Math Finance summary
Mich ’10
7
H.6
Asset price behaviour
The cornerstone of a great many models in mathematical finance
is the assumption that asset prices
S
(
t
),
t
≥
0 (at least for nondividend paying stocks) are
distributed as a
geometric Brownian motion
with
drift
μ
and
volatility
σ
. Specifically,
log
S
(
u
+
t
)
S
(
u
)
∼
N
(
(
μ
−
σ
2
/
2)
t, σ
2
t
)
for all
u
≥
0,
t >
0 and additionally
S
(
u
+
t
)
/S
(
u
) is independent of
S
(
v
),
v
≤
u
.
There is considerable empirical evidence that this model is often reasonable (Taylor’s
theorem for the log function links this model to our earlier observation about relative price
changes when they are small). There is also much research into improving it e.g. by making
the volatility time dependent.
Tuning the binary tree model
It is possible to choose
u
,
d
and
p
to make the binary
tree model emulate the geometric BM model of stock prices. There are various (essentially
equivalent) ways to do this. Divide time period
T
into a large number
n
timesteps of length
Δ =
T/n
(with interest rate per time step equal to
r
Δ) and set
u
= exp(
σ
√
Δ + (
μ
−
σ
2
/
2)Δ)
,
d
= exp(
−
σ
√
Δ + (
μ
−
σ
2
/
2)Δ)
,
p
=
1
2
+
o
(Δ)
where
μ
is the drift and
σ
2
the volatility of the geometric BM stock price model.
The parameters can be made risk neutral by setting (1+
r
Δ
−
d
)
/
(
u
−
d
) =
p
= 1
/
2+
o
(Δ)
which leads in the limit to a geometric BM model with volatility
σ
but with
risk neutral drift
H.12
r
in place of
μ
. Assuming the limit can be passed through the expectation (correct but we
do not prove it in this course) this leads to the call option price formula
C
(
K, T, σ, S, r
) =
e
−
rT
E
r
(
[
Se
W
−
K
]
+
)
where
S
(0) =
S
and
E
r
indicates that we use
W
∼
N
((
r
−
σ
2
/
2)
T, σ
2
T
).
3
The BlackScholes formula
The expectation in the formula for the European call price can be evaluated in the form
H.8.5
C
=
S
Φ(
d
1
)
−
Ke
−
rT
Φ(
d
2
)
(3
.
1)
where
d
1
=
(
r
+
σ
2
/
2)
T
+ log(
S/K
)
σ
√
T
,
d
2
=
d
1
−
σ
√
T
=
(
r
−
σ
2
/
2)
T
+ log(
S/K
)
σ
√
T
and Φ denotes the standard Normal cdf. This is established by standard integration methods.
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 Spring '10
 DrI.M.MacPhee
 Math, Strike price, Option style

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