Question 1

Step 1: Calculate $d_1$

Accepted Answer

$d_1 = \frac{\ln(100 / 100) + (0.05 + \frac{0.20^2}{2}) \cdot 1}{0.20 \cdot \sqrt{1}}$
Since $\ln(1) = 0$, the equation simplifies:
$d_1 = \frac{0 + (0.05 + 0.02) \cdot 1}{0.20 \cdot 1} = \frac{0.07}{0.20} = 0.35$

Question 2

Step 2: Calculate $d_2$

Accepted Answer

$d_2 = 0.35 - 0.20 \cdot \sqrt{1} = 0.35 - 0.20 = 0.15$

Question 3

Step 3: Find the Cumulative Normal Probabilities $N(d_1)$ and $N(d_2)$

Accepted Answer

Using a standard normal distribution table (or statistical software):
$N(0.35) = 0.6368$ (There is a 63.68% probability factor for the stock price)
$N(0.15) = 0.5596$ (There is a 55.96% probability factor for the strike price)

Question 4

Step 4: Calculate the Call Price ($C$)

Accepted Answer

$C = (100 \cdot 0.6368) - (100 \cdot e^{-0.05 \cdot 1} \cdot 0.5596)$
First, calculate the present value of the strike price: $100 \cdot e^{-0.05} = 100 \cdot 0.9512 = 95.12$
$C = 63.68 - (95.12 \cdot 0.5596)$
$C = 63.68 - 53.23 = 10.45$

Question 5

Can I use the Black-Scholes model to price options on individual US stocks?

Accepted Answer

Strictly speaking, the standard Black-Scholes model is designed for European options, while almost all individual U.S. stock options are American-style (meaning they can be exercised early). If the stock does not pay a dividend, the Black-Scholes price will be highly accurate, because it is rarely mathematically optimal to exercise a non-dividend option early. However, if the stock pays a significant dividend, the Black-Scholes model will incorrectly price the option. In professional settings, American options are priced using the Binomial Tree or Bjerksund-Stensland models.

Question 6

What is Implied Volatility (IV) and why does the calculator need it?

Accepted Answer

Implied Volatility is the market's expectation of how much the stock price will fluctuate over the life of the option. Unlike the stock price or the time to expiration, future volatility is unknown. If you are calculating the theoretical price of an option, you must input your own estimate of volatility (often based on historical data). Conversely, if you know the option's current market price, you can run the formula backward to discover the Implied Volatility. It is the single most important metric for determining if an option is relatively cheap or expensive.

Question 7

Why does my calculated Black-Scholes price differ from the actual market price?

Accepted Answer

There are several reasons for a discrepancy. First, the market price reflects supply and demand, not pure mathematics; if there is a panic, buyers will bid the price up far beyond theoretical value. Second, your volatility input might differ from the market's consensus implied volatility. Third, the option might be American-style, or the stock might have an upcoming dividend that your calculation did not account for. Finally, the Black-Scholes model relies on assumptions (like lognormal distributions and constant interest rates) that the real market routinely violates.

Question 8

What happens to the option price if the volatility input is zero?

Accepted Answer

If you input a volatility of exactly 0%, you are telling the model that the stock price will never change from its current spot price. In this scenario, the time value of the option drops to absolute zero. If the option is "out of the money" (e.g., a call strike of $105 when the stock is at $100), the calculated price will be exactly $0.00. If the option is "in the money" (e.g., a call strike of $95 when the stock is at $100), the price will simply equal its intrinsic value ($5.00), minus a tiny fraction for the risk-free interest rate discount.

Question 9

How do interest rates (the risk-free rate) actually affect the option price?

Accepted Answer

Interest rates impact options pricing through the concept of the "cost of carry." Buying a call option is essentially a leveraged substitute for buying 100 shares of the stock. By buying the cheaper option instead of the expensive stock, you keep cash in your bank account, which earns interest. Therefore, as interest rates rise, the mathematical advantage of holding cash increases, which makes call options more valuable. Conversely, higher interest rates decrease the value of put options, because shorting the stock directly would generate cash that could earn high interest, making the put option a less attractive substitute.

Question 10

What is the "Volatility Smile" and does Black-Scholes account for it?

Accepted Answer

The Volatility Smile is a graphical pattern showing that options with strike prices deep in-the-money or deep out-of-the-money have higher implied volatilities than at-the-money options. The standard Black-Scholes model absolutely does not account for this. The model mathematically assumes that volatility is constant across all strike prices (which would draw a flat, horizontal line on a graph). The existence of the volatility smile in the real market is actual proof that traders know the Black-Scholes assumption of perfectly normal, bell-curve market returns is flawed, and they demand higher premiums to protect against extreme market crashes.

Black-Scholes Options Calculator

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