Pricing Derivatives

The secret to understanding financial markets is to understand how derivatives are priced. Options and futures are examples of derivatives that derive their value from some underlying asset and are priced using mathematical models based on several economic variables. Here's a breakdown of the pricing process:

1. Futures Pricing

The cost-of-carry model then applies to the valuation of a futures contract at its current price plus carry costs for holding the underpinning asset until the date of expiration. The formula is as follows:

F = S \times e^{(r - d) \times t}

F=S×e(r−d)×t

Where:

• F: Futures price
  
• S :Spot price or current price of the underlying.
  
• r: The risk-free rate of interest per period (annualised)
  
• d: Dividend yield (if any)
  
• t: Time to expiration in years
  
• e: Euler's number, (approximately equal to 2.718)
  

Example:

• Spot price = ₹1,000
  
• Risk-free rate = 6%
  
• Dividend yield = 2%
  
• Time to expiration = 3 months (0.25 years)
  

F=1000×e(0.06−0.02)×0.25≈₹1,010.05

F = 1000 \times e^{(0.06 - 0.02) \times 0.25} \approx ₹1,010.05

F=1000×e(0.06−0.02)×0.25

F≈₹1,010.05

This calculates the futures price for the stock given the inputs.

2.Options Pricing

Options prices have greater complexity, mainly by resorting to specific models such as the Black-Scholes Model for European options or the Binomial Option Pricing Model for American ones.

Black-Scholes Formula for European Call Option:

C = S \times N(d_1) - X \times e^{-r \times t} \times N(d_2)

C=S×N(d1)−X×e−r×t×N(d2)

Where:

• C: Call option price
  
• S: Spot price of the underlying asset
  
• X: Strike price
  
• r: Risk free interest rate
  
• t: Time to expiration, in years
  
• N(d1)N(d_1)N(d1) and N(d2)N(d_2)N(d2): Cumulative standard normal distribution values
  
• d1d_1d1 and d2d_2d2: Intermediate calculations involving volatility (σ\sigmaσ) and other factors.
  

Example Inputs:

• Spot price = ₹ 1,000
  
• Strike price = ₹1,050
  
• Risk-free rate = 6%
  
• Time to expiry = 1 month ¼, = 0.0833 years
  
• Volatility = 20%
  

By calculating d1d_1d1 and d2d_2d2 and using a normal distribution table, you can find N(d1)N(d_1)N(d1) and N(d2)N(d_2)N(d2) to determine the call option price.

For a Put Option, the put-call parity formula is applied:
P=C+X×e−r×t−SP = C + X \times e^{-r \times t} SP=C+X×e−r×t−S

3. Intrinsic and Time Value

The value of an option has two parts:

• Intrinsic Value: The in-the-money amount, i.e., spot price minus strike price.
  
• Time Value: Time to expiration along with volatility holds the potential for future gain in option.
  

4. Pricing:

Market Influences A number of factors influence derivative pricing, including:

• Volatility - The higher the volatility, the higher is the price of an option. There is more uncertainty.
  
• Interest Rates: These affect the cost-of-carry pertinent to the futures and options.
  
• Dividends: Dividend-paying assets have their impact on the derivative pricing, especially options.
  

Calculation Tools Excel:

Leverage built-in financial functions or user-designed models. Financial Calculators Online tools can make some of these calculations much easier:. Advanced modeling is also provided by libraries such as QuantLib, NumPy in Python, among others.

Pricing derivatives might sound daunting, but with the correct tools and understanding, the process is quite straightforward. These concepts will eventually let you make informed trading decisions.

Conclusion:

Mastering the derivative pricing basics provides a sufficient grounding for treading into financial markets. In the next chapter, we will look in detail, among other things, at the factors that affect a derivative's price: volatilities, interest rates and models such as Black-Sholes.