Notation: is the price of 1 Euro in USD
Note that
Setup: -asset denominated in ; are two independent Brownian motions
where .
The volatility vector of :
The volatility vector of :
Denote the money markets as: , and the corresponding discount process as .
Question: what are possible for currency (denoted as )
we have . (MRP1)
Second asset:
Therefore (MPR2)
The unique set implies the complete market, and there is a unique . The to transform from to is
Under , we have
We can derive that
Suppose we have a portfolio trading shares of , financed by money market in :
The portfolio value in A is
Example: price a call , denominated in
where
Question: What is the risk-neutral measure in Euro, i.e. ?
γ
€ | Money market in A | S | Money market in B |
Currency A | |||
In share | 1 | ||
Currency B | |||
In share | 1 |
To derive the β shareβ from the β shareβ, we can divide by :
These are quotients of martingales under .
We can use to transform from to .
Record:
Recall: the volatility vector of is , the volatility vector of is .
Two examples under :
(1)
(2) Call-put duality
Price of 1 call on = Pirce of puts on .
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