Vanilla Options Pricing

Pricing Frameworks for European-Style Index Options

OLTA is evaluating multiple pricing methodologies to ensure robust, fair, and consistent valuation of its vanilla index options. Each approach accounts for the unique dynamics of on-chain execution, index NAV tracking, and liquidity variability.

Pricing inputs would reference OLTA’s NAV, including TWAP and slippage-aware models, and would be executed transparently via smart contracts.

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1. Black-Scholes-Merton Model

The Black-Scholes-Merton which OLTA will priorize provides a closed-form solution for pricing European options on non-dividend-paying assets. It is widely used in TradFi and serves as a foundational model.

Call Option Price:

$$C = S_0 N(d_1) - K e^{-rT} N(d_2)$$

Put Option Price:

$$P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$$

Where:

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

And:

• S₀ = Spot price of the index

• K = Strike price

• T = Time to maturity (in years)

• r = Risk-free rate (may be assumed near 0% in crypto)

• σ = Implied volatility of the index

• N(x) = Standard normal cumulative distribution function

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2. Tri-linear Interpolation

In cases where implied volatility data is discrete (based on strike ladders or expiry points), OLTA would apply tri-linear interpolation:

• Across strike prices

• Across time to maturity

• Across asset-specific volatility surfaces

$$\text{Interpolated Price} = f(\text{Strike}, \text{Maturity}, \text{Volatility})$$

This model ensures smoothed price estimation and compatibility with fragmented on-chain data.

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3. Monte Carlo Simulation

To model options with more complex dynamics such as varying volatility, execution slippage, or liquidity constraints OLTA would use Monte Carlo methods:

Generalized Approach:

1. Simulate thousands of potential price paths for the index using geometric Brownian motion:

Stochastic Differential Equation (GBM)

$${\,dS_t = \mu\,S_t\,dt \;+\; \sigma\,S_t\,dW_t\,}$$

where

• Sₜ = Simulated index price at time t

• μ = drift (often set to the risk‑free rate (r))

• σ = Volatility of the index

• Wₜ = standard Brownian motion wheres

$$dW_t \sim \mathcal N(0,\,dt)$$

Closed‑Form Solution

Integrating the SDE yields the familiar exponential form:

$$S_t = S_0 \exp\left( \left(r - \frac{\sigma^2}{2}\right)t + \sigma W_t \right)$$

Where:

• S₀ = Initial index price

• Sₜ = Simulated index price at time t

• r = Risk-free rate

• σ = Volatility of the index

• Wₜ is the Standard Brownian motion (random component) defined as following:

$$W_t = \sqrt{t}\,Z,\quad Z \sim \mathcal N(0,1)$$

where all increments are independants:

$$W_{t+\Delta t} - W_t \sim \mathcal N\bigl(0,\,\Delta t\bigr)$$

withW(0) = 0

Discrete‑Time Simulation Scheme

For a time‑step (\Delta t):

$$\begin{aligned} W_{t+\Delta t} &= W_t + \sqrt{\Delta t},\varepsilon, \ S_{t+\Delta t} &= S_t \exp!\Bigl[\bigl(\mu - \tfrac12\sigma^2\bigr)\Delta t + \sigma\sqrt{\Delta t},\varepsilon\Bigr], \end{aligned} \qquad\varepsilon \sim \mathcal N(0,1).$$

1. Compute the average discounted payoff across all simulated paths.

This method allows for high-fidelity modeling of tail risks and dynamic pricing behavior.

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Implementation Notes

• Final option prices would reference OLTA’s slippage-aware NAV and real-time liquidity profile.

• Execution would occur via smart contracts with deterministic pricing.

• Governance would define acceptable pricing thresholds and fallback methods.