Expected payoff and weighted average price

Question

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Let you're trading a security whose probability to be equal to $S_{T}$ at time $T$ follows a p.d.f. like the ones in the picture below.

(That is just an example found with Google images, assume you're considering just one of the expiry dates shown above, e.g. $T=60$).

You are trading with a capital equal to $M$ dollars; your main goal is to maximise your expected payoff at time $T$, and you have some constraints to achieve this result:

• you cannot go short, this is long only;
• your capital, $M$, must be split in a number of parts which is equal or less than $k$, that is, you cannot average your position more than $k-1$ times in addition to the first entry. This means that your position average price will be the weighted average of a maximum of $k$ security prices;
• you cannot borrow money, then no leverage involved;
• at least one entry must be made.

Below an instance in which using the whole capital, $M$, at the first trade was the best choice possible:

From such a description my guess is that what a trader should try to do is to take advantage of the security's volatility: if my average price at $t=T$ is below $S_{T}$ I am earning moneys, then I am involved in a trade off between a greater but less likely profit and a smaller but more likely profit.

Possible objection

Hey, mate, if $E[S_{T}]$ (expected value) is above $S_{0}$ you've just to buy it now and keep it until $T$, isn'it?

I don't think so. Consider the following security's path:

It's easy to see that if the trader does not use the whole capital, $M$, buying the stock at $t=0$, but instead he uses the remainder to buy the valley (about June), his performance will be better. Yes, he's also running the risk to buy surging prices, as well...

Question

1. What the analytical formula of the expected payoff would be? I guess it cannot be the sum of probability-weighted returns due to overlapping events;
2. once the aforementioned payoff formula has been discovered, how to maximise your expected payoff varying the $k$ entry prices along the time which goes from $t=0$ to $t=T$?
3. If an analytical solution did not exist, what a possible simulative approach would be? Could some Monte Carlo simulations help?

Hint

Do not let a "classic" p.d.f. trick you: what if the security p.d.f. was something like this...

Answer 1

There is no advantage to spreading your entry into multiple tranches, unless you have private information (i.e. not priced-in to the market) that prices will fall first, before rising. If the security is following a random walk, with each incremental return being independent, it doesn't matter whether the price has risen or fallen since t=0.

I think you should look at each day's return (or whatever sub-increment you wish to use) as an independent investment. On each day, your total return is that day's return times the amount of invested capital on-or-before that day. If you think that there is a positive expectation -- that prices will on average rise for each of these days -- then you should invest all capital at t=0. Any capital not invested at t=0 is forgoing a day's worth of that expected return.

While you were focused only on expected return, the answer doesn't change when you look at a risk-adjusted metric like Sharpe. You are building a portfolio of 1-day returns. If they all have the same expected return, and all have the same expected volatility, and all are independent, then an efficient portfolio is an equal weighting of all days. I.e., invest it all at t=0.