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January 22.docx

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Actuarial Science
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Jack Pitt

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January 22, 2014 Volatility σ = standard deviation (annualized) of continuously compound underlying stock return u = e (r-δ)h + σ√h d = e (r-δ)h – σ√h r = risk-free rate δ = dividend yield (continuously compounded) h = time interval (Δt) T = total time elapsed Brownian motion Variance of Brownian motion is proportional to the elapsed time 2 var(t) = σ t sd(t) = σ√t American options For a non-dividend stock, early exercise of a call is unwise, but early exercise of a put can be optimal. For dividend stock, might be optimal for call too. At expiration: C AT) = C (E) P (T) = P (T) A E since there is no choice left at T. Dividends are generally announced/anticipated. Ex 12 (McD) American put K = 40 T = 1 H = 1/3 year -0.08/3 + 0.3√(1/3) u = 1.221246 = e d = 0.8636926 = e -0.08/3 - 0.3√(1/3) no dividends: δ = 0 41 30.858 E (8.363) 40-30.858 = 9.142 => compare at any node, S-K or K-S vs option value Dividends & binomial d/u = e (r-δ)h ± σ√h dividends as if c/c (not real world) Review 3 cases of discrete dividends to continuous: - Consistent, regular (eg. 50c/share each quarter) - Consistent growth (eg. 50c now, 10% increase per quarter) -
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