w******l
发帖数: 34 | 1 绿书156页B有没有通俗一点的解释啊
题目是
long a call option and you hedged the position by shorting the stock, such
that the portfolio is delta neutral. If there is an immediate increase or
decrease in the stock price, what will happen to the value of the portfolio?
Any arbitrage exists? | w******i
发帖数: 503 | 2 arbitrage have to think. but the portfolio value should increase... | r**a
发帖数: 536 | 3 他的解释已经比较通俗了。首先考率portfolio的value是不是升高。这个很简单，就是
对portfolio的value做taylor展开w.r.t.股票价格。delta neutral意味着一阶导数是
零，但是他还有二阶导数呢。你要看看二阶导数的符号。这个正好是恒正。所以value
一定上升。
第二的问题，你要搞明白啥是arbitrage在这个case下。非arbitrage意味着value的回
报率为r。那么arbitrage意味着value的回报率不为r。你算一下他的回报率就知道了。
因为这里面都是典型的BS model的假设，没有jumping，所以这时候回报率一定是r。这
个无非就是已知BS equation反推arbitrage free。
portfolio?
【在 w******l 的大作中提到】
: 绿书156页B有没有通俗一点的解释啊
: 题目是
: long a call option and you hedged the position by shorting the stock, such
: that the portfolio is delta neutral. If there is an immediate increase or
: decrease in the stock price, what will happen to the value of the portfolio?
: Any arbitrage exists?
| w******i
发帖数: 503 | 4 i think immedicate increase or decrease is jump...
so there is arbitrage..
value
【在 r**a 的大作中提到】
: 他的解释已经比较通俗了。首先考率portfolio的value是不是升高。这个很简单，就是
: 对portfolio的value做taylor展开w.r.t.股票价格。delta neutral意味着一阶导数是
: 零，但是他还有二阶导数呢。你要看看二阶导数的符号。这个正好是恒正。所以value
: 一定上升。
: 第二的问题，你要搞明白啥是arbitrage在这个case下。非arbitrage意味着value的回
: 报率为r。那么arbitrage意味着value的回报率不为r。你算一下他的回报率就知道了。
: 因为这里面都是典型的BS model的假设，没有jumping，所以这时候回报率一定是r。这
: 个无非就是已知BS equation反推arbitrage free。
:
: portfolio?
| r**a
发帖数: 536 | 5 Yes, if u view the immedicate increase or decrease as a jump, then there
should be an arbitrage possibility.
【在 w******i 的大作中提到】
: i think immedicate increase or decrease is jump...
: so there is arbitrage..
:
: value
| w******l
发帖数: 34 | 6 谢谢， 讲的很清楚
不过delta neutral 就是risk free的意思吗
我觉得只是portfolio对于S变动的一阶效果是0，
但是还有 gamma>0 啊， 这样还算risk free吗，还是我概念混淆了啊？
value
【在 r**a 的大作中提到】
: 他的解释已经比较通俗了。首先考率portfolio的value是不是升高。这个很简单，就是
: 对portfolio的value做taylor展开w.r.t.股票价格。delta neutral意味着一阶导数是
: 零，但是他还有二阶导数呢。你要看看二阶导数的符号。这个正好是恒正。所以value
: 一定上升。
: 第二的问题，你要搞明白啥是arbitrage在这个case下。非arbitrage意味着value的回
: 报率为r。那么arbitrage意味着value的回报率不为r。你算一下他的回报率就知道了。
: 因为这里面都是典型的BS model的假设，没有jumping，所以这时候回报率一定是r。这
: 个无非就是已知BS equation反推arbitrage free。
:
: portfolio?
| r**a
发帖数: 536 | 7 Actually, this is a really good question. First, we need to know the
definition of risk free portfolio or how to capture the risk in terms of
math. In my opinion, you have to go to the VaR. The risk free portfolio
should be defined by VaR_1(L)=0, which means that we have 100% confidence
that the probability that the loss L exceeds 0 is no larger than 0. Here the
loss L is defined by $L=V(t, ...)-V(t+\delta t,...)$. Note there are
several different kinds of definations of L, e.g. discounted loss function.
Here I just give u some basic sense. In fact, in my opinion the discounted
loss function is more reasonable in which we need discount V(t+\delta t,...)
back to time t by terms of bond or cash account. The details can be found
in more proffesional textbooks regarding risk management, e.g. "market risk
analysis, Vol.4" by Carol Alexander.Furthermore, in terms of discounted loss function, the risk free portfolio is equivalent to the equation: dV=rVdt.
Next, the natural idea is to do the Taylor expansion of $L$ at time t.
Normally it contains the partial derivatives of V w.r.t. time, and other
parameters. In your case, at least it contains the 1rd and 2nd order
derivatives of V w.r.t. the stock price S. But the there is an essential
difference comparing to the BS model. We know, that in risk neutral world,
in the BS model \delta S is captured by the dynamics of stock under the risk
neutral measure. But when you analyze the risk, the risk should happen in
the real world, in other words, you have to use the actual probability
measure instead of the risk neutral martingale measure. In your case,
although the gamma does not vanish, this portfolio is still risk free w.r.t. the
movement of stock price. You can do the calculation by yourself. But in the real world because the stock dynamics should not strictly follow the BS model. So generally we still need to make a portfolio with vanishing gamma.
The above is my understanding. It might be wrong. If so, please correct me.
【在 w******l 的大作中提到】
: 谢谢， 讲的很清楚
: 不过delta neutral 就是risk free的意思吗
: 我觉得只是portfolio对于S变动的一阶效果是0，
: 但是还有 gamma>0 啊， 这样还算risk free吗，还是我概念混淆了啊？
:
: value
| s******e
发帖数: 1751 | 8 this is a good question that should be asked to applicants of junior trader
position.
the real questions are:
1. what's gamma and theta? what's their relationship?
2. under what circumstance does BSM hold? and what really is Vega? | r**a
发帖数: 536 | 9
trader
Do u have the answer of the first part of the above 2nd question? I am also
wondering under what circumstance the BSM holds. If i remember rightly, the
stock price should normally follow student's t-distribution with degrees of
freedom 3 (or 4?). Does this implies that the BSM can't hold at all in any
cases? Here I naively assume that the volatility is fixed as a constant
during a very short time, which implies that in the Taylor expansion of the
loss function the term containing Vega is vanishing.
【在 s******e 的大作中提到】
: this is a good question that should be asked to applicants of junior trader
: position.
: the real questions are:
: 1. what's gamma and theta? what's their relationship?
: 2. under what circumstance does BSM hold? and what really is Vega?
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