Expected Return forumla for Binary Options

[lbazer]

Hey Guys,
I’ve been thinking how to mathmatically express the Expected Return (ER) for the Binary Options world.
Using intuition the Expected return is negative in the long run, but i would like to know if you can come up with a formula or something.
I was thinking to use something like a binomial tree asuming that the probability of a Up or Down move of the underlying is 50% but the factor when winning is 1.7 (70% is what you can win in a trade) and the loosing factor being 100% (if the underlying is OTM you loose the whole trade).
I’m really not a fan of this product but i would love to have a mathematical explanation to why binary options are a loosing instrument.
Tanks for your help!

 

[S2000magician]

This is really a Level II question: at Level II the curriculum covers binomial trees for equity valuation.
Of particular interest is that the up and down weights (I hate calling them probabilities) will likely not be 50-50.
If you’re really interested in how to use binomial trees to value stock options, I wrote an article on the subject: http://www.financialexamhelp123.com/binomial-pricing-trees-for-options/
Full disclosure: as of 4/25/15 there is a charge to view the articles on my website.
I don’t discuss binary options per se, but the methodology I present can be used for binary options; you have only to change the payoff values.

 

[krokodilizm]

Given that UP and DOWN moves are equally possible (which some binary options advocates may disagree with) and that a down move is terminal (you can’t have an up move after a down move) I can think of the following algorithm:
The expected payoff after an up move is (1+0.7)/0.5=0.85
The expected payoff after you win first leg and reinvest all principal is 1.7*1.7/0.25=0.7225
So a generalized expression is
EP = (1+r)^t / 2^t where t is the number of successful bets.
Given that r<1, (1+r) is always less than 2 so the payoff tends to zero at an exponential rate!
P.S. The reason I assume up/down moves are equally possible is because of volatility, i.e. you may catch an upward trend but still find yourself in a temporary down trend the moment your option expires. Since timeframes are less than a day in binary options market levels are never too far from your entry level…

 

[S2000magician]

krokodilizm wrote:Given that UP and DOWN moves are equally possible (which some binary options advocates may disagree with) and that a down move is terminal (you can’t have an up move after a down move) I can think of the following algorithm:
The expected payoff after an up move is (1+0.7)/0.5=0.85
The expected payoff after you win first leg and reinvest all principal is 1.7*1.7/0.25=0.7225
So a generalized expression is
EP = (1+r)^t / 2^t where t is the number of successful bets.
Given that r<1, (1+r) is always less than 2 so the payoff tends to zero at an exponential rate!
P.S. The reason I assume up/down moves are equally possible is because of volatility, i.e. you may catch an upward trend but still find yourself in a temporary down trend the moment your option expires. Since timeframes are less than a day in binary options market levels are never too far from your entry level…

As I wrote above, this is a Level II topic, and the weights assigned to the up and down movements are almost certainly not equal, and they’re not probabilities (despite the fact that they’re called risk-neutral probabilities).