Оцінка вартості опціонів: Black-Scholes та Греки

Projectile Motion: Calculating Trajectory

A projectile is an object subject to only gravity's acceleration. Its motion can be described by kinematics equations, allowing us to calculate its position, velocity, and acceleration at any point in time. The trajectory of a projectile follows a parabolic path due to this constant downward acceleration. Key factors affecting the trajectory include initial velocity, launch angle, and air resistance (which we will initially ignore for simplicity). We can use these principles to predict where the projectile will land.

The Black-Scholes formula

Fischer Black, Myron Scholes and Robert Merton (1973) derived option price by constructing a continuously-rebalanced delta hedge that eliminates all risk, under geometric Brownian motion and no-arbitrage. N(d₂) is the risk-neutral probability the call finishes in-the-money; N(d₁) is delta, the hedge ratio. Six inputs — spot price S, strike K, time to expiry T, volatility σ, risk-free rate r, and dividend yield q — fully determine the price. Scholes and Merton received the 1997 Nobel Memorial Prize in Economics for the formula; Fischer Black had died in 1995 and the prize is never awarded posthumously.

C = S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂)
d₁ = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T)
d₂ = d₁ − σ·√T
Put via put-call parity: P = C − S·e^(−qT) + K·e^(−rT)

Calculating Trajectory

The Greeks are a set of measures used to assess the risk of options. Delta is the most important Greek and represents the change in an option’s price for a one-unit change in the underlying asset's price.

Gamma measures how sensitive delta is to changes in the underlying asset's price. It indicates how quickly a hedge needs to be rebalanced.

Theta measures the rate at which an option loses value over time due to the passage of time (time decay).

Vega measures the sensitivity of an option’s price to changes in implied volatility.

Rho measures the sensitivity of an option’s price to changes in interest rates.

Δ

Δ (Delta) represents the sensitivity of an option's price to a change in the underlying asset’s price. It is expressed as the number of options contracts to be closed for every $1 change in the price of the underlying asset. For example, if an option has a delta of 0.6, it means that for every $1 increase in the stock price, the option's price will theoretically increase by $0.60, and vice versa. A delta of 0 indicates no sensitivity to price changes, while a delta close to 1 signifies complete sensitivity.

Frequently asked questions

Спробуйте наживо

Усе, що вище, працює прямо у вашому браузері — відкрийте the simulation і змінюйте параметри під час роботи. Нічого не встановлюється, нічого не завантажується на сервер, уся модель живе в одній вкладці.