Options Pricing Calculator

This tool calculates the price of European call and put options, using either the Black-Scholes formula or the binomial pricing model. The models are based on the book "Options, Futures, and Other Derivatives" by John C. Hull (5th edition). Additionally, you can plot each variable against the options call or put price to visualize the impact of each variable on the option price.

Screenshots

Variables

• s: Spot price (current market price of the underlying asset)
• k: Strike price (price at which the option can be exercised)
• t: Time to expiration (in years)
• r: Risk-free rate (annualized), set as the current 10-year US Treasury yield by default, obtained from the alpha-vantage API
• v: Volatility (standard deviation of the underlying asset's price) = 0.2

Pricing models

Black-Scholes Formula

The Black-Scholes formula calculates the price of a European call and put option.

Formula for (d_1) and (d_2):

$$ d_1 = \frac{\ln(\frac{s}{k}) + (r + \frac{v^2}{2}) \cdot t}{v \cdot \sqrt{t}} $$

$$ d_2 = d_1 - v \cdot \sqrt{t} $$

Call Option Price:

$$ C = s \cdot N(d_1) - k \cdot e^{-rt} \cdot N(d_2) $$

Where:

• (N(d_1)) and (N(d_2)) are the cumulative distribution functions of the standard normal distribution.

Put Option Price:

$$ P = k \cdot e^{-rt} \cdot N(-d_2) - s \cdot N(-d_1) $$

Binomial Pricing Model

The binomial pricing model calculates the price of European call and put options using a binomial tree approach.

Steps:

1. Set Up the Binomial Tree:

   • Number of Time Steps (n): Defines the depth of the tree, set to 100.
   • Time Step Length (dt): Time per step, calculated as $ \frac{t}{n} $.
   • Up Factor (u): Represents the factor by which the asset price increases in one time step, calculated as $ e^{v \cdot \sqrt{dt}} $.
   • Down Factor (d): Represents the factor by which the asset price decreases in one time step, calculated as $ e^{-v \cdot \sqrt{dt}} $.
   • Risk-Neutral Probability (q): Probability of an up move under the risk-neutral measure, calculated as $ \frac{e^{r \cdot dt} - d}{u - d} $.
   • Discount Factor (disc): Factor to discount future values to present value, calculated as $ e^{-r \cdot dt} $.

2. Initialize Asset Prices at Maturity:

   • Compute all possible asset prices at the option's maturity based on different paths through the tree.

3. Initialize Option Values at Maturity:

   • Compute the option values at maturity for call and put options based on the asset prices.

4. Step Backwards Through the Tree:

   • For each time step, calculate the option values by taking the risk-neutral expected value of the option values at the next step and discounting it to the current step.

Call Option Price:

$D$ = Discount Factor
$c_{up}$ = Call Value at the up node
$c_{down}$ = Call Value at the down node

$$ C = d \times ( q \cdot c_{up} + (1 - q) \cdot c_{down}) $$

Put Option Price:

$D$ = Discount Factor
$p_{up}$ = Put Value at the up node
$p_{down}$ = Put Value at the down node

$$ P = Discount Factor \times ( q \cdot Put Value_{up} + (1 - q) \cdot Put Value_{down}) $$