In Tomas Bjork's Arbitrage Theory in Continuous Time (or here), $\exists$ this proposition
Proposition 2.9 Suppose that a claim X is reachable with replicating portfolio h. Then any price at t=0 of the claim X, other than $V_0^{h}$ will lead to an arbitrage possibility.
My prof uses $V_0({\phi})$ instead of $V_0^{h}$, but $\phi$ still refers to the portfolio.
Let $\Pi(t;x)$ be the price of the contingent claim at time t. Then, $\Pi(0;x)$ must = $V_0({\phi})$.
This is the proof written on the board:
Suppose $\Pi(0;x)$ > $V_0({\phi})$.
The arbitrage strategy is:
Sell (or short sell) the claim for $\Pi(0;x)$, and obtain the portfolio $\phi$ worth $V_0({\phi})$.
Left over amount is $\Pi(0;x)$ - $V_0({\phi})$.
At t = 1, the payoff for the claim X w/c you will be liable for will be covered the value of the portfolio $V_1({\phi})$ at t=1.
Suppose $\Pi(0;x)$ < $V_0({\phi})$.
The arbitrage strategy is:
Sell (or short sell) the portfolio worth $V_0({\phi})$. Use that amount to buy claim worth $\Pi(0;x)$.
Left over amount is $V_0({\phi}) - \Pi(0;x)$.
At t = 1, you will get payoff X, w/c is equal to $V_1({\phi})$.
Soooo I tried constructing the arbitrage strategy for the first part to see the exact profit, but I seem to be missing a step.
Suppose $\Pi(0;x)$ > $V_0({\phi})$.
At t = 0,
transaction $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ cash flow
1 short/sell claim for $\Pi(0;x)$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ +$\Pi(0;x)$
2 Buy $\phi$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ -$V_0({\phi})$
3 Invest $\Pi(0;x)$ - $V_0({\phi})$ at R until t=1 $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ -($\Pi(0;x)$ - $V_0({\phi})$)
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ 0
At t = 1,
transaction $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ cash flow
1 collect investment $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ +($\Pi(0;x)$ - $V_0({\phi})$)(1+R)
2 Portfolio grows in value $\phi$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ +$V_1({\phi})$
3 [...]? $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ -$V_1({\phi})$
4 Close short position if needed $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ 0
Profit $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ +($\Pi(0;x)$ - $V_0({\phi})$)(1+R)
Where does the $V_1({\phi})$ go? I was thinking that we were supposed to borrow $\frac{V_1({\phi})}{1+R}$ at t=0 so it would look something like:
Suppose $\Pi(0;x)$ > $V_0({\phi})$.
At t = 0,
transaction $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ cash flow
1 short/sell claim for $\Pi(0;x)$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ +$\Pi(0;x)$
2 Buy $\phi$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ -$V_0({\phi})$
3 Borrow $\frac{V_1({\phi})}{1+R}$ at R until t=1 $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ +$\frac{V_1({\phi})}{1+R}$
4 Invest $\Pi(0;x)$ - $V_0({\phi})$ + $\frac{V_1({\phi})}{1+R}$ at R until t=1 $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$-(\Pi(0;x)$ - $V_0({\phi})$ + $\frac{V_1({\phi})}{1+R})$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ 0
At t = 1,
transaction $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ cash flow
1 collect investment $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$(\Pi(0;x)$ - $V_0({\phi})$ + $\frac{V_1({\phi})}{1+R})(1+R)$
2 Pay debt $\phi$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ -$V_1({\phi})$
3 Close short position if needed $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ 0
Profit $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ +($\Pi(0;x)$ - $V_0({\phi})$)(1+R)
Or is there some action for the missing step that makes both of them equivalent?
