Key Formulas in Financial Derivatives

Financial derivatives derive their value from an underlying asset, such as stocks, bonds, commodities, or currencies. Understanding their pricing requires knowledge of fundamental formulas. Here’s a look at some crucial ones:

Forward Contracts

The forward price (F) is the price agreed upon today for the future delivery of an asset. The formula accounts for the spot price (S), the risk-free rate (r), time to maturity (T), and any storage costs (U) or dividends (D) paid during the life of the contract:

F = S * e^{(r + U – D)T}

This formula uses continuous compounding. For discrete compounding, it would be: F = S * (1 + r + U – D)^T

Futures Contracts

Futures pricing is similar to forwards, but incorporates complexities due to marking-to-market and daily settlement. While the theoretical formula is the same as the forward, in practice, minor adjustments are needed to reflect these factors. The cost of carry model is often used as a practical approximation.

Options Contracts – Black-Scholes Model

The Black-Scholes model is a cornerstone for pricing European-style options (exercisable only at expiration). It provides a theoretical value based on several inputs:

• S: Current stock price
• K: Strike price of the option
• T: Time to expiration (in years)
• r: Risk-free interest rate
• σ: Volatility of the stock price

The formulas are as follows:

Call Option Price (C) = S * N(d1) – K * e^-rT * N(d2)

Put Option Price (P) = K * e^-rT * N(-d2) – S * N(-d1)

Where:

d1 = [ln(S/K) + (r + σ²/2) * T] / (σ * √T)

d2 = d1 – σ * √T

N(x) represents the cumulative standard normal distribution function.

Assumptions and Limitations: The Black-Scholes model relies on several assumptions, including constant volatility, a constant risk-free rate, no dividends, and efficient markets. In reality, these assumptions rarely hold perfectly, leading to discrepancies between the model’s output and actual market prices. Modifications and more complex models exist to address these limitations.

Put-Call Parity

Put-call parity establishes a relationship between the prices of European call and put options with the same strike price and expiration date. It’s based on the principle of no arbitrage.

C + K * e^-rT = P + S

Where:

• C: Call option price
• P: Put option price
• K: Strike price
• S: Current stock price
• r: Risk-free interest rate
• T: Time to expiration

This equation indicates that a portfolio consisting of a call option and a risk-free bond that pays the strike price at expiration is equivalent to a portfolio containing a put option and the underlying asset. Any deviation from this parity presents an arbitrage opportunity.

These formulas represent a starting point for understanding derivative pricing. Real-world applications often involve more sophisticated models and adjustments to account for market conditions and specific instrument characteristics.