08-25-2005, 03:46 PM
This was posted in the Science, Math and Philosphy forum, but it doesn't look like I'm going to get an answer there, so I posted here (apologies to the wrong-forum nits)
So I'm reading one of my textbooks, and I read something that if I understand correctly basically says:
Suppose that you have $1. You bet some one on a coin flip, even money. If you lose, you stop. If you win, 1/2 a second later you bet $2. If you lose you stop. If you win, 1/4 a second later you bet $4. If you lose you stop. If you win, 1/8 of a second you bet $8. etc...
So, the authors of the book conclude the probability that you will go broke after 1 second is 1. I agree.
However they conclude that your EV after 1 second is -$1. I disagree, I say it's $0.
What do you guys think? It may just be me not understanding a definition some where, I dunno
Anyways, here's some of the book:
Quote:
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Example 3.1 The following example of a suicide strategy is borrowed from Harrison and Pliska (1981). It can be modified easily to provide an example of an arbitrage opportunity in an unconstrained Black-Scholes setting. For simplicity, we take r = 0, T=1, and So=1. For a strictly positive constant b>0, we consider the following trading strategy:
y1(t) = {-b 0 =< t <= r(b), 0 otherwise}
y2(t) = {1+b, 0 =< t <= r(b), 0 otherwise)
where
r(b) = inf{t: St = 1 + 1/b} = inf{t:V(y,t) = 0}
--------------------------------------------------------------------------------
here V(y,t) is defined as:
y1(t)*St + y2(t)*Bt
St = Stock Price
Bt = Bond Price (risk free investment, like money in a savings account (means Bt>0) or loaned from a bank (means Bt<0))
Quote:
--------------------------------------------------------------------------------
In financial interpretation, an investor starts with one dollar of wealth, sell b shares of stock short, and buys 1+b bonds. Then he holds the portfolio until the terminal date T=1, or goes bankrupt, whichever comes first. The probability of bankruptcy under this strategy is equal to p(b) = P{r(b)<1}, so that it increases from zero to one as b increases from zero to infinity. By selling short a very large amount of stock, the investor makes his failure almost certain, but he will probably make a great deal of money if he survives. The chance of survival can be completely eliminated, however, by escalating the amount of stock sold short in the following way.
To show this, we shall modify the strategy as follows. On the time interval [0,1/2], we follow the strategy above with b = 1. The probability of bankruptcy during [0,1/2] thus equals p = P{r() =< 1/2}. If r(1) >1/2, the amount of stock sold short is adjusted to a new level b1 at time 1/2. Simultaneously, the number of bonds held is revised in a self-financing fashion. The number b1 is chosen so as to make the conditional probability of ruin during the time interval (1/2,3/4], given that we have survived up to time 1/2, equal to p again.
In general, if at any time t(n) = 1 - (1/2)^n we still have positive wealth, then we readjust (typically increase) the amount of stock sold short, so that the conditional probability of bankruptcy during (t(n), t(n+1)] is always p. To keep the strategy self-financing, the amount of bonds held must be adjusted at each time t(n) as well. The probability of survival until time t(n) is then (1-p)^n, and it vanishes as n tends to 0 (so that t(n) tends to 1). We thus have an example of a piecewise constant, self-financing strategy, (y1,y2) say, with V(y,t) = y1(0)So + y2(o) = 1,
V(y,t) = y1(t)St + y2(t)>=0, for any t in [0,1)
--------------------------------------------------------------------------------
by So they mean St at time t=0
Quote:
--------------------------------------------------------------------------------
and V(y,1) = 0. To get a reliable model of a security market we need, of course, to exclude such examples of doubling strategies from the market model.
--------------------------------------------------------------------------------
So basically I say V(y,1) = 1, not 0.
This is from "Martingale Methods in Financial Modelling" by Marek Musiela and Marek Rutkowski
I typed this (yes the whole damn thing) so there many or may not be some errors, also, I changed a few symbols (like y instead of psi)
So I'm reading one of my textbooks, and I read something that if I understand correctly basically says:
Suppose that you have $1. You bet some one on a coin flip, even money. If you lose, you stop. If you win, 1/2 a second later you bet $2. If you lose you stop. If you win, 1/4 a second later you bet $4. If you lose you stop. If you win, 1/8 of a second you bet $8. etc...
So, the authors of the book conclude the probability that you will go broke after 1 second is 1. I agree.
However they conclude that your EV after 1 second is -$1. I disagree, I say it's $0.
What do you guys think? It may just be me not understanding a definition some where, I dunno
Anyways, here's some of the book:
Quote:
--------------------------------------------------------------------------------
Example 3.1 The following example of a suicide strategy is borrowed from Harrison and Pliska (1981). It can be modified easily to provide an example of an arbitrage opportunity in an unconstrained Black-Scholes setting. For simplicity, we take r = 0, T=1, and So=1. For a strictly positive constant b>0, we consider the following trading strategy:
y1(t) = {-b 0 =< t <= r(b), 0 otherwise}
y2(t) = {1+b, 0 =< t <= r(b), 0 otherwise)
where
r(b) = inf{t: St = 1 + 1/b} = inf{t:V(y,t) = 0}
--------------------------------------------------------------------------------
here V(y,t) is defined as:
y1(t)*St + y2(t)*Bt
St = Stock Price
Bt = Bond Price (risk free investment, like money in a savings account (means Bt>0) or loaned from a bank (means Bt<0))
Quote:
--------------------------------------------------------------------------------
In financial interpretation, an investor starts with one dollar of wealth, sell b shares of stock short, and buys 1+b bonds. Then he holds the portfolio until the terminal date T=1, or goes bankrupt, whichever comes first. The probability of bankruptcy under this strategy is equal to p(b) = P{r(b)<1}, so that it increases from zero to one as b increases from zero to infinity. By selling short a very large amount of stock, the investor makes his failure almost certain, but he will probably make a great deal of money if he survives. The chance of survival can be completely eliminated, however, by escalating the amount of stock sold short in the following way.
To show this, we shall modify the strategy as follows. On the time interval [0,1/2], we follow the strategy above with b = 1. The probability of bankruptcy during [0,1/2] thus equals p = P{r() =< 1/2}. If r(1) >1/2, the amount of stock sold short is adjusted to a new level b1 at time 1/2. Simultaneously, the number of bonds held is revised in a self-financing fashion. The number b1 is chosen so as to make the conditional probability of ruin during the time interval (1/2,3/4], given that we have survived up to time 1/2, equal to p again.
In general, if at any time t(n) = 1 - (1/2)^n we still have positive wealth, then we readjust (typically increase) the amount of stock sold short, so that the conditional probability of bankruptcy during (t(n), t(n+1)] is always p. To keep the strategy self-financing, the amount of bonds held must be adjusted at each time t(n) as well. The probability of survival until time t(n) is then (1-p)^n, and it vanishes as n tends to 0 (so that t(n) tends to 1). We thus have an example of a piecewise constant, self-financing strategy, (y1,y2) say, with V(y,t) = y1(0)So + y2(o) = 1,
V(y,t) = y1(t)St + y2(t)>=0, for any t in [0,1)
--------------------------------------------------------------------------------
by So they mean St at time t=0
Quote:
--------------------------------------------------------------------------------
and V(y,1) = 0. To get a reliable model of a security market we need, of course, to exclude such examples of doubling strategies from the market model.
--------------------------------------------------------------------------------
So basically I say V(y,1) = 1, not 0.
This is from "Martingale Methods in Financial Modelling" by Marek Musiela and Marek Rutkowski
I typed this (yes the whole damn thing) so there many or may not be some errors, also, I changed a few symbols (like y instead of psi)