Janssens Hedge -- Proof
Imagine that "World 1" is a world where we did fork, FORK=yes. World 2 is a world where did not fork, FORK=no.
We buy a portfolio today, and we want the return to be the same in both worlds.
It is desired that Portfolio_1_return = Portfolio_2_return.
1. return_1 = return_2
final_revenue_1 final_revenue_2
2. --------------- = -----------------
portfolio_cost portfolio_cost
The cost of each portfolio should be the same in both cases, since we're buying it today.
3. revenue_1 = revenue_2
4. shares_owned * final_prices_in_case_1 = shares_owned * final_prices_in_case_2
Reminder:
Short Long
Upper Row (Fork) q5 q7
Lower Row (No Fork) q6 q8
This portfolio only ever owns shares of two kinds (q6 and q7). Therefore, we can easily expand this out. p6_1 will denote "the final per share price of token 6, if we end up in world 1".
5. (q6 * p6_1) + (q7 * p7_1) = (q6 * p6_2) + (q7 * p7_2)
We know that, in World 1 (Yes Fork), the price of q6 is equal to 0, because everything in the lower row is worth zero in such a case. In other words, p6_1 = 0.
We know that, in World 2 ( No Fork), the price of q7 is equal to 0, because everything in the upper row is worth zero in such a case. In other words, p7_2 = 0.
Finally, we know that, in World 2, the price of q6 is equal to 1, because the SegWit2x fork never existed and therefore it has a representative price of zero. In other words, we know that p6_2 = 1.
Let us substitute what we know.
6a. (q6 * 0 ) + (q7 * p7_1) = (q6 * 1 ) + (q7 * 0 )
6b. q7 * p7_1 = q6
Rearrange.
7. q7 / q6 = 1 / p7_1
We now see that our hedged portfolio should contain shares that are purchased in a ratio of ( q7 / q6 ), and that this ratio will actually equal that of the ratio 1:p7_1 .