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.
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).
First off: Under general conditions, call options have a positive expected return, whilst puts have a negative expected return.
The mathematical expression would be the expected pay-off of the option divided the option price. Note that the option price is (or can be) calculated by using risk-neutral probabilities (q), while the expected pay-off should be calculated using real-world probabilities (p). That’s because the expected return using risk-neutral valuation is the risk-free rate, while the underlying (stock) is a risky asset that includes a risk premium.
The difference between q and p is explained by the marginal rate of substitution. If we ‘deflate’ the real-world probability using the marginal rate of substitution (also called stochastic discount factor) we can derive the risk-neutral probability. This is because a risk-averse investor is willing to give up more in the upstate than to lose in the downstate in terms of utility. If p > q (which is the case for risk-averse investors), we have that (1-p) < (1-q), hence the investor is putting more weight on the downstate than the upstate.
To come back to my first point, why does a put option have a negative expected return? You are buying a financial instrument that protects you against a downstate, whilst the expected return on the underlying is positive (given a positive risk premium). A call option has a positive expected return because the underlying asset is expected to increase in value.
All rather technical but I hope it clears things up. Note that this explanation is based on academic literature, not the CFA curriculum per se.