The roles of N(d1) and N(d2) in the Black-Scholes model in layman’s terms…
Recall that the Black-Scholes formula for a non dividend paying European call option is given by:
C = S (t)* N(d1) - K * e^(-r * T) * N(d2)
- C: Option value (non dividend paying call option price)
- S(t): Stock price at time t
- N(d1): Cumulative distribution function of the standard normal distribution evaluated at d1
- K: Strike price
- r: Risk-free interest rate
- T: Time to maturity
- N(d2): Cumulative distribution function of the standard normal distribution evaluated at d2
- N(d1) represents the cumulative standard normal distribution of the value d1 (*)
- It captures the probability that the option will finish in-the-money (ITM) and the underlying asset's price will be above the strike price at expiration.
- This probability is then multiplied by the normal probability density function (PDF) to incorporate the continuous nature of the distribution.
- In other words, N(d1) gives the probability-weighted likelihood that the option will be exercised profitably.
NB: The chart below helps visualize N(d1) for a call option struck at 80 with a stock price of 100, a volatility of 20%, a 0% interest rate and T =1
- N(d2) represents the cumulative standard normal distribution of the value d2 (*)
- It captures the probability that if the option is exercised, we would have to pay the strike price to acquire the underlying asset, considering the present value of the strike price.
- N(d2) also takes into account the risk-free rate and the time to expiration.
- This probability is used to discount the expected payoff from exercising the option.
In summary, N(d1) captures the probability that the option ends in-the-money and adjusts for the continuous probability distribution using the normal PDF. N(d2) captures the probability-weighted amount we would have to pay if the option is exercised. Both N(d1) and N(d2) contribute to the calculation of the option's theoretical value by influencing the expected payoff and discounting it to the present value.
(*) In the Black-Scholes model, the assumption about the distribution of stock prices is that they follow a log-normal distribution. This assumption is central to the model and plays a key role in calculating option prices.
A log-normal distribution means that the logarithm of the stock price is normally distributed. This distribution is characterized by its positive skewness, which reflects the tendency of stock prices to experience larger positive movements than negative movements.
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