Hedging a Cash-or-Nothing Stock Payoff with Options
Summary
The document considers how to hedge a short, fixed-cash payoff that is triggered when a stock finishes above a specified strike. It identifies the contract as a binary call option and notes that, under Black–Scholes assumptions, its theoretical value can be calculated using the risk-neutral probability term.
One practical approximation is a narrow call spread: buy a call at a lower strike and sell a call at the trigger strike, with quantities sized to approximate the fixed payout. The spread can be valued with Black–Scholes, but its payoff is not identical to the binary contract; around the lower strike it may over-hedge the liability. If an equivalent binary option is available, buying the opposite position can hedge directly. The discussion is conceptual and provides no market quotes or evidence about execution, liquidity, or model accuracy.
Key ideas
- A fixed payment triggered above a strike is a binary call payoff.
- A narrow call spread can approximate the binary payoff using vanilla options.
- The spread's payoff differs from the binary contract and can over-hedge in some price regions.
- An available binary option with matching terms can directly offset the short exposure.
- Black–Scholes valuation depends on the model assumptions used for the stock and option.
Tags
Full text
# How can I theoretically hedge a bet, which is about whether or not price of a stock will get above a number? # 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? ## Answer by David Duarte (score 2) https://quant.stackexchange.com/a/55367 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. ## Answer by Oscar (score 1) https://quant.stackexchange.com/a/54579 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.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.