15
Barriers: Simple European Options
Barrier options are like simple options but with an extra feature which is triggered by the stock price passing through a barrier. The feature may be that the option ceases to exist (knock-out) or starts to exist (knock-in) or is changed into a different option. These are the archetypal exotics and constitute the majority of exotic options sold in the market (Reiner and Rubinstein, 1991a).
The general topic is a large one and we have chosen to spread it across two chapters (plus a fair chunk of the Appendix), rather than concentrating everything into one indigestible monolith.
If the reader is approaching the subject for the first time, he may feel daunted by the sizes of the formulas and by the number of large integrals; but he should make a point of stepping back to understand the underlying principles rather than drowning in the minutiae. There are in fact only a couple of integrals which are just applied over and over again.
This chapter lays out the basic principles and is a direct continuation of the analysis of the Black Scholes model, given in Chapter 5. The following chapter applies these principles to a number of more complex situations; it finishes with an explanation of how to apply trees to pricing barrier options numerically.
15 Barriers: Simple European Options
(ii) We will now apply these results to stock price movements. Consider a stock with a starting price ST in the presence of a barrier K. Closely following the Black Scholes analysis of Section 5.2, we write xT =ln(ST/S0) and note that xT is normally distributed with mean mT and varianceσ2T, wherem=r−q−12σ2. We use the notationb=ln(K/S0), so that A=exp(2mb/σ2)=(K/S0)2m/σ2.
In the remainder of this section, various knock-in options will be evaluated. These will involve a transformation from the variableST to either of the variableszT orzT, which were defined in the last subsection by
ST =S0emT+σ√T zT =S0emT+2b+σ√T zT
When setting up the integral for evaluating a call option, we integrate with respect toST from X to∞. On transforming to the variableszT orzT, the integrals will run fromZX to∞or fromZX to∞, where
ZX =ln(X/S0)−mT σ√
T ; ZX =ln(X/S0)−mT −2b σ√
T Analogous limits of integrationzK andzK are defined by
ZK = ln(K/S0)−mT σ√
T ; ZK =ln(K/S0)−mT −2b σ√
T
S0
K X
f00 F 0f0f0f
return
F F0
Figure 15.1 Down-and-in call;X<K (iii)Explicit Calculations:In this section we calculate two
specific examples in order to illustrate how the formu- las for prices are obtained. It would be repetitive and boring to do this for every possible knock-in option.
However, generalized results for all options are given later in the chapter.
Example (a): Down-and-in Call;X <K. The option is explained schematically in Figure 15.1. The proba-
bility density functionFcrossersis different on each side of the barrier as shown.
The price of the option is written Cd−i(X <K)=e−r T
+∞
0
(ST −X)+FcrossersdST =e−r T +∞
X
(ST −X)FcrossersdST
=e−r T K
X
(ST −X)F0dST +e−r T ∞
K
(ST −X)FreturndST
The first integral on the right-hand side can be split into two manageable parts as follows:
e−r T K
X
(ST−X)F0dST =e−r T ∞
X
(ST−X)F0dST−e−r T ∞
K
(ST−X)F0dST
=[BSC]−[GC]
The first integral here is just the Black Scholes formula for a call with strike X. The second integral is the formula for a gap option which was described in Section 11.4.
15.1 SINGLE BARRIER CALLS AND PUTS
To evaluate the second integral in the expression forCd−i(X <K), we make the transfor- mation to the standard normal variatezTdescribed in subsection (ii) and use the integral result of equations (A1.7):
[JC]=e−r T ∞
K
(ST−X)FreturndST =e−r T ∞
zK
(S0emT+2b+σ√T zT −X)An(zT) dzT
= Ae−r T
S0e2b+(m+12σ2)TN[σ√
T −ZK]−XN[−ZK] The value of this option can then be written
Cd−i(X <K)=[BSC]−[GC]+[JC]
S0 K X
return
F
F0
Figure 15.2 Up-and-in put;K<X Example (b): Up-and-in Put;K <X. The reasoning
in this example is precisely analogous to that of the last example (see Figure 15.2). The reader is asked to pay particular attention to the signs of the various terms:
Pu−i(K <X)=e−r T +∞
0
(X−ST)+FcrossersdST
=e−r T K
0
(X−ST)FreturndST
+e−r T X
K
(X−ST)F0dST
The second integral on the right may be written e−r T
X K
(X−ST)F0dST =e−r T X
0
(X−ST)F0dST −e−r T K
0
(X−ST)F0dST
=[BSP]−[GP]
As in the previous example, the first term is the Black Scholes formula (for a put option this time) while the second term is again a gap option.
The first integral is solved by making the same transformation as in the last example and using the integral result of equations (A1.7):
[JP]=e−r T K
0
(X−ST)FreturndST =e−r T ZK
−∞ (X−S0emT+2b+σ√T zT)An(zT) dzT
= Ae−r T
XN[ZK]−S0e2b+(m+12σ2)TN[ZK−σ√ T]
The value of the option is written
Pu−i(K <X)=[BSP]−[GP]+[JP]
(iv)Generalizing the Results:If the reader compares the results of the last two examples he will be struck by how similar they are. The essential differences are:
r
The first example is for a call while the second is for a put. Each of the terms reflects this difference, which can be accommodated by the use of the parameterφ(= +1 for a call179
15 Barriers: Simple European Options
and−1 for a put); this was explained in Section 5.2(iv) where we wrote a general Black Scholes formula which could be used for either a put or a call.
r
If we make use of the parameterφ, we can almost write a general expression which could be applied to either of the last two examples. There is, however, still a difference in the term [J]: the signs of the arguments of the cumulative normal functions are reversed. This is essentially due to the fact that the limits of integration wereZKto+∞in the first example and−∞to ZK in the second; the difference comes because the stock price had tofallto reach the barrier in the first example butrisein the second.Therefore a factorψ(= +1 for rise-to-barrier and−1 for fall-to-barrier) multiplying the argu- ments of the cumulative normal function of [J] would allow us to write a general expression which prices eitherCd−i(X<K) orPu−i(K <X).