05 05 032 365 45 q 09946 90 03 365 r 60 d2 09946

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Unformatted text preview: is: 60 ln 50 + (0.05 + 0.5 · 0.32 ) 365 45 q = 0.9946 90 0.3 365 r 60 d2 = 0.9946 0.3 = 0.8730 365 N (d1 ) = N (0.99) = 0.8389, N (d2 ) = N (0.87) = 0.8078 d1 = C = 50 · 0.8389 45 · e 60 0.05· 365 · 0.8078 = 5.892 The 30-day call expires at a value of 50 The final investment is: 5.892 45 = 5 5 = 0.892 The 30-day holding period profit is: 0.892 30 0.874e 0.05· 365 = 0.014 19 / 25 Example 11.3. Solution (cont.) On the other hand, if the stock price is 45, the value of the 90-day call, which is now a 60-day call, is: 60 ln 45 + (0.05 + 0.5 · 0.32 ) 365 45 q = 0.1284 90 0.3 365 r 60 d2 = 0.1284 0.3 = 0.0068 365 N (d1 ) = N (0.13) = 0.5517, N (d2 ) = N (0.07) = 0.5040 d1 = C = 45 · 0.5517 45 · e 60 0.05· 365 · 0.5040 = 2.332 The 30-day call expires without value The final investment is: 2.332 The 30-day holding period profit is: 2.332 30 0.874e 0.05· 365 = 1.454 20 / 25 Volatility: general and BS framework In general, volatility of a stock St at time t : q (St , Xt , t ) = lim Var (ln (St +h /St )) /h h !0 is stochastic volatility, because it depends on St , t or Xt , where Xt include all other factors that a↵ect the volatility. However, the BS framework assumes volatility is constant. 21 / 25 Two methods of estimating volatility There are two methods for estimating volatility: implied 1. Historical Volatility. Start with option prices and a pricing model, and back out from the option prices. Historical 2. Implied Volatility. Start with historical stock prices and calculate the standard deviation of the logged changes in price over short periods of time. We have already learned how to estimate historical volatility. Backing out the implied volatility from BS requires an iterative, because appears in both d1 and d2 . We’ll look at the simplest case of d2 = d1 . 22 / 25 Example 11.4 For a European call options on a non-dividend paying stock: (i) The stock price is 50 (ii) Time to expiry is t (iii) The strike price is 50e 0.04t (iv) The continuously compo...
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This document was uploaded on 02/26/2014.

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