Installment Options and Static Hedging
Mark H. A. Davis
Department of Mathematics, Imperial College,
London SW7 2BZ, England
Walter Schachermayer and Robert G. Tompkins
Financial and Actuarial Mathematics Group,
Technische Universit¨at Wien
Wiedner Hauptstrasse 8, A-1040 Wien, Austria
January 8, 2001
Abstract
An installment option is a European option in which the premium,
instead of being paid up-front, is paid in a series of installments. If all
installments are paid the holder receives the exercise value, but the holder
has the right to terminate payments on any payment date, in which case
the option lapses with no further payments on either side. We discuss
pricing and risk management for these options, in particular the use of
static hedges to obtain both no-arbitrage pricing bounds and very effective
hedging strategies with almost no vega risk.
1 Introduction
In a conventional option contract the buyer pays the premium up front and
acquires the right, but not the obligation, to exercise the option at a fixed time
T in the future (for European-style exercise) or at any time at or before T
(for American-style exercise). In this paper we consider an alternative form
of contract in which the buyer pays a smaller up-front premium and then a
sequence of “installments”, i.e. further premium payments at equally spaced
time intervals before the maturity time T . If all installments are paid the buyer
can exercise the option, European-style, at time T . Crucially, though, the buyer
has the right to ‘walk away’: if any installment is not paid then the contract
terminates with no further payments on either side. This provides useful extra
optionality to the buyer while, as we shall see, the seller can hedge the option
using simple static hedges that largely eliminate model risk.
The case of two installments is equivalent to a compound option, previously
considered by Geske [7] and Selby and Hodges [8]. Let C(t, T, S, K) denote the
Black-Scholes value at time t of a European call option with strike K maturing
at time T when the current underlying price is S (all other model parameters are
constant). Installments p
0
, p
1
are paid at times t
0
, t
1
and final exercise is at time
T > t
1
. At time t
1
the holder can either pay the premium p
1
and continue to hold
the option, or walk away, so the value at t
1
is max (C(t
1
, T, S(t
1
), K) − p
1
, 0).
The holder will pay the premium p
1
if this is less than the value of the call
1
option. The value of this contract at t
0
is thus the value of a call on C with
‘strike’ p
1
.
Another way of looking at it, that will be useful later, is this: the holder
buys the underlying call at time t
0
for a premium p = p
0
+ e
−r(t
1
−t
0
)
p
1
(the
NPV of the two premium payments where r denotes the riskless interest rate)
but has the right to sell the option at time t
1
for price p
1
. The compound call is
thus equivalent to the underlying call option plus a put on the call with exercise
at time t
1
and strike price p
1
. The value p is thus greater than the Black-Scholes
value C(t
0
, T, S(t
0
), K), the difference being the value of the put on the call.
A similar analysis applies to installment options with premium payments
p
0
, p
1
, . . . , p
k
at times t
0
, . . . , t
k
< T . The NPV of the premium payments is
p =
P
k
i=0
p
i
e
−r(t
i
−t
0
)
and the installment option is equivalent to paying p at
time t
0
and acquiring the underlying option plus the right to sell it at time
t
j
, 1 ≤ j ≤ k at a price q
j
=
P
k
i=j
p
i
e
−r(t
i
−t
j
)
(all subsequent premiums are
‘refunded’ when the right to sell is exercised). The installment option is thus
equivalent to the underlying option plus a Bermuda put on the underlying option
with time-varying strike q
i
.
In the next section we discuss pricing in the Black-Scholes framework. As
for American options a finite-difference algorithm must be used. In section
3, simply-stated ‘no-arbitrage’ bounds on the price are derived valid for very
general price process models. As will be seen, these depend on comparison with
other options and suggest possible classes of hedging strategies. In section 4
we introduce and analyse static hedges for installment options, concentrating
on a specific example to illustrate our point. Concluding remarks are given in
Section 5.
This paper is largely a summary of our companion paper [5], which also
contains a discussion of ‘continuous-installment options’, not considered here.
2 Pricing in the Black-Scholes framework
Consider an asset whose price process S
t
is the conventional log-normal diffusion
dS
t
= rS
t
dt + σS
t
dw
t
, (1)
where r is the riskless rate and w
t
a standard Brownian motion; thus (1) is the
price process in the risk-neutral measure. We consider a European call option
on S
t
with exercise time T and payoff
[S
T
− K]
+
= max(S
T
− K, 0). (2)
The Black-Scholes value of this option at time 0 is of course
p
BS
= Ee
−rT
[S
T
− K]
+
. (3)
p
BS
is the unique arbitrage-free price for the option, to be paid at time 0. As
an illustrative example we will take T = 1 year, r = 0, K = 100, S
0
= 100 and
σ = 25.132%, giving p
BS
= 10.00.
In an installment option we choose times 0 = t
0
< t
1
< · · · < t
n
= T
(generally t
i
= iT/n to a close approximation). We pay an upfront premium p
0
at t
0
and pay an ‘installment’ of p
1
at each of the n − 1 times t
1
, . . . , t
n−1
. We
also have the right to walk away from the deal at each time t
i
: if the installment
2
due at t
i
is not paid then the deal is terminated with no further payments on
either side. The pricing problem is to determine what is the no arbitrage value of
the premium p
1
for a given value of p
0
. The present value of premium payments
– assuming they are all paid – is
p
0
+ p
1
n−1
X
i=1
e
−rt
i
, (4)
and this must exceed the Black-Scholes value in view of the extra optionality.
Computing the exact value is straightforward in principle. Let V
i
(S) denote
the net value of the deal to the holder at time t
i
when the asset price is S
t
i
= S.
In particular
V
n
(S) = [S − K]
+
. (5)
At time t
i
we can either walk away, or pay p
1
to continue, the continuation
value being
E
t
i
,S(t
i
)
[e
−r(t
i+1
−t
i
)
V
i+1
(S
t
i+1
)]. (6)
Thus
V
i
(S) = max(0, E
t
i
,S
[e
−r(t
i+1
−t
i
)
V
i+1
(S
t
i+1
)] − p
1
). (7)
In particular, V
n−1
is just the maximum of 0 and BS − p
1
, where BS denotes
the Black-Scholes value of the option at time t
n−1
. The unique arbitrage free
value of the initial premium is then
p
0
= V
+
0
(S
0
, p
1
) := E
t
0
,S
0
h
e
−r(t
1
−t
0
)
V
1
(S
t
1
)
i
.
For fixed p
1
, V
+
0
(S
0
, p
1
) is easily evaluated using a binomial or trinomial
tree and this determines the up-front payment p
0
. If we want to go the other
way round, pre-specifying p
0
, then we need a simple one-dimensional search to
solve the equation p
0
= V
+
0
(S
0
, p
1
) for p
1
. A similar search solves the equation
bp = V
+
0
(S
0
, bp) giving the installment value bp when all installments, including
the initial one, are the same.
Figure 1 shows the price bp at time 0 for our standard example with 4 equal
installments. For comparison, one quarter of the Black-Scholes value is also
shown. At S
0
= 100, bp = 3.284, which is 31% greater than one quarter of the
Black-Scholes value. Figure 2 shows the value at time t
1
; when S(t
1
) ≤ 98.28 it
is optimal not to continue and the option has value zero.
3 No-arbitrage bounds derived from static hedges
The pricing model of the previous section makes the standard Black-Scholes as-
sumptions: log-normal price process, constant volatility. By considering static
super-replicating portfolios, however, we can determine easily computable bounds
on the price valid for essentially arbitrary price models. We need only assume
that for any s ∈ [t
0
, T ] there is a liquid market for European calls with maturities
t ∈ [s, T ], the price being given by
C(s, t, K) = E
Q
h
e
−r(t−s)
(S
t
− K)
+
F
s
i
(9)
3
0
2
4
6
8
10
12
60 70 80 90 100 110 120 130 140
Price S(t0)
O p t i o n p r e m i u m
Installment
option
premium
1/4 X BS premium
Figure 1: Fair installment value and Black-Scholes value
0
5
10
15
20
25
30
35
60 70 80 90 100 110 120 130 140
Price S(t
1
)
O p t i o n v a l u e
Figure 2: Value of installment option at time t
1
as function of price S(t
1
)
where Q is a martingale measure for the process S and F
s
denotes the infor-
mation available at time s. By put-call parity this also determines the value of
put options P (s, t, K). We know today’s prices C(t
0
, t, K) and that is all we
know about the process S and the measure Q. We ignore interest rate volatility,
assuming for notational convenience that the riskless rate is a constant, r, in
continuously compounding terms. We also assume that no dividends are paid.
Let us first consider a 2-installment, i.e. compound, option, with premiums
p
0
, p
1
paid at t
0
, t
1
for an underlying option with strike K maturing at T = t
2
.
The subsequent result provides no-arbitrage bounds on the prices p
0
, p
1
which are independent of the special choice of the model S and the equivalent
martingale measure Q.
Proposition 3.1 For the compound option described above, there is an arbi-
trage opportunity if p
0
, p
1
do not satisfy the inequalities
C
t
0
, T, K + e
r(T −t
1
)
p
1
> p
0
> C(t
0
, T, K)−e
−r(t
1
−t
0
)
p
1
+P (t
0
, t
1
, p
1
). (10)
4
Proof. Denote K
0
= K + e
r(T −t
1
)
p
1
and suppose we sell the compound option
with agreed premium payments p
0
, p
1
such that p
0
≥ C(t
0
, T, K
0
). We then
buy the call with strike K
0
and place x = p
0
− C(t
0
, T, K
0
) ≥ 0 in the riskless
account. If the second installment is not paid, the value of our position at time
t
1
is xe
r(t
1
−t
0
)
+ C(t
1
, T, K
0
) ≥ 0, whereas if the second installment is paid
we add it to the cash account, and the value at time T is then xe
r(T −t
0
)
+
p
1
e
r(T −t
1
)
+ C(T, T, K
0
) − C(T, T, K) ≥ 0. This is an arbitrage opportunity,
giving the left-hand inequality in (10).
Now suppose the compound option is available at p
0
, p
1
satisfying
p
0
+ e
−r(t
1
−t
0
)
p
1
≤ C(t
0
, T, K) + P (t
0
, t
1
, p
1
). (11)
We buy it, i.e. pay p
0
, and sell the two options on the right (call them
b
C,
b
P ),
so our cash position is
b
C +
b
P − p
0
≥ e
−r(t
1
−t
0
)
p
1
. At time t
1
the cash position
is therefore x ≥ p
1
, and we have the right to pay p
1
and receive the call option.
We exercise this right if C(t
1
, T, K) ≥ p
1
. Then our cash position is x − p
1
≥ 0,
b
C is covered and
b
P will not be exercised because p
1
< C(t
1
, T, K) ≤ S(t
1
) (the
call option value is never greater than the value of the underlying asset). On the
other hand, max(p
1
−C(t
1
, T, K), 0) ≥ max(p
1
−S(t
1
), 0), so if p
1
> C(t
1
, T, K)
we do not pay the second installment and still have enough cash to cover
b
C and
b
P . Thus there is an arbitrage opportunity when the right-hand inequality in
(10) is violated.
Of course, for practical purpose P (t
0
, t
1
, p
1
) will be a negligible quantity
(and there will be no liquid market as typically p
1
S
0
) but for obtaining
theoretically sharp bounds one must not forget this term.
In fact, the above inequalities are sharp: it is not hard to construct exam-
ples of arbitrage-free markets such that the differences in the left (resp. right)
inequality in (10) become arbitrarily small.
Finally let us interpret the right hand side of inequality (10) by using the
interpretation of the compound option given in the introductory section: the net
present value p
0
+ e
−r(t
1
−t
0
)
p
1
of the payment for the compound must equal —
by no-arbitrage — the price C(t
0
, T, K) of the corresponding European option
plus a put option to sell this call option at time t
1
at price p
1
. Denoting the
latter security by Put(Call) we obtain the no-arbitrage equality.
p
0
= C(t
0
, T, K) − e
−r(t
1
−t
0
)
p
1
+ Put(Call). (12)
In the proof of proposition 3.1 we have (trivially) estimated this Put on the
Call from below by the corresponding Put P (t
0
, t
1
, p
1
) on the underlying S. We
now see that the difference in the right hand inequality of (10) is precisely equal
to the difference Put(Call) − P (t
0
, t
1
, p
1
) in this estimation.
Similar arguments apply for n installments, where holding the installment
option is equivalent to holding the underlying option plus the right to sell this
option at any installment date at a price equal to the NPV of all future in-
stallments. The value of the Bermuda option on the option is greater than the
equivalent option on the stock. This gives us the following result.
Proposition 3.2 For the n-installment call option with premium payment p
0
at time t
0
and p
1
at times t
1
, . . . , t
n−1
there is an arbitrage opportunity if p
0
,
5