Variance Optimal Hedging in the Black-Scholes Model for a given Number of Transactions
Variance Optimal Hedging in the Black-Scholes Model for a given Number of Transactions
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Black-Scholes 模型中给定交易数量的方差最优对冲
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
Christophe Patry
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文献类型：
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作者：
C. Martini;Christophe Patry
在Black-Scholes期权定价范式中，假设做市商设计了连续时间套期保值。从实际的角度来看，这是不现实的。我们在Black-Scholes模型中引入了交易限制，即只允许对冲一定次数，只有次数是固定的，做市商可以自由选择（停止）次数和对冲比率。我们确定的战略，最大限度地减少跟踪误差的方差为一个给定的初始值的投资组合。最小方差被证明是一个序列的最佳停止问题的解决方案。证明了存在性和唯一性。我们设计了一个复杂度为N3（N为格点个数）的格点算法来求解相应的Cox-Ross-Rubinstein离散问题。该计划的收敛依赖于粘性解决方案的参数。数值结果和动态模拟。
In the Black-Scholes option pricing paradigm it is assumed that the market-mak- er designs a continuous-time hedge. This is not realistic from a practical point of view. We introduce trading restrictions in the Black-Scholes model in the sense that hedging is only allowed a given number of times-only the number is fixed, the market-maker is free to choose the (stopping) times and hedge ratios. We identify the strategy which minimizes the variance of the tracking error for a given initial value of the portfolio. The minimal variance is shown to be the solution to a sequence of optimal stopping problems. Existence and uniqueness is proved. We design a lattice algorithm with complexity N3 (N being the number of lattice points) to solve the corresponding discrete problem in the Cox-Ross-Rubinstein setting. The convergence of the scheme relies on a viscosity solution argument. Numerical results and dynamic simulations are provided.