# How can I theoretically hedge a bet, which is about whether or not price of a stock will get above a number?

For example, the bet may be about someone giving me P dollar and I will pay that person 300 dollars if tomorrow Tesla gets above 860 dollars.

While assuming the Black-Scholes model hold, I can simply use the risk-neutral measure to calculate the fair amount of P.

However, is there a way for me to hedge the risk away?

• barrier options investopedia.com/terms/b/barrieroption.asp – amdopt Jun 1 '20 at 13:48
• @amdopt Sounds more like a binary option, no? – Oscar Jun 1 '20 at 13:51
• @Oscar the question is unclear - it is a barrier option if Tesla above 860 at any time triggers the pay off, it is a binary option if the price of Tesla at expiry is the trigger. – Attack68 Jun 1 '20 at 14:31

To value and hedge your short position on a binary option, you could approximate it using vanilla options and the Black Scholes Model.

You need a fixed payoff amount and not a payoff relative to the spot value, so you could buy an appropriate amount of a call spread with a very small difference in strikes.

Let's say you buy 300 calls @ 859 and sell 300 calls @ 860. Effectively, if the stock turns out to be over 860, you will have the 300 dollars to pay off your short position.

The value of the binary option should not be too diferent from this call spread, so you can value these two vanilla options with the Black Scholes model to get a rough idea.

This is an approximation with tradeable instruments, and it's not exactly the same thing. For one, your hedge is better for you than your short position so it's natural for it to be more expensive (imagine if the stock finishes at 859.5) but hopefully it's usefull to get the rationale.

Theoreticaly, you can of course just use the N(d2) from the Black Scholes formula to get the value of the binary.

You've just described a binary call option with payoff 300 dollars and maturity tomorrow. So yes, since you effectively just sold someone a binary call option with Strike 860 and payoff 300 dollars you can hedge the contract by simply entering the opposite side of the contract on the open market by buying a binary call option with the same strike, maturity and payoff.