The lower bound of a European put option is defined as the greater of zero or the present value of the exercise price minus the spot price. This is because a rational investor would not pay more for a put than the amount they could earn by exercising it at maturity in the worst-case scenario. If the spot price is above the strike, the put is worthless, hence the floor at zero. If it’s below, the minimum fair value must reflect this intrinsic benefit.

When the current stock price (S₀) equals the present value of the strike price (PV(X)), the intrinsic value of both a European call and put is zero. Therefore, any premium paid for either option must come solely from time value. This situation reflects a classic case of at-the-money options, where neither has immediate exercise value, but both may gain value before expiration depending on future price movements.

Time value, or theta, declines as the option nears expiration. This decline is:

• A cost to the buyer of an option (like a long put), because the option loses value over time if nothing else changes.

• A benefit to the seller of an option (like a short call), because the option they sold becomes cheaper to buy back or expires worthless.

Therefore, option time decay favors the seller and works against the buyer, making it a benefit to the call-selling strategy and a cost to the put-buying strategy.

The upper bound of a call option’s price is the current market price of the underlying asset. No rational investor would pay more for a call option than the cost of directly buying the asset itself.

At maturity, the time value of an option is zero—only intrinsic value remains.

• The put option protects against downside but allows for upside; the maximum loss is limited to the premium paid.

• The forward contract locks in a price, so the upside is capped, and if the stock rises significantly, the investor misses out—this can result in a larger opportunity cost (economic loss) than the fixed premium paid for the put.

Thus, if the stock price exceeds the forward price by more than the put premium, the forward strategy underperforms.

The value of a European put option decreases as the risk-free interest rate increases, because the present value of the strike price becomes smaller, lowering the option’s intrinsic value.

• If the underlying price falls, the put becomes more valuable.

• If volatility increases, the put becomes more valuable due to higher uncertainty and potential for profit.
  Only an increase in the risk-free rate reliably reduces a European put's value.

Dividends reduce the expected future price of the underlying asset, since the asset's value drops when dividends are paid. Because a call option benefits from a higher future price, higher expected dividends reduce the value of the European call. Unlike American calls, European calls cannot be exercised early to capture dividend payouts, making them especially sensitive to dividend effects.

The value of a European call option increases as time to expiration increases because there is more opportunity for the underlying asset to move favorably. A higher exercise price decreases the value of a call, and expected dividend payments typically reduce the value of a call because the underlying asset’s price is expected to drop when dividends are paid.