Financial calculus Introduction to Financial Option Valuation 8 potx
Financial calculus Introduction to Financial Option Valuation_2 pdf
... characterized by (6 .8) and (6.9) is the solution to an SDE.
Reasonably accessible SDE texts are (Gard, 1 988 ; Mao, 1997; Øksendal, 19 98) ,
although all require some background in stochastic processes ... familiar to anybody who has seen stock market
data displayed in graphical form.
To examine this data, it is reasonable to treat it on the same level as the output
from a pseudo-ran...
Financial calculus Introduction to Financial Option Valuation_3 doc
... relation (8. 3). [Hint: use Exer-
cise 3.7.]
8. 2. Show that (8. 21) can be replaced by (8. 22).
8. 3. Confirm that C(S, t) in (8. 19) satisfies (8. 16), (8. 17) and (8. 18) . [Hint:
to deal with (8. 16), ... transformed into such
aproblem. The other constraints, (8. 17) and (8. 18) , are known as boundary condi-
tions.
8. 5 Black–Scholes formulas
Imposing (8. 16), (8. 17) and (8. 18)...
Financial calculus Introduction to Financial Option Valuation_4 pptx
... explicit formulas (8. 19) and (8. 24), we may differentiate with
respect to S to obtain the required asset holding A
i
in (8. 10). This partial derivative
∂V/∂ S is called the delta of an option, and the ... Black–Scholes option valuation formulas (8. 19) and (8. 24) depend upon S, t
and the parameters E, r and σ .Inthis chapter we derive expressions for partial
derivatives of the o...
Financial calculus Introduction to Financial Option Valuation_5 doc
... call option,
and
p :=
P
S
, for a put option.
In these new variables, d
1
and d
2
in (8. 20) and (8. 21) simplify to
d
1
=
m
τ
+
τ
2
and d
2
=
m
τ
−
τ
2
, (11.1)
and, from (8. 19) and (8. 24), ... possible, for exam-
ple, to design a hybrid algorithm that uses a safe method, like bisection, until
the iterates are close to an x
and then switches to Newton’s method to get th...
Financial calculus Introduction to Financial Option Valuation_6 ppt
... 340
5425 280
1
2
5525 226
5625 179
1
2
5725 139
582 5 105
5100 5200 5300 5400 5500 5600 5700 580 0 5900
0.172
0.174
0.176
0.1 78
0. 18
0. 182
0. 184
0. 186
0. 188
0.19
0.192
Exercise price
Implied volatility
FTSE ... costly. To reduce the error to, say,
10
−4
would take of the order of 10
8
samples, and to reduce it to 10
−6
would take
of the order of 10
12
samples; see Exerc...
Financial calculus Introduction to Financial Option Valuation_8 potx
... to the option, and hence
for each S, t one of ( 18. 1) and ( 18. 2) is at equality. ( 18. 3)
The three components ( 18. 1), ( 18. 2) and ( 18. 3) are the key features in the the-
ory of American option valuation. ... the option value V
0
0
.
The option value output by ch 18. m is 1.01 58. The validity of the result will be confirmed by
ch24 in Chapter 24.
PROGRAMMING EXERCISES
P 18....
Financial calculus Introduction to Financial Option Valuation_9 potx
... σ
is computed in order to value an option, then a widely quoted rule of
thumb is to make the historical data time-frame Mt equal to that of the option:
to value an option that expires in six ... references 209
20 40 60 80 100 120 140 160 180
0
0.2
0.4
0.6
0 .8
1
Days
Vol
Daily
20 40 60 80 100 120 140 160 180 200
0
0.2
0.4
0.6
0 .8
1
Weeks
Vol
Weekly
Fig. 20.1. Historical v...
Financial calculus Introduction to Financial Option Valuation_10 ppt
... widths
10
2
[1 .88 41, 2.0752] [1.9623, 2.0004] 0 .89 5.0
10
3
[1.95 38, 2.0 087 ] [1.9937, 2.00 48] 0 .88 4.9
10
4
[1. 989 0, 2.0062] [1.9993, 2.0027] 0 .88 5.0
10
5
[1.9969, 2.0023] [1.9994, 2.0005] 0 .88 5.0
var(e
√
U
− ... Ratio of widths
10
2
[1 .88 41, 2.0752] [1. 987 5, 2.0012] 14.0
10
3
[1.95 38, 2.0 087 ] [1.9976, 2.0017] 13.4
10
4
[1. 989 0, 2.0062] [1.9997, 2.0010] 13.5
10
5
[1.996...
An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_13 pot
... is to truncate the domain to S ∈ [0, L], where
L is some suitably large value. Using (8. 17) and (8. 18) , this gives call boundary
conditions
C(0,τ) = 0andC(L,τ) = L. (24.4)
Similarly, from (8. 26) ... (Iserles, 1996; Mitchell and Griffiths,
1 980 ; Morton and Mayers, 1994; Strikwerda, 1 989 ).
The texts (Clewlow and Strickland, 19 98; Kwok, 19 98; Wilmott, 19 98; Wilmott
et al., 1995; Se...
An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_14 pot
... Index
dividends, 49, 182
double barrier option, 191
down-and-in call, 188 , 189
down-and-in put, 190
down-and-out call, 187 – 189 , 260–261, 265
down-and-out put, 190
drift, 54, 105, 1 98
efficient market ... 1 98
geometric Brownian motion, 57, 61
geometrically declining weights, 2 08, 210
Greeks, 99–102
grid, 239
heat equation, 2 38 239, 262, 265
hedging, 74, 76– 78, 82 , 87 –93, 106,...
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