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ACT370H1 (9)
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January 29.docx

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University of Toronto St. George
Actuarial Science
Jack Pitt

January 29, 2014 Leverage – inherently risky small amount of cash -> large effect Dividends Corporation has profit: - Keep and reinvest - Pay out (dividends) Theoretical stock price = 1/δ, assuming constant dividends (rate δ) forever Constant growth rate g -> 1/δ-g Dividend situations Regular dividends - Constant - Growing Special dividends Corporation dividend categories - No dividends - Irregular dividends (special, ad hoc) - Regular (“widows and orphans”) All of our work here is based on predictable dividends and mostly on regular payments -> δ or δ-g Option value = IV + TV IV = intrinsic value (S-K or K-S) as if we could exercise the option right now (for European, pretend you can exercise now) TV = time value – comes from the belief/hope that there will be favourable conditions by exercise time TV decays to zero as we approach expiration date The Greeks Partial derivatives showing how an investment price responds to observable variables Black-Scholes (B-S) equation: C = f(S, K, r, σ, t) ∂C/∂σ, ∂C/∂S, etc. “Investment” above: - Single (eg. Option or stock) - Portfolio In this course, we focus on singles (for options) However, Hull uses portfolio notation (Π) but discusses options Π portfolio could consist of just one investment ∂Π/∂S could represent ∂C/∂S or ∂P/∂S Δ (delta) = ∂Π/∂S (underlying stock price) 2 2 Γ (gamma) = ∂ Π/∂S = ∂Δ/∂S θ (theta) = ∂Π/∂t (calendar time) or -∂Π/∂T (remaining time) Κ (vega) = ∂Π/∂σ (volatility) ρ (rho) = ∂Π/∂r (rate) Q: If stock price goes up, will call price a) Go up b) Go down c) Indeterminate? a, ∂Π/∂S is positive for a “vanilla” call vanilla = simple, European no dividends Can we generalize about signs and boundaries of the Greeks? For θ, t is calendar time (increasing as we approach expiration), measured in days Δ = ∂Π/∂S Δ C 0 Δ
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