We consider the problem of finding investment strategies that maximize the average growth-rate of the capital of an investor. This is usually achieved through the so-called Kelly criterion, which in a dynamic setting where investment decisions are adjusted over time, prescribes a constant optimal fraction of capital that should be re-invested at each time, i.e. the investor's optimal leverage . We generalize this problem by accounting for the effects of market impact, that is the fact that prices respond to trading activity. In particular, we assume that the value of an investment portfolio should be measured in terms of the cash-flow that can be generated by liquidating the portfolio, rather than by its mark-to-market value . We formulate the problem in terms of a stochastic process with multiplicative noise and a non-linear drift term that is determined by the specific functional form of market-impact . We solve the stochastic equation for two classes of market-impact functions (power laws and logarithmic), and in both cases we compute optimal leverage trajectories . We further test numerically the validity of our analytical result .
We consider the problem of finding optimal strategies that maximize the average growth-rate of multiplicative stochastic processes. For a geometric Brownian motion the problem is solved through the so-called Kelly criterion, according to which the optimal growth rate is achieved by investing a constant given fraction of resources at any step of the dynamics . We generalize these finding to the case of dynamical equations with finite carrying capacity, which can find applications in biology, mathematical ecology, and finance . We formulate the problem in terms of a stochastic process with multiplicative noise and a non-linear drift term that is determined by the specific functional form of carrying capacity . We solve the stochastic equation for two classes of carrying capacity functions (power laws and logarithmic), and in both cases compute optimal trajectories of the control parameter . We further test the validity of our analytical results using numerical simulations .
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investment
optimal
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the capital of an investor. This is usually achieved
multiplicative stochastic processes. For a geometric Brownian motion the problem is solved
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which in a dynamic setting where investment decisions are adjusted over time, prescribes a constant optimal fraction of capital that should be re-invested at each time, i.e. the investor's optimal leverage
according to which the optimal growth rate is achieved by investing a constant given fraction of resources at any step of the dynamics
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this problem by accounting for the effects of market impact, that is the fact that prices respond to trading activity. In particular, we assume that the value of an investment portfolio should be measured in terms of the cash-flow that can be generated by liquidating the portfolio, rather than by its mark-to-market value
these finding to the case of dynamical equations with finite carrying capacity, which can find applications in biology, mathematical ecology, and finance
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market-impact
carrying capacity
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market-impact
carrying capacity
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we compute optimal leverage trajectories
compute optimal trajectories of the control parameter
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numerically
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result
results using numerical simulations