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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 30day call expires at a value of 50
The ﬁnal investment is: 5.892 45 = 5 5 = 0.892 The 30day 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 90day
call, which is now a 60day 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 30day call expires without value
The ﬁnal investment is: 2.332
The 30day 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 nondividend 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.
 Fall '14

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