• Global Frameworks: GARP FRM Part 2 (Market Risk Modeling), Technical Finance Standards.

1. The One-Period Binomial Option Pricing Model
The binomial model uses a discrete tree framework, assuming that over a single time step, a stock price can move up to an increased price (u × S) or down to a decreased price (d × S).
                    ┌─── u × S (Upward Price Jump)
                    │
[Initial Spot Price S]
                    │
                    └─── d × S (Downward Price Jump)

To price the option, analysts construct a risk-free portfolio containing a long position in the stock and a short position in the option. Since the portfolio is risk-free, it must earn the risk-free interest rate. This allows analysts to determine the option’s value using Risk-Neutral Probabilities:

π = (e^(r × Δt) − d) / (u − d)

Current option value:
C = e^(−r × Δt) × [π × C_u + (1 − π) × C_d]

2. The Black-Scholes-Merton (BSM) Closed-Form Equation
As the time steps in a binomial model approach zero, the model transitions into the continuous-time Black-Scholes-Merton formula for pricing European call options:

C = S₀ × N(d₁) − X × e^(−r × T) × N(d₂)

d₁ = [ln(S₀/X) + (r + σ²/2) × T] / (σ × √T)

d₂ = d₁ − σ × √T

  • Core Inputs: The model relies on five variables: Spot Price (S₀), Strike Price (X), Risk-free Rate (r), Time to Maturity (T), and Asset Volatility (σ). Volatility is the only variable that cannot be directly observed in the market.
  • Statistical Meaning: N(d₂) represents the risk-neutral probability that the option will expire in-the-money. N(d₁) represents the delta-adjusted replication weight needed to hedge the option position.

3. The Option Greens Matrix
The Option Greens measure how sensitive an option’s price is to changes in the underlying model variables:
  • Delta (Δ): Measures the change in the option price for a $1 change in the underlying asset’s spot price. Call options have positive deltas (0 to +1.0); put options have negative deltas (-1.0 to 0).
  • Gamma (Γ): Measures the acceleration rate of Delta per $1 shift in the stock price. It highlights the stability of a delta hedge.
  • Vega: Measures option price sensitivity to changes in the underlying asset’s volatility (σ).
  • Theta (Θ): Measures the rate of option value decay over time (Time Decay). As expiration approaches, an option’s time value drops at an accelerating rate.

 
PreviousNext