Digital Options
We investigate here options of the type “cash or nothing,” which we call digital options. We did not attempt, in the previous chapter, to be so general as to cover digital options every time a general treatment was possible. Yet this chapter will make man
- PDF / 443,003 Bytes
- 31 Pages / 439.36 x 666.15 pts Page_size
- 39 Downloads / 210 Views
Digital Options
9.1 Introduction and Main Results 9.1.1 Digital Options 9.1.1.1 Definition A “cash-or-nothing,” or digital, option is a contract by which a seller agrees to pay a buyer a fixed amount D if, at a given exercise time T , the market price S(T ) of a share of the underlying stock is higher – resp. lower – than an agreed strike K , leading to a digital call – resp. put. It should be noted that one could, without loss of generality, let D = 1 and, for an option with a different D, assume that D options have been traded. Yet, we retain D in our development to emphasize that it is not dimensionless but a currency value. As a consequence, the payment function M(s) can be expressed with the help of the Heaviside function 0 if s < 0, ϒ (s) = 1 if s ≥ 0, as
M(s) =
Dϒ (s − K ) for a digital call, Dϒ (K − s) for a digital put.
(9.1)
In our definition, we have ruled out payments “in kind.” Such “one-stock-ornothing” options exist. We will not consider them here. And, as in the vanilla case, we will concentrate on call options. The case of put options is very similar and symmetrical.
P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 9, © Springer Science+Business Media New York 2013
167
168
9 Digital Options
9.1.1.2 Terminal Payment At exercise time, the seller must close out his position and pay M(S(T )) to the buyer. Hence his total terminal cost is N(u(T ), v(T )) with M(u) − c− v if v > 0, N(u, v) = (9.2) M(u) − c+ v if v < 0. We can write this as in (8.4) ˇ u) − v, N(u, v) = w(T, ˇ u) + cε v(T,
(9.3)
with v(T, ˇ u) = 0,
w(T, ˇ u) = M(u),
(9.4)
and M(u) still given by (9.1). Proposition 9.1. The function v → N(u, v) is convex for all u ∈ R+ . Note, however, that N is no longer convex in u, or even continuous.
9.1.2 Main Results These are the main results for digital options. They are on several counts less complete and more complicated (less elegant?) than for vanilla options. To avoid complicated statements, we limit ourselves here to call options. Put options are very similar, the symmetry with call options leading to a reversal of the sign of v.
9.1.2.1 DQVI We consider the same differential quasivariational inequality (DQVI) (8.7) as previously: ∀(t, u, v) ∈ [0, T ) × R+ × R, ∂W ∂W ∂W ∂W ε ∂W + − −τ u+ −1 v , − +C , +C max − = 0, ∂t ∂u ∂v ∂v ∂v ∀(u, v) ∈ R+ × R,
W (T, u, v) = N(u, v),
but where the function N is now given equivalently by (9.2) or its representation (9.3) and (9.4). All of Sect. 8.1.4 applies to the current problem. In particular, the same proof via the Joshua transformation yields the following theorem.
9.1 Introduction and Main Results
169
Theorem 9.2. The Value function W is a (discontinuous) viscosity solution (VS) (see [22]) of the DQVI. Its uniqueness is not proved, however, and we will need a conjecture. Conjecture 9.1. The discontinuous VS (in the sense of Barles [22]) of the DQVI is unique. For digital options, we can only stat
Data Loading...
