Consider a Black-Scholes model with ? = 0.3, S(0) = 50, r = 0.05, ? = 0.03. Suppose you use a bull spread, buying a European Call option with K = 48 and selling a Call with K = 52. Both options expire at T = 1.

Find the aggregate Delta, Gamma, Vega, Theta and Rho of the option portfolio.

Recall that the Greeks are partial derivatives of the portfolio value with respect to the given parameter. Instead of using calculus to find the derivative, we may use a finite-difference approximation. For example, to approximate Vega we can use Taylor expansion:

for a small perturbation . This means you compute the portfolio value plugging in ?0 for volatility, then plugging-in ?0 + for volatility and look at the difference.

Use the above method to approximate the Vega of the above bull-spread portfolio. Use = 0.002. Compare to the true Vega in part 1.