SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
Black and Scholes: Option pricing in continuous time
• Two unrealistic features of binomial model:
– In reality shares can change their values to any positive
number, not only two possible numbers over a specified time
period.
– In reality trade in shares and other securities can happen
all the time, not at given time intervals.
• Will amend both problems simultaneously.
• Define h ≡ t/n as the interval length.
• t measures calendar time until expiration. Fixed.
• n is number of periods (intervals) we divide t into.
• Let n → ∞, h → 0.
• Consider what happens to share and option values.
• S ∗ becomes product of many independent variables, e.g.,
S ∗ = d · d · u · d · u · u · d · u · u · S0.
• Better to work with ln(S ∗/S0), rewritten


S∗ 

= j ln(u) + (n − j) ln(d).
ln
S0
where j is a binomial random variable.
• Central limit theorem: When n → ∞, the expression ln(S ∗/S0)
approaches a normally distributed random variable.
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
Normal and lognormal distributions
When ln(S ∗/S0) = ln(S ∗) − ln(S0) is normally distributed:
• ln(S ∗) also normally distr., since ln(S0) is a constant.
• S ∗/S0 and S ∗ are lognormally distributed.
• ln(S ∗) can take any (real) value, positive or negative.
• S ∗ can take any positive value.
• Graphs show normal and lognormal distributions:
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
The transition to the continuous-time model
• Any positive S ∗ value will have positive probability.
• But over a short period of time (e.g., one week), large (and very
small) S ∗/S0 will have very low probability.
• Black and Scholes wrote model in continuous time directly.
• To exploit arbitrage: Must adjust replicating pf. all the time.
• Relies on (literally) no transaction costs.
• Sufficient that someone have no transaction costs: They will
use arbitrage opportunity if available.
• Small fixed costs of “each transaction” destroys model.
• New unrealistic feature introduced (?)
• Consistent, consequence of assuming no transaction costs.
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
Mathematics of transition to continuous time
• Will need to make u, d, r, q functions of h (or n).
• If not: Model would “explode” when n → ∞.
• Specifically, rn → ∞.
• Also, if u > d, then E[ln(S ∗/S0)] → ∞.
• Instead, start with some reasonable values for rt,
µt ≡ E[ln(S ∗/S0)],
and
σ 2t ≡ var[ln(S ∗/S0)].
• Then choose u, d, r̂, q for binomial model as functions of h
such that as h → 0, the binomial model’s E[ln(S ∗/S0)] and
var[ln(S ∗/S0)] approaches the “starting” values (above).
• Easiest:
r̂n = rt ⇒ r̂ = rt/n = rh.
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
u, d, q as functions of h
Remember


∗
S
u
ln   = j ln
+ n ln(d),
S0
d
with j binomial, the number of u outcomes in n independent draws,
each with probability q.
Let µ̂, σ̂ belong in binomial model:
µ̂n ≡ E[ln(S ∗/S0)] = [q ln(u/d) + ln(d)]n,
σ̂ 2n ≡ var[ln(S ∗/S0)] = q(1 − q) [ln(u/d)]2 n.
• Want h → 0 to imply both µ̂n → µt and σ̂ 2n → σ 2t.
• Free to choose u, d, q to obtain two goals.
• Many ways to achieve this, one degree of freedom.
• Choose that one which makes S a continuous function of time.
• No jumps in time path.
• Necessary in order to be able to adjust replicating portfolio.
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
u, d, q as functions of h, contd.
√
σ h
Let u = e
work:
√
−σ h
,d=e
,q=
1
2
+
1µ
2σ
√
h. Will show these choices
√
√
1 1 µ√ 


+
h · 2σ h − σ h n
µ̂n =
2 2σ





t
t
t
+µ −σ 
 n = µt,
n
n
n



√
√
µ
1
µ
1
1
σ̂ 2n = 
h  −
h · 4 · σ 2hn
2σ
2 2σ


1 µ2 t 
1
=  − 2  · 4σ 2t → σ 2t when n → ∞.
4 4σ n

=
σ
Observe that our choices make u and d independent of the value of
µ. Thus the option value, which depends on u and d, but not of q,
will be independent of µ.
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
Example of convergence, h → 0
Let t = 2 and assume that we want convergence to
2µ = E[ln(S ∗/S0)] = 0.08926, 2σ 2 = var[ln(S ∗/S0)] = 0.09959
so that µ = 0.04463 and σ = 0.2231.
Choose
1 1 µ√
1
= , q= +
h.
u=e , d=e
u
2 2σ
Will show what the numbers look like when n = 2 and when n = 4.
√
σ h
√
−σ h
For n = 2 (i.e., h = t/n = 1) we find
1 1µ
1
= 0.8, q = +
= 0.6,
u
2 2σ
which yields, in the binomial model,
u = eσ = 1.25, d =






S ∗ 
S ∗ 
2




µ̂n = E ln
= 0.08926, σ̂ n = var ln
= 0.09560.
S0
S0
For n = 4 (i.e., h = t/n = 1/2) we find
1 1 µ√
1
= 1.1709, d = = 0.8540, q = +
0.5 = 0.5707,
u=e
u
2 2σ
which yields, in the binomial model,
√
σ 0.5






∗
∗
S
S
2
µ̂n = E ln   = 0.08926, σ̂ n = var ln   = 0.09759.
S0
S0
Observe that as h → 0, we have u → 1, d → 1, q → 0.5, and
σ̂ 2n → σ 2t.
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
Convergence of option value to Black and Scholes
• Have constructed meaningful convergence of share price model.
• Now: What happens to option values?
• Consider “binomial n-period option value formula.”
• Show it converges to Black and Scholes formula.
Binomial formula (writing S instead of S0):
C = SΦ(a; n, p) − Kr−nΦ(a; n, p),
where p ≡ (r − d)/(u − r), p ≡ pu/r, and






n
n

ln(K/Sdn ) 
 ln(K/Sd ) ln(K/Sd )


.
a ≡ min 
,
+
1
i
∈
N;
i
≥
∈


ln(u/d) 
ln(u/d)
ln(u/d)
Black and Scholes:
√
C = SN (x) − Kr−tN (x − σ t),
where N () is the standard normal cumulative distribution function,
and
ln(S/Kr−t) 1 √
√
+ σ t.
x≡
2
σ t
As h → 0, the two Φ() expressions will converge towards the N ()
expressions, respectively. Will only show this for the second expression.
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
Convergence of Φ() expressions to N () expressions
• Central limit theorem implies: Binomial variable → normal.
• But: Need exact expression, argument in N () function.
Observe that
Φ(a; n, p) = Pr(j ≥ a),
when the probability is p in each of n draws. Then


1 − Φ(a; n, p) = Pr(j ≤ a − 1) = Pr 

j − np
np(1 − p)

≤
a − 1 − np 

np(1 − p)

,
rewritten so that the left hand (stochastic) side of the inequality
has expectation 0 and variance 1.
Thus, as h → 0, the left hand side approaches a standard normal random variable. Remains to show that the right hand side
approaches the expression in Black and Scholes’ formula.
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SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
Convergence of Φ() expressions, contd.
In order to rewrite


Pr 

j − np
np(1 − p)

≤
a − 1 − np 

np(1 − p)

,
define the variables µ̂p, σ̂p by
µ̂pn ≡ Ep[ln(S ∗/S)] = [p ln(u/d) + ln(d)]n,
σ̂p2n ≡ varp[ln(S ∗/S)] = p(1 − p) [ln(u/d)]2 n.
These are similar to µ̂, σ̂, except that p is substituted for q. Now
observe that
ln(K/Sdn )
+ 1 − ε,
a=
ln(u/d)
for some ε ∈ (0, 1]. The right hand side of the inequality becomes
a − 1 − np
np(1 − p)
=
ln(K/Sdn ) − ε ln(u/d) − np ln(u/d)
ln(u/d) np(1 − p)
ln(K/S) − µ̂pn − ε ln(u/d)
√
.
σ̂p n
10
=
SØK460/ECON460 Finance Theory, Diderik Lund, 23 October 2002
Convergence of Φ() expressions, contd.
A separate handout shows that as h → 0, we get
p→
Using
1 ln(r) − σ /2 t
+
.
2
2σ
n
2
√
√
µ̂pn = np ln(u/d) + n ln(d) = 2pσ nt − σ nt,
we then get the convergence


ln(r) − σ /2
1

µ̂pn → 2  + 
2
2σ
2




√
t  √
σ
nt−σ
nt = (ln(r)−σ 2/2)t.

n
This gives the limit for the right-hand side of our inequality:
ln(K/S) − (ln(r) − σ 2/2)t
ln(K/S) − µ̂pn − ε ln(u/d)
√
√
,
→
σ̂p n
σ t
so that


−t
1 √
 ln(Kr /S)
√
+ σ t ,
1 − Φ(a; n, p) → N 
2
σ t
which implies


−t
1 √
 ln(S/Kr )
√

− σ t ,
Φ(a; n, p) → N
2
σ t
since 1 − N (z) = N (−z) for all z.
The convergence of the first Φ expression is proved in a similar way.
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