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Hi All,
I have a problem as follows, and was looking for some help:
There are two assets A and B, of unit price
and respectively, which are traded on the open market with a price ratio of:The instantaneous price ratio at any time is determined by the market but is always between these bounds. i.e. we always have:
Let's suppose I start by depositing an initial number of units of asset A into an account,
, having a value ofEvery time I deposit a number of units of asset A into the account, I am gifted an equal number of units of asset B,
, having a value of:Where:
I am free to withdraw asset B ONLY, and may trade it on the market in return for an equal-valued amount of asset A. In other words, I can sell all of my asset
, and receive more asset A, of the same value:I may then deposit the new amount of asset A,
, back into the account, where I will again be gifted an equal amount of units of asset B, which again I may withdraw trade, and re-deposit. I may do this as many times as I like, let's say N times, to increase my amount of asset A in the account from what I initially deposited.For this problem, to simplify, let's suppose we have a price ratio:
that is CONSTANT and does not change each time we do this.Note also here that unless
, then we always have and therefore . So to simplify again let's suppose , then we can see that if I repeat this process an infinite number of times (N tends to infinity), the amount of asset A in the account will converge to some fixed amount, , which is greater than the initial amount deposited, and the amount of asset B gifted to me each time I deposit will tend to 0. So Asset A and B will basically each have a convergent geometric sequence in terms of N.What I want is a formula for each asset, where I can input:
, and outputAdditionally I'd like to find a formula for
, given , and I know that to get this I would have to take the limit as N tends to infinity of the formula for asset A.I've been messing around for a while on a spreadsheet and paper, but my college maths is a bit rusty, and I can't seem to arrive at a solution that works for any arbitrary N. (Also, apologies for the formatting error in the post - LaTeX seems to add a new-line each time I insert something, and I'm not sure how to prevent it doing this?)
Many Thanks
EDIT:
I think I've worked out that the general formulas for Asset A and asset B will be:
Where:
is initial deposit of asset AI still can't find a formula where I can input e.g. n=10, R_P =0.5, A_0 = 100, and find the amount of A a the end. (seems to require some recursive calculations - maybe will write an excel macro to calculate)
Also cannot find a formula for the value that asset A converges to when n tends to infinity. Think there is some algebraic solution to this kind of problem?
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