If we knew exactly how the price was moving...
You could sell this law and buy half a planet. And don't bother with strategies.
then the problem becomes an ordinary manager's problem... which is optimal: more often but less, or less often but more...
for an analogy, take the optimal delivery problem, taking into account storage costs and the cost of delivery of each batch
Sergey Akhtershev "Tasks for maximum and minimum"
p.73 (sort of...)
The question is really more of a rhetorical question, but still.
What if we know exactly the law of price movement (suppose we know the exact formula for the probability density distribution of its differences/increases).
Knowing the probability density function in no way means knowing the law of price movement.
"The question is actually rather rhetorical" - not because it doesn't need to be answered, but because it is completely wrong.
P.S. You must have recently started reading Terwer?
Knowing the probability density function in no way means knowing the law of price movement.
"The question is actually rather rhetorical" - not because it doesn't need to be answered, but because it is completely wrong.
P.S. You must have recently started reading Terver?
In fact, I myself don't really know what I want, so I'll try to formulate it without using clever terminology.
Suppose we trade on a completely abstract market, the quotes on which are generated by computer. We know for sure that its price difference distribution will not be bell-shaped with thick tails or even classical Gaussian, but, for example, triangular or saddle-shaped (so we have a "market of crooked mirrors") and we know in advance the distribution formula with all its parameters.
We go to trade at such a market and basically all we want is to make as much money as possible. Such artificial market will open tomorrow and will work for N ticks.
The task is to develop such a trading strategy based on the a priori knowledge of the price distribution function and available history to maximise the expectation of our deposit size after N ticks.
If we assume that the difference process is stationary, then it's a decent problem to solve. I don't know how to solve it, but I think one should also know the autocorrelation function of the process. It is said that there are even difurcas that describe the optimal strategy.
If we assume that the difference process is stationary, then the problem is well worth solving. I do not know how to solve it, but I think one should also know the autocorrelation function of the process. It is said that there are even difurcations describing the optimal strategy.
Knowing the distribution of price differences is not enough. You also need a model. If the model is a random walk, even if stationary, but independent increments with any distribution, then a profitable strategy is impossible.
If one assumes that the difference process is stationary, then the problem is quite decent. I don't know how to solve it, but I think one should also know the autocorrelation function of the process. It is said that there are even diffurcas that describe the optimal strategy.
Mathemat, you have long noticed that you place too much emphasis, in my opinion, on the stationarity of the difference process. In essence, it is a stochastic ripple on a wave. An acceptable analogy here could be the FPC equations, with their drift and diffusion coefficients.
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The question is really more of a rhetorical question, but still.
What if we know the exact law of price movement (let's say we know the exact formula for the probability density distribution of its differences/increases). How can we calculate a trading strategy that is optimal according to some criterion that we specify for a given distribution law?