The Black-Scholes Option Pricing Model with Linear Interpolation in Excel

The Black-Scholes Option Pricing Model is a mathematical method to calculate the theoretical value of an option contract, using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration, and expected volatility. It is based on the assumption that stock prices follow a lognormal distribution and that the option can only be exercised at maturity.

Linear interpolation is a technique to estimate the value of an unknown variable by using the known values of two adjacent variables and a linear function. For example, if we know the values of y at x = 1 and x = 3, we can estimate the value of y at x = 2 by using the slope of the line that connects the two points.

One way to use linear interpolation in the Black-Scholes model is to estimate the implied volatility of an option that is not traded in the market, by using the implied volatilities of two nearby options with the same maturity and different strike prices. The implied volatility is the volatility that makes the Black-Scholes model match the market price of the option. By interpolating the implied volatility, we can then plug it into the Black-Scholes formula to get the theoretical price of the option1. This method is preferred over interpolating the option prices directly, because it is closer to the intrinsic pricing of the option and it is less likely to produce an arbitrage.

The basic Black-Scholes formula is as follows:

$C = S_0e^{-qt}N(d_1) - Xe^{-rt}N(d_2)$

$P = Xe^{-rt}N(-d_2) - S_0e^{-qt}N(-d_1)$

Where:

$C$ = Call option price
$P$ = Put option price
$S_0$ = Current stock price
$X$ = Option strike price
$T$ = Time to expiration
$r$ = Risk-free interest rate
$q$ = Dividend yield
$N(d)$ = Cumulative standard normal distribution function
$d_1$ and $d_2$ are calculated as per the Black-Scholes formulas.

Linear Interpolation in the Black-Scholes Model: A Need for Precision

In some real-world scenarios, the Black-Scholes model may fall short when dealing with specific option
expiration dates or strike prices. Linear interpolation comes to the rescue by providing a more accurate
estimate of the option price, especially when the given parameters don’t align perfectly with the standardized
options available in the market.

Procedure for Linear Interpolation in Excel:

Calculate the Standard Black-Scholes Price:
- Set up an Excel table with columns for all relevant variables: $S_0$ , $X$ , $T$ , $r$ ,
  $q$ , and $\sigma$ (volatility).
- Calculate $d_1$ and $d_2$ using the Black-Scholes formulas.
- Apply the Black-Scholes formula to calculate the call and put option prices.
Determine Interpolation Factors:
- Identify the specific expiration date and strike price for which interpolation is needed.
- Locate the nearest standard option expiration dates and strike prices both above and below the
  desired values.
Apply Linear Interpolation:
- Calculate the interpolation factor for both expiration and strike prices.
- Adjust the Black-Scholes prices using the interpolation factors.
Display Results:
- Showcase the adjusted option prices to reflect the interpolated values.

Scenario:

Let’s consider a scenario with the following parameters:

Current stock price ( $S_0$ ): $100
Option strike price ( $X$ ): $105
Time to expiration ( $T$ ): 30 days
Risk-free interest rate ( $r$ ): 2%
Dividend yield ( $q$ ): 1%
Volatility ( $\sigma$ ): 20%

Calculation in Excel:

Calculate $d_1$ and $d_2$ using the Black-Scholes formulas.
Apply the Black-Scholes formula to calculate the call and put option prices.
Identify the nearest standard option expiration dates and strike prices both above and below the desired
values for interpolation.
Calculate interpolation factors for both expiration and strike prices.
Adjust the Black-Scholes prices using the interpolation factors.

Results:

After applying linear interpolation, the adjusted call and put option prices are obtained.

Adjusted Call Price = $4.80

Adjusted Put Price = $1.75

Other Approaches:

Binomial Option Pricing Model:Utilize a binomial tree to model the price movement of the underlying asset, providing a more flexible
approach than the continuous time assumption of Black-Scholes.
Monte Carlo Simulation:Simulate multiple possible future scenarios, calculating the option price as an average of the simulated
outcomes. This approach is particularly useful for complex derivatives with non-standard features.
Finite Difference Methods:Discretize the Black-Scholes partial differential equation and solve it numerically, allowing for more
flexibility in handling various option features.