Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy, has many desirable properties such as maximizing the asympotic long run growth of capital. However, it has considerable short run risk since the utility is logarithmic, with essentially zero Arrow-Pratt risk aversion. Most investors favor a smooth wealth path with high growth. In this paper we provide a method to obtain the maximum growth while staying above a predetermined ex-ante discrete time smooth wealth path with high probability, with shortfalls below the path penalized with a convex function of the shortfall so as to force the investor to remain above the wealth path. This results in a lower investment fraction than the Kelly strategy with less risk, and lower but maximal growth rate under the assumptions. A mixture model with Markov transitions between several normally distributed market regimes is used for the dynamics of asset prices. The investment model allows the determination of the optimal constrained growth wagers at discrete points in time in an attempt to stay above the ex-ante path

Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy, has many desirable properties such as maximizing the asympotic long run growth of capital. However, it has considerable short run risk since the utility is logarithmic, with essentially zero Arrow-Pratt risk aversion. It is common to control risk with a Value-at-Risk constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is less growth than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within regime. The stochastic investment model is reformulated as a deterministic program which allows the calculation of the optimal constrained growth wagers at discrete points in time

Optimal liquidation strategies and their implications

This paper studies optimal liquidation when the selling price depends on the rate of liquidation, transaction time, volume, and the asset\u27s intrinsic value. A generic closed-form solution for maximizing the discounted liquidation proceeds is derived. To obtain financial insights, three parametric specifications that proxy for increasingly realistic market conditions are examined. In our framework, maximizing liquidation proceeds and minimizing liquidity costs are equivalent. The optimal strategies imply more rapid liquidations in less liquid markets. We also show that volatility is stochastic when market liquidity is unpredictable

Gröbner Bases and Syzygy Modules

The theory of Gröbner bases has become a useful tool in computational commutative algebra. In this paper, we outline some basic results, as they are found in [1], including the concepts of terms ordering, multivariable polynomial division, Gröbner bases, Buchberger\u27s algorithm, and syzygy modules. Specially, we present several equivalent definitions for Gröbner bases and prove how to compute a Gröbner basis for an ideal I of A = k[x1, x2, • • • , xn] generated by {fl, f2, • • • , f8} through Buchberger\u27s algorithm. As an application of Gröbner bases, we present a standard method (see [1]) to compute the syzygy module of a set {f1, f2, • • • , f8} of polynomials, illustrated with original examples. Finally, we implement these examples on the computer using the Mathematica package of [4]

On Leland's option hedging strategy with transaction costs

Nonzero transaction costs invalidate the Black-Scholes (1973) arbitrage argument based on continuous trading. Leland (1985) developed a hedging strategy which modifies the Black-Scholes hedging strategy with a volatility adjusted by the length of the rebalance interval and the rate of the proportional transaction cost. Leland claimed that the exact hedge could be achieved in the limit as the length of rebalance intervals approaches zero. Unfortunately, the main theorem (Leland 1985, P1290) is in error. Simulation results also confirm opposite findings to those in Leland (1985). Since standard delta hedging fails to exactly replicate the option in the presence of transaction costs, we study a pricing and hedging model which is similar to the delta hedging strategy with an endogenous parameter, namely the volatility, for the calculation of delta over time. With transaction costs, the optimally adjusted volatility is substantially different from the stock's volatility under the criterion of minimizing the mean absolute replication error weighted by the probabilities that the option is in or out of the money. This model partially explains the phenomenon that the implied volatilities with equity options are skewed. Data on S&P500 index cash options from January to June 2002 are used to illustrate the model. Option prices from our model are highly consistent with the Black-Scholes option prices when transaction costs are zero

Dynamic investment models with downside risk control

Mean-variance analysis has been broadly used in the theory and practice of portfolio management. However, the continuous analogy is not fully studied either academically or in practice. This thesis provides a similar efficient frontier to Markowitz (1952) and a general solution using martingale method employed in Cox and Huang (1989). Comparisons between the expected utility approach and the mean-variance analysis have been made. Traditional utility maximization cannot be used for explicit risk control of downside losses. An adjusted investment objective function by the worst case outcome is incorporated in the investment model. The problem can be divided into two subproblems as in Cox and Huang (1989). Closed form solution is derived for geometric Brownian motion and HARA utility setting. An interesting result is that the investor's decision is governed by a single "security" - a call option on a dynamic mutual fund. A similar strategy, Risk Neutral Excess Return(RNER), to Portfolio Insurance is discussed. With geometric Brownian motion, the RNER strategy has a payoff structure similar to a straddle option strategy. Compare to the strategic asset allocation methods, such as Buy and Hold, Fixed Mix, and Portfolio Insurance , the new approach appears to be superior under a popular risk measure, Value at Risk(VaR). A new objective function is defined for applying stochastic programming to financial investment under uncertainty. Incomplete market conditions are considered in implementing this model. The risk neutral probability is fully studied using stochastic programming techniques.Business, Sauder School ofGraduat