Unformatted text preview: parameters: no dividend (1)
We choose the tree parameters p, u, and d to fit the
We
and
real/empirical mean & volatility of the stock price
changes:
changes:
Recall our original setting (no dividend, no
no
adjustment term):
δ S /S = r δ t + σ ε √δ t
and Sδ t = S exp(η), where η ~ N(rδ t, σ√δ t)
),
σ√
The average stock price (i.e., p S u + (1– p ) S d)
The
should equal equilibrium mean S erδt (why S erδt? – the
the
return of holding the stock should equal the riskfree
rate)
rate)
The volatility of stock price (i.e., the square root of
The
p S 2 u 2 + (1– p ) S 2 d 2 – [p S u + (1– p ) S d ]2) should
should
equal the equilibrium volatility S eσ√δ t
1541 Equation of tree parameters: no dividend (2)
Mean: S erδt = p S u + (1– p ) S d
Mean:
= > erδt = p u + (1– p ) d
t
Variance: S 2 e2 (σ √δ t)) = p S 2 u 2 + (1– p ) S 2 d 2 – [p S
u + (1– p ) S d ]2
Two equations with three unknown (p, u, and d )
and
So, to solve the problem, we let d =1/u and
So,
tedious derivations lead to the following solution
for parameters:
for
1. u = eσ√δt (i.e., d = eσ√δt , the intuition is simple: the
stock price will either go up or down by 1 st. dev.)
stock
2. p = (erδt – d )/(u – d)
2.
)/(
1542 Equation of tree parameters: with dividend (1)
We choose the tree parameters p, u, and d to fit the
We
and
real/empirical mean & volatility of the stock price
changes:
changes:
Recall our original setting (with dividend rate q, no
with
no
adjustment term):
δ S /S = (r − q )δ t + σ ε √δ t
and Sδ t = S exp(η), where η ~ N((r − q ) δ t, σ√δ t)
),
σ√
The average stock price (i.e., p S u + (1– p ) S d)
The
should equal equilibrium mean S e(rq)δt (why S e(rq)δt? –
the return of holding eqδt share of the stock (that will
grow to 1 share) should equal the riskfree rate)
grow
The volatility of stock price (i.e., the square root of
The
p S 2 u 2 + (1– p ) S 2 d 2 – [p S u + (1– p ) S d ]2) should
should
equal the equilibrium volatility S eσ√δ t
1543 Equation of tree parameters: with dividend (2)
Mean: S e(rq)δt = p S u + (1– p ) S d
Mean:
= > e(rq)δt = p u + (1– p ) d
t
Variance: S 2 e2 (σ √δ t)) = p S 2 u 2 + (1– p ) S 2 d 2 – [p S
u + (1– p ) S d ]2
Two equations with three unknown (p, u, and d )
and
So, to solve the problem, we let d =1/u and
So,
tedious derivations lead to the following solution
for parameters:
for
u = eσ√δt (i.e., d = eσ√δt , the intuition is simple: the
stock price will either go up or down by 1 st. dev.)
stock
p = (e(rq) δt – d )/(u – d)
)/(
1544...
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This document was uploaded on 03/03/2014.
 Spring '09

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