Stock options are influenced by a variety of factors, each playing a pivotal role in determining their value. Here are the six key factors:

**Current Stock Price (S0)****Strike Price (K)****Time to Expiration (T)****Volatility of the Stock Price****Risk-Free Interest Rate (r)****Expected Dividends**

Changes in any of these factors, while keeping the others constant, can significantly impact option prices. The effects of these changes on European and American call and put options are diverse:

**Stock and Strike Prices**: For call options, an increase in the stock price or a decrease in the strike price raises its value. For put options, it’s the opposite.**Time to Expiration**: American options generally increase in value as the expiration time extends. European options usually follow this trend, but exceptions exist, especially in scenarios involving dividends.**Volatility**: Increased volatility raises the value of both call and put options.**Risk-Free Interest Rate**: Rising interest rates typically increase the value of call options and decrease the value of put options.**Dividends**: Expected dividends can decrease the value of call options and increase the value of put options.

These relationships are crucial in understanding the dynamics of option trading.

## Summary of Stock Option Price Influences

**Upper Bounds of Option Prices:**

**For Call Options:**The price of both American and European call options can never exceed the current stock price, as owning the option should not be more expensive than owning the stock itself.**For American Put Options:**The maximum value of an American put option is the strike price, since this is the highest price the holder can get for the stock.**For European Put Options:**The upper limit is the present value of the strike price, discounted at the risk-free interest rate.

**Lower Bounds of Option Prices:**

**For European Call Options on Non-Dividend-Paying Stocks:**The lower bound is the difference between the current stock price and the present value of the strike price. If the price falls below this, arbitrage opportunities arise.**For European Put Options on Non-Dividend-Paying Stocks:**The lower bound is the difference between the present value of the strike price and the current stock price.

These bounds are important for traders to understand the range within which options should logically be priced, considering market conditions and the risk-free interest rate. Arbitrageurs closely watch these bounds to capitalize on any discrepancies that occur in the market.

**Exploring Put-Call Parity**

Using these properties, we can now derive the *put-call parity*, which is a significant principle in options trading, linking the prices of European put and call options with identical strike prices and expiration dates. It revolves around comparing two portfolios:

**Portfolio A**: Comprises a European call option and a zero-coupon bond, with the bond maturing at the option’s strike price at time T.**Portfolio C**: Contains a European put option and a stock.

At expiration, both portfolios end up having the same value, whether the stock price is above or below the strike price. This leads to the fundamental put-call parity equation:

\(c + Ke^{-rT} = p + S_0 \)

Here, \(c \) is the price of the call option, \(Ke^{-rT} \) is the present value of the strike price, \(p \) is the price of the put option, and \(S_0 \) is the current stock price. This equation implies that the value of a European call option can be determined from the value of a European put option with the same strike price and maturity, and vice versa.

**American vs. European Options on Non-Dividend-Paying Stocks**

**American Call Options**: It’s never optimal to exercise American call options on non-dividend-paying stocks before their expiration. Holding the option rather than exercising it early can earn interest on the strike price, and there’s always a chance that stock prices may fall. Selling the option rather than exercising it is usually more profitable.**European Call and Put Options**: European options have specific lower bounds for their prices. The lower bound for a call is the difference between the current stock price and the present value of the strike price. For a put, it’s the difference between the present value of the strike price and the current stock price.**American Put Options**: Contrary to calls, it can be optimal to exercise American put options early, especially if they are deep in the money. The value of an American put option can equal its intrinsic value (the difference between strike price and stock price) when the stock price is very low.**Impact of Dividends**: Dividends change the dynamic. American call options might be optimally exercised just before an ex-dividend date. For European options, dividends adjust the lower bounds for calls and puts.**Put-Call Parity with Dividends**: With dividends, the put-call parity equation adjusts to include the present value of dividends. This equation helps link the values of European calls and puts, considering dividends.

These insights offer a nuanced understanding of when and why certain options might be exercised or held, and how dividends can influence option strategies. Continue to the next part of this guide here.