Exotic Options - Computational Methods for Quantitative Finance - page 82

6.4 Swing Options

Although swing options appear in various forms, many of them are mathematically
of the same type, namely optimal multiple stopping time problems. For example, in
energy markets, the delivery of a commodity is limited by capacity constraints usu-
ally resulting in a pre-specified refracting time for contracts with several exercise
rights. It can be agreed that the refraction period δ which is greater than the minimal
delivery time is constant. This separation of two exercise times not only represents
an important contract constraint, but also prevents the case of single optimal stop-
ping time problems where all rights are exercised at once.
Let us denote by T t,T the set of all stopping times for S t with values in (t, T )

and by T t, ∞ the set of all stopping times with values greater or equal than t .For

stopping time problems with p exercise rights, constant refracting period δ and

maturity T the following sets are defined.

∈ N

Definition 6.4.1 The set of admissible stopping time vectors with length p ∈ N and

refracting time δ> 0 is defined by

(p)

τ (p) T := { = (τ 1 ,...,τ p ) | τ i ∈ T t, ∞ with τ 1 ≤ T a.s. and τ i + 1 − τ i ≥ δ

t

for i = 1 ,...,p − 1 } .

Consider a continuous time-dependent payoff function g : R + × R + → R +

where we assume that g(t, 0for t>T . The finite horizon multiple stopping

time problem with maturity T and p

· ) =

∈ N exercise rights is defined as

s .

p

V (p) (t, s) e − r(τ i − t) g(τ i ,S τ i )

:= E |

S t =

sup
τ (p)

(6.9)

(p)

∈ T i = 1

t

Itisshownin[32] that the multiple stopping time problem can be reduced to a
cascade of single stopping time problems, in particular we have

e − r(τ − t) g (p) (τ, S τ )

s ,

V (p) (t, s) =

τ ∈ T t,T E

sup | S t =

with

g(t,s)

E V (p − 1 ) (t s e − rδ

+ + δ,S t + δ ) | S t = if t ≤ T − δ,

g (p) (t, s) :=

g(t,s) if t ∈ (T − δ,T ] ,

V ( 0 ) (t, s)

:=

0 .