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MartinGale7
Topic Author
Posts: 152
Joined: April 1st, 2011, 7:42 am

Making Money on Zero Drift LogNormal Random Walks

I'm not sure how obvious this brain teaser will be for the group. Here goes anyhow -In hypothetical 'Quant Village', the local stock exchange trades with 100 stock. Each follow a Log-Normal Random walk and are for all intents and purposes divorced from reality. These stock are all independent (from each other and every other external factor) and have zero drift (mu=0) and a constant 10% volatility (sigma=0.10). You can assume that your own trading would have no effect on the path of the stock (ie you are too small to significantly influence price movement). Also you can assume that stocks are completely liquid and there are no transaction costs, slippage etc. We are under purely textbook conditions.The question is, as a systematic trader, can I make money in Quant Village, or should I move on?

crmorcom
Posts: 452
Joined: April 9th, 2008, 4:09 pm

Making Money on Zero Drift LogNormal Random Walks

The sneaky answer - though possibly not the one you want - is "yes". If the log price process has no drift d(lnS) = sigma dw, the price process does have a drift because of convexity: dS/S = 0.5 sigma^2 dt + sigma dw. In this case, simply holding a diversified portfolio of all the stocks would get you a positive return.

MartinGale7
Topic Author
Posts: 152
Joined: April 1st, 2011, 7:42 am

Making Money on Zero Drift LogNormal Random Walks

That's the answer I was looking for.

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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Making Money on Zero Drift LogNormal Random Walks

With that strategy, since there are only 100 stocks, I say there's an approximately 31% chance that you'll lose money.

MartinGale7
Topic Author
Posts: 152
Joined: April 1st, 2011, 7:42 am

Making Money on Zero Drift LogNormal Random Walks

Over one day? Over one year? Over 30 years? Including rebalancing and how often?

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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Making Money on Zero Drift LogNormal Random Walks

My answer is for T=1 (one round), where T is measured in the same units as $\sigma$. For smaller (larger) T, but no rebalancing, the loss probability will be larger (smaller).As for multiple rounds M with rebalancing every T, haven't worked that one out.

Costeanu
Posts: 189
Joined: December 29th, 2008, 5:33 pm

Making Money on Zero Drift LogNormal Random Walks

Another way of answering your question is that no, you can't make money. The stated assumptions are that the stocks themselves are driftless, so their SDE would look like dS = sigma S dW, not that their logs are driftless. If the stocks themselves are martingales, there is no way to systematically make money.

theRedBaron
Posts: 117
Joined: October 13th, 2010, 8:13 pm

Re: Making Money on Zero Drift LogNormal Random Walks

To MartinGale7 and Alan: so, what exactly is the strategy to make money? Are you just net long the portfolio? How do you decide which stocks to buy and how/when to rebalance?

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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Re: Making Money on Zero Drift LogNormal Random Walks

If you are asking about MartinGale7's question, my answer is that there's no way to make money risklessly in that putative market.

However, your expected return would be positive with vanishing risk in the limit where you approach owning an infinite number of (uncorrelated) stocks with vanishing individual weights. Since the orig. puzzle said '100 stocks', that limit was excluded.

Also, I am excluding roulette-style doubling strategies theoretically possible in continuous-time ('textbook') trading when you can borrow arbitrary amounts. So, my answer presumes discrete-time trading by a mortal individual and/or some kind of borrowing limit. Otherwise, you would be allowed to just keep doubling a bet by borrowing (perhaps with an infinitesimal holding period) until you win a \$1 and then quit.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: Making Money on Zero Drift LogNormal Random Walks

I'm not sure how obvious this brain teaser will be for the group.  Here goes anyhow -In hypothetical 'Quant Village', the local stock exchange trades with 100 stock.  Each follow a Log-Normal Random walk and are for all intents and purposes divorced from reality.  These stock are all independent (from each other and every other external factor) and have zero drift (mu=0) and a constant 10% volatility (sigma=0.10).  You can assume that your own trading would have no effect on the path of the stock (ie you are too small to significantly influence price movement).  Also you can assume that stocks are completely liquid and there are no transaction costs, slippage etc.  We are under purely textbook conditions.The question is, as a systematic trader, can I make money in Quant Village, or should I move on?
As far as market is formalized in term of GBM equations it makes sense to express the intention to 'make money' formally,ie in a formula. Otherwise we basically rely on our intuition.