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今天突发奇想，假如市场上所有的证券都服从同一种随机过程，volatility等参数都一样，假设就是最标准的布朗运动，r=0, 期限无限长，没有交易费，交易不会影响价格，等等等等。我能不能设置一个策略，随便买入证券，在证券涨5％的时候就抛，跌30％的时候再抛。假设先涨到5％的几率大于先跌30％的几率的6倍，我就赚了。但是我想了半天也相不出合适的数学证明。遍了一上午的VBA也失败了。
求高手指点。

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关键词：交易策略 Volatility 突发奇想 随机过程 布朗运动 突发奇想 数学 影响 证券

if the stock is a brownian motion, this does not satisfy your condition, since it is a martingale, your profit

p(t)=int_0_t b(S)dw(s) is a also a martingale (50:50 up and down)

the process satisfies your condition should be a sub-martingale I guess.

我编出来了，似乎确实趋近于0，但运行速度太慢，算不了足够大的数。
可是有一点我想不明白。假如我的每一个增量都附和均匀分布而不是正态分布，比如在110和90的区间里，这样我不管怎么设上下限，比如我设上10下5，先触到上10的概率就是先触到下5的1／2，profit＝10＊1／3－5＊2／3＝0，不管怎么样都是零，因为这里面有一个线性关系。可是正态分布没有这种线性关系，比如说,设定一个时限，我在价值超过101.645的时候卖，在低于100－2.326＝97.674的时候stop－loss，我在我有5％的概率触1.645的线, 1%的概率触2.326的线，所以我的收益期望值是1.645＊5％－2.236＊1％>0。然后对于剩下94％的情况，我继续持有到下一阶段。
这样的话套利存在吧？

schwereburg 发表于 2014-1-19 03:24
我编出来了，似乎确实趋近于0，但运行速度太慢，算不了足够大的数。
可是有一点我想不明白。假如我的每一个 ...

OK, very very good question.

I think it for a while and my answer is as follows:

There is a very important theorem for martingale: optional sampling theorem.

If X_t is a martingale, we know that E(X_t)=X0.  The theorem tells you that X_T is also a martingale as long as T is a bounded stopping time.

To answer your question I give an example:

if X_t now is a standard brownian motion, which is of course a martingale and X0=0. Then what's the probability that X_t first hits 2 or -1?

in this problem we are considering the first passage time/hitting time which is an example of stopping time. let the first passage time be tau. By the stopping time theorem, X_min{t,tau} is a martingale. So that E(X_min{t,tau})=X_0=0,  

let P(X_tau=2)=P,

2*P+(-1)*(1-P)=0, So P=1/3 and 1-P=2/3

this is satisfies the linear rule you mentioned.

It should be noticed that we are not considering the problem at a fixed time point s. If you look at X_t at the fixed time point s, then of course, the linear rule does not hold. We are now looking at the whole path, you have to consider the time.

In general, as long as you trading a martingale (not only brownian), your expected profit should be zero. The linear rule holds. This also guarantees no-arbitrage. The expected profit is zero no matter how many times you play and what time point you choose to enter or leave.

best,

That is truly an exciting and delightful answer.
I thought I should long a stock,long a american down and in put, and short a american up and in call, but through your proof and my simulation, the profit is 0.
I should change them to UIC and DIP with maturity of T, right?