The payoff of a European call option at expiration is:
Payoff = max(S_T - X, 0)
where S_T is the stock price at expiration and X is the exercise price.
Components in Black-Scholes Model:
1. Exercise of the Strike Price (N(d2)):
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The probability of the option being in the money (i.e., S_T > X) is N(d2).
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The present value of paying the exercise price X, discounted at the risk-free rate, is X * e^(-rt) * N(d2).
2. Receipt of the Stock (N(d1)):
The expected value of receiving the stock, considering the option is in the money, is :
S * N(d1).
N(d1) adjusts for both the probability of exercise and the expected stock price, ensuring the expected payoff is greater than the current stock price. Note that N(d1) will be always greater than N(d2) because if N(d2) is the probability of being exercise, aka E(St) then the probability the expected value of receiving the stock, considering the option is in the money, aka E(S_T|S_T>X) will be inflated by an higher weighted probability because we now consider not a simple probability but a conditional expectation with less possible outcomes.
3. Discounting at the Risk-Free Rate:
Ensures the option's value respects the no-arbitrage principle by discounting future expected payoffs to their present value.
Combining these components, the value of the call option C is:
C = S * N(d1) - X * e^(-rt) * N(d2)
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S * N(d1): Present value of the expected stock price received if the option is exercised.
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-X * e^(-rt) * N(d2): Present value of paying the exercise price.
Intuition:
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N(d2) is the probability the option will be in the money.
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N(d1) accounts for both this probability and the higher expected stock price if exercised, ensuring no-arbitrage pricing.
Relative Value of N(d1) and N(d2):
In the Black-Scholes model, N(d1) and N(d2) are cumulative standard normal distribution functions representing different probabilities:
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N(d2): The probability that the option will be in the money at expiration (S_T > X).
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N(d1): Adjusts for this probability and also incorporates the expected stock price if the option is exercised.
Ensuring Non-Negative Value for the Option
The Black-Scholes formula for a European call option is:
C = S * N(d1) - X * e^(-rt) * N(d2)
Why N(d1) > N(d2) Ensures Non-Negative Option Value
1. Probability Adjustment:
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N(d2) reflects the probability of the option being in the money.
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N(d1) not only reflects this probability but also adjusts for the expected payoff from the stock, making it higher.
2. Present Value Consideration:
- The term S * N(d1) represents the present value of the expected stock price, while X * e^(-rt) * N(d2) represents the present value of the exercise price payment.
- Since N(d1) > N(d2), the present value of the stock received (adjusted by N(d1)) will always be greater than the present value of the exercise price payment (adjusted by N(d2), ensuring C >= 0.
The intrinsic value (S - X) is positive when S > X, which is reflected by N(d2). The time value is reflected by the difference between N(d1) and N(d2), representing the potential for the stock price to exceed the exercise price before expiration.
The relative values of N(d1) and N(d2) in the Black-Scholes model ensure a non-negative option value by appropriately adjusting for both the probability of being in the money and the conditional expectation of the stock price, thus capturing both the intrinsic value and time value of the option.
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