Derivatives Theory

Increase in share price

• Call Option: Increases as share price rises. A higher stock price raises the likelihood and magnitude of finishing in-the-money (max(S−K,0)).

 

• Put Option: Decreases as share price rises. A higher stock price reduces the payoff from selling at a fixed strike (max(K−S,0)).

Increase in strike price

• Call Option: Decreases as strike price rises. A higher strike makes the right to buy at that price more expensive and less likely to finish deep in-the-money.

 

• Put Option: Increases as strike price rises. A higher strike makes the right to sell at that price more valuable and increases payoff potential.

Increase in time of expiry

• Call Option: Increases with more time to expiry. Longer horizons provide more time for the stock to appreciate above the strike, giving greater upside with protected downside.

 

• Put Option: Generally increases with more time to expiry. More time allows greater potential price swings below the strike. 

Increase in volatility

• Call Option: Increases as volatility increases. Greater volatility widens the distribution of outcomes; option holders capture upside while downside losses remain capped at zero.

• Put Option: Increases as volatility increases. The asymmetric payoff structure benefits from large downward swings while floor protection caps downside loss.

Risk-Free Interest Rate (r)

• Call Option: Increases slightly as interest rates rise. Higher rates lower the present value of the strike price you pay in the future (Ke−rT) and increase the forward price of the stock.

• Put Option: Decreases slightly as interest rates rise. Higher rates discount the present value, reducing the present value of the future cash inflow and lowering the value of the put

Arbitrage: Overpriced European Call (CE<=S0) - T=0

1. Short call → (+CE)

2. Buy stock in the stock market → (-S0)

3. Invest proceeds

Arbitrage: Underpriced European Call ( CE>= S0-Xe^-rt)

1. Long 1 call → -CE

2. Short sell → +S0

3. Invest in bank

Arbitrage: Overpriced European Put (Pe<=Xe^-rt)

1. Sell put (+PE)

2. Invest proceeds in bank (-PE)

Arbitrage: Underpriced European Put (PE>= Xe^-rt -S0)(

1. Buy put (-PE)

2. Buy stock (-CE)

3. Borrow from bank