A forward price is calculated by determining where a synthetic long option position would break even. A synthetic long is a combination of a long at-the-money (ATM) call and a short ATM put. This option strategy replicates a long stock position in that its payoff diagram matches a long stock diagram, except that the fundamentals or pricing discrepancies of the options vs. the stock may shift the chart to a higher or lower breakeven point.
For example, suppose it’s Thursday and a stock were trading at $100. The stock is going ex-dividend for $5 tomorrow, meaning that it will open at $95. Buying the stock would mean a breakeven at $100 – $5 = $95 for the 1-day term if bought right now at $100. The comparable 1-day synthetic long position would not benefit from the dividend because the owner of the call would not receive the dividend since they don’t actually own the stock, so its pricing would account for that. As a result, today’s 1-day ATM call option at 100 might cost $1.15, whereas the 1-day ATM put might sell for $6.05. Entering the option position would result in a $4.90 credit. However, after the stock goes ex-dividend on the last day before expiration, the put will carry a $5 liability and the call will be well out of the money. In this scenario, the forward price calculated on Thursday would be $100 – $4.90 = $95.10.
In another scenario, consider a stock under significant short pressure. As it becomes hard to borrow for shorting, traders will often turn to options. This increased demand will drive up the price of puts (because they want to buy them) and drive down the price of calls (because they want to sell them). As a result, the forward price calculated will end up being significantly lower than the current price of the stock. For example, a heavily shorted stock may be trading at $100, which is at a premium partly due to short-driven reasons. Since many traders are unable to easily borrow the stock to short, they turn to puts, which drives a 100 put up to $20. At the same time, they may also be selling the 100 call for $5. Entering this synthetic long would result in a $15 credit, which would result in a breakeven of $100 – $15 = $85 at expiration.
The video below discusses an example.