How are binary prices calculated?

Hi there,

How do the fixed-odds companies go about calculating the prices in binary bets?

I assume that they use something like Black-Scholes to calculate the fair option value, and then do something with it to munge it into a binary. It's the munging bit that interests me - anyone know how it's done? Can anyone point out any example calculations?

Also, I know that spread betting firms can lay off bets on the futures market if they don't want to take on the risk themselves. Can fixed-odds be laid off on the options market in a similar way, or are most binaries too short a timescale for that to be feasible? (Most binaries are less than 24 hours, aren't they?)

Many thanks,

Geoff

Have a look at

http://www.mathfinance.de/seminars/risk/barriers2002/efficientHedgingBarriers-print.pdf

However, for short term binaries (< 1 day), I think that assuming a normal distribution is actually no worse than assuming a lognormal distribution because the fractional price movement will be small. The price can then be calculated just by integrating the normal probability distribution over the range required to give the probability of the binary expiring true. I've found that to within the bid-offer spread, and given the errors involved in determining the intraday volatility, most binary betting prices on sites follow this reasonably closely (with the volatility upped a little in the wings perhaps).

For hedging, my guess is that the betting sites either don't bother for their daily options or use futures to offset the worse of the risk. I can't believe they'd use options because the expiries are so far away and the strikes so far apart that they're not going to provide enough gamma to be useful.

Where S = spot, X = strike, b = carry, sigma = implied vol, T = time to exp as proportion of year, r = risk free rate, CND is the cumulative normal distribution and W = win amount if the binary pays out.


Binary call price = W * Exp(-r * T) * CND(d)
so it follows that a binary put must be: W * Exp(-r * T) * CND(-d)

Where, as in Black Scholes:

d = (Log(S / X) + (b - sigma^ 2 / 2) * T) / (sigma * Sqr(T))

Simply stated, it is the present value of the payout if the bet is won multiplied by the probability of winning.

Hedging? Not sure how they each do it. There are a number of possible methods.