Derivatives Pricing

Derivatives are financial instruments whose value is derived from an underlying asset. Understanding how to price derivatives is central to quantitative finance, with applications ranging from hedging strategies to complex structured products.

Forward and Futures Contracts

Forward Contracts

A forward contract obligates the buyer to purchase (and the seller to sell) an asset at a predetermined price $K$ on a future date $T$.

Forward price (no arbitrage): For an asset with spot price $S_0$: $F_0 = S_0 e^{rT}$

For assets paying continuous dividend yield $q$: $F_0 = S_0 e^{(r-q)T}$

Value of forward position at time $t$: $V_t = (F_t - K)e^{-r(T-t)}$

Futures vs. Forwards

• Futures: Standardized, exchange-traded, daily settlement (marking to market)
• Forwards: Customizable, over-the-counter, settlement at maturity

Daily settlement means futures prices differ slightly from forward prices when interest rates are stochastic.

Options Fundamentals

Option Types

Call option: Right (not obligation) to buy at strike price $K$ Put option: Right (not obligation) to sell at strike price $K$

European options: Exercisable only at expiration American options: Exercisable any time before expiration

Payoffs at Expiration

Call payoff: $\max(S_T - K, 0) = (S_T - K)^+$ Put payoff: $\max(K - S_T, 0) = (K - S_T)^+$

Put-Call Parity

For European options on non-dividend-paying stocks: $C - P = S_0 - Ke^{-rT}$

This no-arbitrage relationship allows pricing puts from calls and vice versa.

The Black-Scholes Model

Model Assumptions

1. Stock price follows geometric Brownian motion
2. No dividends during option life
3. No transaction costs or taxes
4. Constant risk-free rate
5. Continuous trading possible
6. No arbitrage opportunities

The Black-Scholes Equation

Under the risk-neutral measure, option value $V(S,t)$ satisfies: $\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = rV$

Black-Scholes Formulas

European call: $C = S_0 N(d_1) - Ke^{-rT}N(d_2)$

European put: $P = Ke^{-rT}N(-d_2) - S_0 N(-d_1)$

where: $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$ $d_2 = d_1 - \sigma\sqrt{T}$

$N(x)$ is the cumulative standard normal distribution function.

Extension for Dividends

For continuous dividend yield $q$: $C = S_0 e^{-qT} N(d_1) - Ke^{-rT}N(d_2)$

with $d_1$ adjusted by replacing $r$ with $r - q$.

The Greeks

The Greeks measure option price sensitivity to various factors:

Delta ($\Delta$)

$\Delta = \frac{\partial V}{\partial S}$

• Call delta: $N(d_1) \in (0, 1)$
• Put delta: $N(d_1) - 1 \in (-1, 0)$

Delta represents the hedge ratio for delta-neutral portfolios.

Gamma ($\Gamma$)

$\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}$

Gamma is highest for at-the-money options near expiration.

Theta ($\Theta$)

$\Theta = \frac{\partial V}{\partial t}$

Theta measures time decay. Options lose value as expiration approaches (all else equal).

Vega ($\mathcal{V}$)

$\mathcal{V} = \frac{\partial V}{\partial \sigma}$

Sensitivity to implied volatility. Long options have positive vega.

Rho ($\rho$)

$\rho = \frac{\partial V}{\partial r}$

Sensitivity to interest rates.

Implied Volatility

Implied volatility (IV) is the volatility input that makes the Black-Scholes price equal the market price. Since Black-Scholes cannot be inverted analytically for $\sigma$, numerical methods (Newton-Raphson, bisection) are used.

The Volatility Smile

In practice, implied volatility varies with strike price, forming a "smile" or "smirk" pattern. This reflects:

• Fat tails in return distributions
• Jump risk
• Stochastic volatility
• Supply/demand dynamics

Numerical Methods

Binomial Trees

Discretize time into steps. At each node, price moves up (by factor $u$) or down (by factor $d$): $u = e^{\sigma\sqrt{\Delta t}}, \quad d = e^{-\sigma\sqrt{\Delta t}} = 1/u$

Risk-neutral probability: $p = \frac{e^{r\Delta t} - d}{u - d}$

Work backward from terminal payoffs to price the option.

Monte Carlo Simulation

Simulate many paths of the underlying: $S_{t+\Delta t} = S_t \exp\left[(r - \frac{\sigma^2}{2})\Delta t + \sigma\sqrt{\Delta t} Z\right]$

where $Z \sim N(0,1)$.

Average discounted payoffs across paths: $V_0 = e^{-rT} \frac{1}{N}\sum_{i=1}^{N} \text{Payoff}_i$

Monte Carlo is particularly useful for path-dependent options and high-dimensional problems.

Finite Difference Methods

Discretize the Black-Scholes PDE on a grid and solve numerically. Common schemes include explicit, implicit, and Crank-Nicolson methods.

Programming Implementation

Efficient options pricing requires:

• Fast cumulative normal distribution calculations
• Optimized root-finding for implied volatility
• Vectorized Monte Carlo simulations
• Grid management for finite differences
• Calibration routines for model parameters