Actually, in the pricing assumptions used to derive Black Scholes it is the logarithm of the price that undergoes a random walk, not the price itself. So a price has equal probability of growing by a factor of x or 1/x in any time t. (This assumes there is no trend, more generally a random walk is of the form log p = ut + w(t), were u is the trend factor and w(t) is a sum of random variables.) So assuming that log eur/usd is a random walk without a trend, an x% rise in eur/usd is just as likely as an x% rise in usd/eur.
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A truly riskless system?
- Thread starter MathAndLogic
- Start date
Actually, in the pricing assumptions used to derive Black Scholes it is the logarithm of the price that undergoes a random walk, not the price itself. So a price has equal probability of growing by a factor of x or 1/x in any time t. (This assumes there is no trend, more generally a random walk is of the form log p = ut + w(t), were u is the trend factor and w(t) is a sum of random variables.) So assuming that log eur/usd is a random walk without a trend, an x% rise in eur/usd is just as likely as an x% rise in usd/eur.
Quote from rew:
Actually, in the pricing assumptions used to derive Black Scholes it is the logarithm of the price that undergoes a random walk, not the price itself. So a price has equal probability of growing by a factor of x or 1/x in any time t. (This assumes there is no trend, more generally a random walk is of the form log p = ut + w(t), were u is the trend factor and w(t) is a sum of random variables.) So assuming that log eur/usd is a random walk without a trend, an x% rise in eur/usd is just as likely as an x% rise in usd/eur.
interesting, coz in simian's 51-movement backtest, there was no trend, just a sideways market. price ended pretty much where it began.
Correct... But doesn't sambian make an explicit assumption that it's eur/usd, rather than log(eur/usd), that's a random walk? Isn't that fundamentally how he arrives at the conclusion that E[...] > 0?Quote from rew:
Actually, in the pricing assumptions used to derive Black Scholes it is the logarithm of the price that undergoes a random walk, not the price itself. So a price has equal probability of growing by a factor of x or 1/x in any time t. (This assumes there is no trend, more generally a random walk is of the form log p = ut + w(t), were u is the trend factor and w(t) is a sum of random variables.) So assuming that log eur/usd is a random walk without a trend, an x% rise in eur/usd is just as likely as an x% rise in usd/eur.
But that's not the way sambian calculates expected returns for his strategy... My point is that his process assumption is inconsistent with the way he calculates expectation.Quote from heech:
This is exactly why models of asset returns assume log-normality, rather than normality.
Take your 100% EUR/USD gain. Ln(2) = 0.693.
Take your 50% USD/EUR loss. Ln(0.5) = -0.693.
Expected value: 0.
Doesn't this type of low risk system play better at trading straddles.
There are a few more variables but price action on OTM options that lose 1/2 there value as they go further OTM but double in value as they get NTM maybe be favorable for such system.
Exposure is expiration and unwinding costs.
For ES Buy $2000 of puts and calls approx 10 strikes OTM.
Liquidate when market moves 5 strikes.
One side is down roughly $1K the other side is up $2K.
There are a few more variables but price action on OTM options that lose 1/2 there value as they go further OTM but double in value as they get NTM maybe be favorable for such system.
Exposure is expiration and unwinding costs.
For ES Buy $2000 of puts and calls approx 10 strikes OTM.
Liquidate when market moves 5 strikes.
One side is down roughly $1K the other side is up $2K.
Quote from rew:
....Actually, in the pricing assumptions used to derive Black Scholes it is the logarithm of the price that undergoes a random walk, not the price itself....
actually not sure i agree with this. does'nt Black-Scholes assume price is a geometric brownian motion. ie, a random walk? and this leads to a log-normal return. but there's nothing about log price being a random walk.
so sambian, could you explain how you go from:
Letâs assume that we donât know anything about the future price movements.
to:
This means that for us an increase of 100% in eur/usd (A) is as likely as an increase of 100% in usd/eur (B).
Martinghoul, can I ask you to first determine exactly what my error is and then write here? Let me remind you what "errors" you found in chronological order:Quote from Martinghoul:
Sorry, sambian, I wish to clarify, if I may... Your error is in the very first sentence above. What you're defining as a "random walk" simply isn't.
1. The outcomes for my "binomial tree" are defined at will. If I were to try outcomes A) EURUSD rises 100%; and B) EURUSD falls 100%, and assume those are equally probable, I will get something more sensible.
Comment: I will not get anything sensible, of course, and you quickly changed the "error" which you found.
2. My problem is that my choice of numeraire is inconsistent.
Comment: you didn't care to explain my "error" further, and soon came up with a new one.
3. What I'm defining as a "random walk" simply isn't.
Comment: You should say the same thing to Black and Scholes and to the committee which gave them the Nobel Price. Because they use the term "random walk" in the same sense as me. Here is a link - http://riem.swufe.edu.cn/new/techupload/course/200742423245359181.pdf
On page 5 of the pdf you can see that one of the assumptions for their model is:
b) The stock price follows a random walk
And in their model a 100% increase of the price is equally likely as a 50% decrease. You can play around with some Black-Scholes calculator if you don't believe me, just don't forget to set "risk-free rate" to 0.
To sum it up: thank you for the "errors" which you find, but please be more consistent and take your time to explain them. Then go ahead and explain the same "errors" to Black, Scholes etc.