Comprehensive Summary¶

Background¶

Black Scholes Summary

Educational script demonstrating black scholes summary concepts.

Code¶

```python """ Black Scholes Summary

Educational script demonstrating black scholes summary concepts. """

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black_scholes_COMPREHENSIVE_SUMMARY.py¶

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import black_scholes as bs

Model parameters¶

if name == "main": S0 = 100 # Current stock price K = 105 # Strike price T = 0.25 # 3 months to expiration r = 0.05 # 5% risk-free rate sigma = 0.2 # 20% volatility q = 0.02 # 2% dividend yield mu = 0.1

bs.BlackScholes(S0=S0, K=K, T=T, r=r, sigma=sigma, q=q).summary()

```

Exercises¶

Exercise 1. Compute the European call price for $S_0 = 100$, $K = 105$, $T = 0.25$, $r = 0.05$, $\sigma = 0.2$, and $q = 0.02$ using the Black-Scholes formula with continuous dividends.

Solution to Exercise 1

With dividends: $d_1 = \frac{\ln(100/105) + (0.05 - 0.02 + 0.02)(0.25)}{0.2\sqrt{0.25}} = \frac{-0.0488 + 0.0125}{0.1} = -0.363$. Then $d_2 = -0.463$. The call is $C = 100 e^{-0.005} N(-0.363) - 105 e^{-0.0125} N(-0.463) = 99.50(0.358) - 103.69(0.322) = \$2.29$.

Exercise 2. List all five Greeks for the option in Exercise 1 and state their signs.

Solution to Exercise 2

Delta $= e^{-qT} N(d_1) = 0.357$ (positive); Gamma $= e^{-qT} n(d_1) / (S\sigma\sqrt{T}) > 0$ (positive); Vega $= Se^{-qT}\sqrt{T} n(d_1) > 0$ (positive); Theta $< 0$ (negative, time decay); Rho $= KTe^{-rT}N(d_2) > 0$ (positive). Delta is most important for short-term hedging.

Exercise 3. Compute the put price via put-call parity with continuous dividends: $P = C - S_0 e^{-qT} + K e^{-rT}$.

Solution to Exercise 3

\[ P = 2.29 - 100 e^{-0.005} + 105 e^{-0.0125} = 2.29 - 99.50 + 103.69 = \$6.48 \]

Exercise 4. Explain how the dividend yield $q = 0.02$ affects call and put prices compared to $q = 0$, and compute the call price difference.

Solution to Exercise 4

With $q = 0$: $d_1 = (-0.0488 + 0.0175)/0.1 = -0.313$, giving $C = 100(0.377) - 103.69(0.340) = \$2.48$. The dividend reduces the call by $2.48 - 2.29 = \$0.19$ because dividends lower the expected terminal stock price, reducing call values and increasing put values.