Understanding Call and Put Options Prices Theorem with Excel Formulas

The Prices of Call and Put Options Theorem, also known as put-call parity, is a principle that defines the relationship between the prices of European put and call options of the same class. A class of options means that they have the same underlying asset, strike price, and expiration date. European options can only be exercised at the expiration date, unlike American options that can be exercised at any time before the expiration date.

The theorem states that the price of a call option implies a certain fair price for the corresponding put option with the same strike price and expiration, and vice versa. This means that if you know the price of one option, you can calculate the price of the other option using a simple formula. The formula also shows that a portfolio consisting of a long call option and a short put option should have the same value as a forward contract on the same underlying asset, with the same expiration and strike price. A forward contract is an agreement to buy or sell an asset at a fixed price in the future.

The theorem is based on the assumption that there are no arbitrage opportunities in the market, meaning that there is no way to make a risk-free profit by exploiting price differences. If the theorem is violated, then arbitrageurs can take advantage of the situation and buy the cheaper option and sell the more expensive one, until the prices converge and the theorem holds again.

The theorem can be used to understand the relationship between different types of options and the underlying asset, as well as to price options and hedge risk. It can also help to explain some phenomena in the options market, such as volatility smiles and skews, which are patterns of implied volatility across different strike prices and maturities.

Basic Theory:

Call Option: A financial contract that gives the holder the right (but not the obligation) to buy an asset at a predetermined price (strike price) before the expiration date.
Put Option: Similar to a Call option but gives the holder the right (not obligation) to sell an asset at a predetermined price (strike price) before the expiration date.

Procedures:

Black-Scholes Model: One of the most widely used models for pricing options. The Black-Scholes formula for Call option price is:

$C = S_0 e^{-qt} N(d_1) - X e^{-rt} N(d_2)$

Where:

$C$ = Call option price
$S_0$ = Current stock price
$X$ = Strike price
$t$ = Time to expiration
$r$ = Risk-free interest rate
$N(d_1)$ and $N(d_2)$ are cumulative distribution functions of the standard normal distribution
$q$ = Dividend yield (if any)

The Put option price ( $P$ ) is then calculated as:

$P = X e^{-rt} N(-d_2) - S_0 e^{-qt} N(-d_1)$

Where:

$N(-d_1)$ and $N(-d_2)$ are the cumulative distribution functions of the standard normal distribution for – $d_1$ and – $d_2$ respectively.
$d_1$ and $d_2$ are calculated as:
$d_1 = \frac{ \ln(S_0 / X) + (r - q + \sigma^2 / 2) t }{ \sigma \sqrt{t} }$
$d_2 = d_1 - \sigma \sqrt{t}$

Comprehensive Explanation:

Let’s consider a scenario with the following values:

Current stock price ( $S_0$ ): $100
Strike price ( $X$ ): $105
Time to expiration ( $t$ ): 1 year
Risk-free interest rate ( $r$ ): 5%
Dividend yield ( $q$ ): 2%
Volatility ( $\sigma$ ): 20%

Excel Calculation:

Calculate and
- $d_1 = \frac{\ln(100/105) + (0.05 - 0.02 + 0.2^2 / 2) \times 1}{0.2 \times \sqrt{1}}$
- $d_2 = d_1 - 0.2 \times \sqrt{1}$
Use $N(d_1)$ and $N(d_2)$ from Excel’s NORM.S.DIST function.
Calculate Call Option Price () and Put Option Price ():
- $C = 100 \times e^{-0.02 \times 1} \times N(d_1) - 105 \times e^{-0.05 \times 1} \times N(d_2)$
- $P = 105 \times e^{-0.05 \times 1} \times N(-d_2) - 100 \times e^{-0.02 \times 1} \times N(-d_1)$

Result:

In Excel, the calculated Call Option Price ( $C$ ) is $4.23 and the Put Option Price ( $P$ ) is $9.64.

Other Approaches:

Binomial Model: A discrete-time model for option pricing that breaks down the time to expiration into a number of steps, providing a more intuitive approach.
Monte Carlo Simulation: Simulating the possible outcomes of the option prices by using random sampling techniques.
Greeks (Sensitivity Measures): Explore Delta, Gamma, Theta, and Vega to understand how the option price changes concerning various factors.