Black Scholes Greeks¶

Background¶

Black Scholes Greeks

Educational script demonstrating black scholes greeks concepts.

Code¶

```python """ Black Scholes Greeks

Educational script demonstrating black scholes greeks concepts. """

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black_scholes/black_scholes_greeks.py¶

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from .black_scholes_base import BlackScholesBase from .black_scholes_utils import delta, gamma, vega

class BlackScholesGreeks(BlackScholesBase): def delta(self): return delta(self.S0, self.K, self.T, self.r, self.sigma, self.q)

def gamma(self):
    return gamma(self.S0, self.K, self.T, self.r, self.sigma, self.q)

def vega(self):
    return vega(self.S0, self.K, self.T, self.r, self.sigma, self.q)

if name == "main": pass ```

Exercises¶

Exercise 1. Define delta ($\Delta$) for a European call and put. Compute $\Delta_{\text{call}}$ for $S_0 = 100$, $K = 100$, $T = 1$, $r = 0.05$, $\sigma = 0.20$, $q = 0$.

Solution to Exercise 1

Delta is $\Delta_{\text{call}} = e^{-qT}\mathcal{N}(d_1)$ and $\Delta_{\text{put}} = -e^{-qT}\mathcal{N}(-d_1)$.

With $q = 0$: $d_1 = (0 + 0.07)/0.20 = 0.35$, so $\Phi(0.35) pprox 0.6368$.

\[ \Delta_{\text{call}} = 0.6368, \quad \Delta_{\text{put}} = -(1 - 0.6368) = -0.3632 \]

The call delta of 0.64 means the option price increases by approximately $\$0.64$ for each $\$1$ increase in the stock price.

Exercise 2. Show that gamma ($\Gamma$) is the same for calls and puts. Derive the formula $\Gamma = \frac{e^{-qT}\phi(d_1)}{S_0\sigma\sqrt{T}}$ where $\phi$ is the standard normal PDF.

Solution to Exercise 2

Since $\Delta_{\text{call}} - \Delta_{\text{put}} = e^{-qT}$ (a constant), their derivatives with respect to $S$ are equal:

\[ \Gamma = \frac{\partial \Delta_{\text{call}}}{\partial S} = \frac{\partial \Delta_{\text{put}}}{\partial S} \]

Computing: $\Gamma = e^{-qT}\phi(d_1)\frac{\partial d_1}{\partial S}$. Since $\frac{\partial d_1}{\partial S} = \frac{1}{S\sigma\sqrt{T}}$:

\[ \Gamma = \frac{e^{-qT}\phi(d_1)}{S_0\sigma\sqrt{T}} \]

Gamma is always positive, peaking when the option is at-the-money.

Exercise 3. Explain the financial interpretation of vega ($ u$). Why is vega not a Greek letter, and why is it important for traders?

Solution to Exercise 3

Vega measures the sensitivity of the option price to changes in implied volatility: $

u = \partial V / \partial \sigma$. The formula is $ u = S_0 e^{-qT}\phi(d_1)\sqrt{T}$.

Vega is not a Greek letter (it comes from the star Vega), but it is included among the "Greeks" by convention. It is crucial for traders because:

1. Volatility is the only BS parameter not directly observable -- it must be estimated or implied.
2. Changes in implied volatility can move option prices substantially.
3. Volatility trading strategies (straddles, strangles) are designed to profit from vega exposure.
4. Vega hedging is essential for market makers to manage their volatility risk.

Exercise 4. Theta ($\Theta$) measures time decay. Explain why theta is typically negative for long option positions and derive the relationship $\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + (r-q)S\Delta = rV$.

Solution to Exercise 4

This relationship comes directly from the Black-Scholes PDE: $\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q)S\frac{\partial V}{\partial S} = rV$. Identifying the partial derivatives as Greeks: $\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + (r-q)S\Delta = rV$.

Theta is typically negative because options lose value as expiration approaches (time value erodes). This is because the probability of a large favorable move decreases with less time remaining. The equation shows that theta is the "cost" of carrying the gamma and delta exposure -- it balances the portfolio's risk-neutral return.