SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
More inequality results on option values
For American call option, will show
S ≥ C ≥ max(0, S − K, S − Kr−t − D+).
Absence-of-arbitrage proofs:
1. C ≥ 0: If not, buy option, keep until expiration. Get something
positive now, certainly nothing negative later.
2. C ≥ S − K: If not, buy option, exercise immediately.
3. C ≥ S − Kr−t − D+: If not, do the following:
Expiration
Now Div. date If S ∗ ≤ K If S ∗ > K
−S ∗
Sell share
S
−D
−S ∗
Buy D+ pf. −D+
≥D
0
0
Buy call
−C
0
0
S∗ − K
0
K
K
Lend
−Kr−t
≥0
≥0
≥0
0
A riskless arbitrage.
4. C ≤ S: If not, buy share, sell (i.e., write) call, receive C − S >
0. Get K > 0 if option is exercised, get S if not.
Important implication: American call option on shares which
certainly will not pay dividends before option’s expiration, should
not be exercised before expiration, since
C ≥ S − Kr−t > S − K.
Worth more “alive than dead.” When no dividends: Value of
American call equal to value of European.
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SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
Summing up some results
Both American and European call options on shares which may pay
dividends:
C ≥ S − Kr−t − D+.
Both American and European call options on shares which for sure
pay no dividends:
C ≥ S − Kr−t > S − K.
American call options on shares which may pay dividends:
C ≥ S − K.
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SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
American calls when dividends possible: More
• For each dividend payment: Two dates.
– One date for announcement, after which D known.
– One ex-dividend date, after which share does not give the
right to that dividend payment.
• Our interest is in ex-dividend dates.
• Owners of shares on morning of ex-div. date receive D.
• Assume there is a given number of ex-div. dates.
• Say, 2 ex-div. dates, td1, td2, before option’s expiration, t∗.
• Can show: C > S − K except just before td1, td2, t∗.
• Assume contrary, C ≤ S − K. Then riskless arbitrage:
• Buy call, exercise just before:
Now Just before next tdi or t∗
Buy call −C
S−K
Sell share S
−S
Lend
−K
Kr∆t
≥0
K(r∆t − 1)
• Riskless arbitrage, except if ∆t ≈ 0, just before.
Implication: When possible ex-dividend dates are known, American call options should never be exercised except perhaps just
before one of these or at expiration.
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SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
More inequality results on option values
• Ch. 4 results robust: Based on few assumptions.
• Inequalities place restrictions on option values.
• But even more useful to have exact formula.
• Exact formula in ch. 5 based on more assumptions.
• Thus exact formula less robust.
• Below: Three inequality results.
• American call options, same share, same expiration date.
• Show C(K) convex with slope ∈ [−1, 0].
• S does not appear, thus dividends in no problem.
1. K2 > K1 ⇒ C(K1) ≥ C(K2).
2. K2 > K1 ⇒ K2 − K1 ≥ C(K1) − C(K2).
3. K3 > K2 > K1 ⇒ C(K2) ≤
Will prove these.
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K3 −K2
K3 −K1 C(K1 )
2 −K1
+K
K3 −K1 C(K3 ).
SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
Proof of C(K) inequality, slope ≤ 0.
(As in all absence-of-arbitrage proofs:) Assume the contrary, C(K1) <
C(K2) or vice versa is observed. (Pick one of these alternatives,
here: C(K1) < C(K2). The proof for the opposite alternative is
analogous.) Will show this cannot be an equilibrium, because it
allows risk free arbitrage:
To exploit the arbitrage (as always, “buy cheaper, sell more expensive”): Buy call with exercise price K1, sell (i.e., write) the other
call, cash in C(K2) − C(K1) > 0 now.
If the owner of the second call exercises it before expiration, then
exercise the first call at the same time. Gives a net of K2 − K1 > 0
at that point in time.
If not, keep until the expiration date. The table shows the outcome
then:
If S ∗ ≤ K1 If K1 < S ∗ ≤ K2 If K2 ≤ S ∗
Having call 1
0
S ∗ − K1
S ∗ − K1
Having written call 2
0
0
K2 − S ∗
0
S ∗ − K1 > 0
K2 − K1 > 0
Whatever happens to S ∗, the net outcome is positive. This completes the proof of the first inequality.
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SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
Proof of C(K) inequality, |slope| ≤ 1.
Assume the contrary, C(K1) − C(K2) > K2 − K1 is observed. To
exploit arbitrage: Sell call with exercise price K1, buy call with K2,
lend K2 − K1 (i.e., buy bonds for that amount).
Expiration, t
Now
If S ∗ ≤ K1 If K1 ≤ S ∗ ≤ K2
If K2 ≤ S ∗
C(K1)
0
K1 − S ∗
K1 − S ∗
−C(K2)
0
0
S ∗ − K2
−(K2 − K1)
(K2 − K1)rt
(K2 − K1)rt
(K2 − K1)rt
C(K1) − C(K2) (K2 − K1)rt (K2 − K1)(rt − 1) (K2 − K1)(rt − 1)
>0
+K2 − S ∗ > 0
>0
−(K2 − K1) > 0
Table shows what will happen if options are kept until expiration.
But owner of option 1 may choose early exercise. (Contd. overleaf.)
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SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
Proof of second C(K) inequality, contd.
What if owner of option 1 chooses early exercise, t from now? Let
r = rt , and define values at the time of the exercise as C2 = C(K2)
and S = S. To ensure the arbitrage, take actions as follows:
1. If C2 > S − K2: Sell option 2, receive in total
K1−S +(K2−K1)r+C2 = (K2−K1)(r−1)+C2 +K2−S > 0.
2. If C2 ≤ S − K2: Exercise option 2, receive in total
K1 − S + (K2 − K1)r + S − K2 = (K2 − K1)(r − 1) > 0.
All positions closed, nothing left until expiration date.
This completes proof of the second inequality.
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SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
Proof of C(K) inequality, convexity.
Observe that if K3 > K2 > K1, then if we define
λ≡
K3 − K2
,
K3 − K1
then λ ∈ (0, 1), and
1−λ≡
K2 − K1
,
K3 − K1
and
K2 = λK1 + (1 − λ)K3.
Thus the inequalities we show are exactly the definition of a convex
function C(K):
C(K2) ≤ λC(K1) + (1 − λ)C(K3).
Proof: Assume instead C(K2) > λC(K1) + (1 − λ)C(K3) is observed. (In shorthand:) Write C(K2), buy λC(K1)+(1−λ)C(K3),
receive positive difference now. (Buying the linear combination
means buying λ units of call 1, 1 − λ units of call 3.) (Contd.
overleaf.)
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SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
If written call is exercised before expiration, exercise the calls 1 and
3, to obtain the required share at the cost λK1 + (1 − λ)K3 = K2,
which is exactly what is collected from the exercise of the written
call 2. This means a net receipt of zero. (It may be possible to
get something strictly positive instead, from selling call 1 in the
market instead of exercising it, if C(K1) > S − K1 at the time of
the exercise of call 2.)
If the written call is not exercised before expiration, the table shows
the result at the expiration date:
S ∗ ≤K1
K1 <S ∗ ≤K2
K2 <S ∗ ≤K3
K2 − S ∗
λ(S ∗ − K1)
K3 ≤S ∗
0
K2 − S ∗
λ(S ∗ − K1)
λ(S ∗ − K1)
0
0
(1 − λ)(S ∗ − K3)
λ(S ∗ − K1) λ(S ∗ − K1)
0
>0
+K2 − S ∗ > 0
In each case you end up with something ≥ 0. This completes the
proof of inequality no. 3.
−C2
λC1
(1 − λ)C3
0
0
0
0
Later we find exact formula for C(S, K, r, t, σ), which is differentiable. It satisfies ∂C/∂K ∈ [−1, 0] and ∂ 2C/∂K 2 < 0. But it
relies on restrictive assumptions. Have shown above that if someone comes up with another formula based on other assumptions,
then the formula’s partial derivatives must nevertheless have these
properties.
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SØK460/ECON460 Finance Theory, Diderik Lund, 16 October 2002
Implication of C(K) inequalities
If it is optimal to exercise one American call option with exercise price K2, then it is also optimal to exercise another which
is similar in all respects, except that it has K1 < K2.
Proof:
Assume contrary: Optimal to exercise option 2, but not option 1.
If optimal to exercise option 2, then C(K2) = S − K2.
If not optimal to exercise option 1, then C(K1) > S − K1.
But from this follows C(K1) − C(K2) > K2 − K1, which we have
shown to be wrong.
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