envelope and that the assumptions on the discounted payoff imply that Z is a supermartingale of class D. As indicated before, the Doob-Meyer decomposition shows that where MZ Z is a martingale that is null at zero and A is a continuous, non-decreasing, adapted process with also null at zero.