Barriers - Simple European Options

First, however, we take note of a simple but powerful relationship: Knock-in Option + Knock-out Option = European Option This result is obvious if we consider a portfolio consisting of t

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

(i) The reader should refer to Appendix A.8 which lays out the framework for this chapter The key

result in this context is given by equation (A8.4) Imagine a Brownian particle starting at x0= 0;

the probability distribution function of just those particles that have crossed a barrier at b is

Fcrossers(x T , T ) =



Freturn(x T , T ) for x T on the same side of the barrier as x0

F0(x T , T ) for x T on the other side of the barrier

F0(x T , T ) is the normal distribution function for a particle starting at x0= 0 and with unrestrained movement, i.e

F0(x T , T ) dx T = 1

σ√2πT exp



−1 2

x T − mT

σ√T

2

dx T =√1

2πe

− 1z2

dz T = n(z T ) dz T

Freturn(x T , T )is the distribution function at time T , for those particles starting at x0= 0,

crossing the barrier at b and then returning back across the barrier before time T It is shown in Appendix A.8(iii) that this can be written Freturn(x T , T ) = AF0(x T − 2b, T ), where the term F0(x T − 2b, T ) is the normal distribution for a particle starting at x0= 2b and

A = exp(2mb/σ2); m is the drift rate of x T We can then write

Freturn(x T , T ) dx T = exp

2mb

σ2

 1

σ√2πT exp



−1 2

x T − 2b − mT

σ√T

2

dx T

= exp

2mb

σ2

 1

√

2πe−

1z2

T dzT = An(z

T ) dzT

(ii) We will now apply these results to stock price movements Consider a stock with a starting

price S T in the presence of a barrier K Closely following the Black Scholes analysis of Section 5.2, we write x T = ln(S T /S0) and note that x T is normally distributed with mean

mT and variance σ2T , where m = r − q −1

2σ2 We use the notation b = 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 variable S T to either of the variables z T or zT, which were defined in the last subsection by

S T = S0emT +σ√T z T = S0emT +2b+σ√T zT When setting up the integral for evaluating a call option, we integrate with respect to S T from

X to ∞ On transforming to the variables z T or zT , the integrals will run from Z X to∞ or

from ZX to∞, where

Z X =ln(X /S0)− mT

X =ln(X /S0)− mT − 2b

σ√T Analogous limits of integration z K and zK are defined by

Z K = ln(K /S0)− mT

K =ln(K /S0)− mT − 2b

σ√T

0 S K

X

0 f0 f0

F 0 f0 f

return F 0

F

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 function Fcrossersis different on each side of the barrier as shown

The price of the option is written

C d −i (X < K ) = e −rT +∞

0

(S T − X)+F

crossersdS T = e−rT +∞

X

(S T − X)FcrossersdS T

= e−rT K

X

(S T − X)F0dS T + e−rT ∞

K

(S T − X)FreturndS T

The first integral on the right-hand side can be split into two manageable parts as follows:

e−rT

 K

X

(S T − X)F0dS T = e−rT ∞

X

(S T − X)F0dS T− e−rT ∞

K

(S T − X)F0dS T

= [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 for C d −i (X < K ), we make the transfor-mation to the standard normal variate zTdescribed in subsection (ii) and use the integral result

of equations (A1.7):

[JC]= e−rT ∞

K

(S T − X)FreturndS T = e−rT ∞

z

K (S0emT +2b+σ√T zT − X)An(z

T ) dzT

= A e −rT

S0e2b +(m+1σ2)TN[σ√T − Z

K]− X N[−Z

K] The value of this option can then be written

C d −i (X < K ) = [BS C]− [GC]+ [JC]

0

return F

0 F

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:

P u −i (K < X) = e −rT +∞

0

(X − S T)+FcrossersdS T

= e−rT K

0

(X − S T )FreturndS T

+ e−rT X

K

(X − S T )F0dS T

The second integral on the right may be written

e−rT

 X

K

(X − S T )F0dS T = e−rT X

0

(X − S T )F0dS T − e−rT K

0

(X − S T )F0dS T

= [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−rT K

0

(X − S T )FreturndS T = e−rT Z

K

−∞ (X − S0emT +2b+σ√T zT ) An(zT ) dzT

= A e −rT

X N[ZK]− S0e2b+(m+1σ2)T N[ZK − σ√T ]

The value of the option is written

P u −i (K < X) = [BS P]− [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:

rThe 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 call

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

rIf 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 were ZKto+∞ in the first example and−∞ to Z

K in the second; the difference comes because the stock price had to fall to reach the barrier in the first example but rise in 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 either C d −i (X < K ) or P u −i (K < X).

The reader should now be in a position to derive a formula for any knock-in option If he really enjoys integration, he can work out the integral results for all the puts and calls with barriers in different positions Without showing all the detailed workings, we give the results in the next subsection First, however, we take note of a simple but powerful relationship:

Knock-in Option + Knock-out Option = European Option

This result is obvious if we consider a portfolio consisting of two options which are the same except that one knocks in and the other knocks out Whether or not the barrier is crossed, the payoff is that of a European option This relationship allows us to calculate all the knock-out formulas from the knock-in results

The following definitions are used:

[BS]= e−rT φS0e(m+1σ2)TN[φ(σ√T − Z X)]− X N[−φZ X]

[G]= e−rT φS0e(m+1σ2)TN[φ(σ√T − Z K)]− X N[−φZ K]

[H]= A e −rT φS0e2b+(m+1σ2)TN[ψ(Z

X − σ√T )] − X N[ψ Z

X] [J]= A e −rT φS0e2b +(m+1σ2)TN[ψ(Z

K − σ√T )] − X N[ψ Z

K]

ψ = +1 up to barrier−1 down to barrier φ = +1 call−1 put

m = r − q −1

2σ2; b = ln(K/S0); A = exp(2mb/σ2)= (K/S0)2m /σ2

Z X =ln(X /S0)− mT

X = ln(X /S0)− mT − 2b

σ√T

Z K =ln(K /S0)− mT

K = ln(K /S0)− mT − 2b

σ√T

The formulas for all the single barrier options are given in Tables 15.1 and 15.2

15.3 SOLUTIONS OF THE BLACK SCHOLES EQUATION

Table 15.1 Single barrier knock-in options

C d −i (X < K ) P u −i (K < X) [BS]− [G] + [J]

C d −i (K < X) P u −i (X < K ) [H]

C u −i (X < K ) P d −i (K < X) [G]+ [J] − [H]

C u −i (K < X) P d −i (X < K ) [BS]

Table 15.2 Single barrier knock-out options

C d −o (X < K ) P u −o (K < X) [G]− [J]

C d −o (K < X) P u −o (X < K ) [BS]− [H]

C u −o (X < K ) P d −o (K < X) [BS]− [G] − [J] + [H]

C u −o (K < X) P d −o (X < K ) 0

(i) The general approach to pricing barrier options has been to use the Fokker Planck equation

to derive an analytic expression for the probability distribution function of particles crossing

a barrier This explicit probability density function is then used to calculate an expression for the value of a knock-in option; the knock-out option prices are obtained from the symmetry relationship which states that the sum of the values of a knock-out and a knock-in option equals the value of the corresponding European option

In Appendix A.4 we discuss the close relationship between the Kolmogorov equations and the Black Scholes equation A reader might well ask why we bothered to go to the trouble of

a two-step solution (first, find the probability distribution function; second, calculate the risk-neutral expected payoff), rather than solving the Black Scholes equation directly The reason

is partly historical: at the time when people first needed to calculate a formula for a barrier option, the expression for the transition probability density function for a Brownian particle

in the presence of an absorbing barrier had already been worked out; it was just a question of looking it up in the right book But there are other good reasons for the approach adopted: it allows a unified approach to all knock-in options with an emphasis on the underlying processes

in terms of probabilities The pure solution of differential equations can be rather sterile, without much reference to underlying processes Furthermore, in some cases, the boundary conditions for the Black Scholes model are rather hard to apply We will therefore content ourselves here

by sketching out the approach to a relatively easy example: the down-and-out call (X < K )

which is the “out” equivalent of the down-and-in call illustrated in Figure 15.1

The approach is identical to that of Section 5.3 where we solved the Black Scholes equation for a European call option The fundamental equation is unchanged We seek a solution in the

range K < S0< ∞ subject to the following initial and boundary conditions:

rC(S0, 0) = max[0, S0− X]; X < K ; K < S0< ∞

rlimS0→K C(S0, T ) → 0

rlimS0→∞C(S0, T ) → S0e−qT − X e −rt

Using the notation and transformations of Section 5.3, the Black Scholes equation becomes

∂v/∂T= ∂2v/∂x2with initial and boundary conditions

rv(x, 0) = max[0, e (k +1)x − X e kx

]; ln X < b; b < x0< ∞; b = ln K

rlimx →b v(x,T)→ 0

rlimx→∞v(x,T ) → e (k+1)x+(k+1)2T

− X e kx +k2T

The solutions of this type of equation are given by equations (A6.8) or (A7.10) in the Appendix:

v(x, T)=

 +∞

b

ekxmax[0, e x − X] 1

2√

πT

 exp



−(y − x)2

4T



−exp



−(y + x + 2b)2

4T



dy

We can replace [0, e x − X] by e x − X since this is always positive in the range of integration.

It then just remains to follow the computational procedures set out in Section 5.3 to work out this integral; unsurprisingly, the answer is the same as that given in Table 15.1

(i) First Passage or Absorption Probabilities: The pseudo-probability of a barrier above being

crossed is straightforward to calculate It is simply the sum of the probabilities of a particle crossing and returning, and a particle crossing and staying across In terms of equity prices, this is written

Pcros sin g =

 ∞

−∞FcrossersdS T =

 K

−∞FreturndS T+

 ∞

K

F0dS T

=

 Z

K

−∞ An(z

T ) dzT +

 +∞

Z K

n(z T ) dz T = A N[Z

K]+ N[−Z K]

There is an analogous expression for the pseudo-probability of crossing a barrier below, and the general expression can be written

Pcros sin g = A N[ψ Z

K]+ N[−ψ Z K]

= exp

2mb

σ2

 N



−ψ (b + mT )

σ√T

 + N



−ψ (b − mT )

σ√T



(15.1)

It should be remembered that this is a pseudo-probability in a risk-neutral world It is not the

probability in the real world that an option will be knocked in or out

(ii) Knock-in Rebate: Occasionally, barrier options are structured so that the purchaser receives a

lump sum payment if his investment strategy does not work For example, if he buys a knock-in option and the stock price does not reach the barrier before maturity, he receives a fixed amount

R at maturity.

The upfront value of this rebate is simply the present value of R multiplied by the pseudo-probability of the barrier not being reached:

Rmaturity= e−rT R(1− Pcros sin g) where Pcros sin gis given in the last subsection

15.5 BINARY (DIGITAL) OPTIONS WITH BARRIERS

(iii) Knock-out Rebate: More common than for knock-in options, rebates are often given as

con-solation prizes with knock-out options However, the calculation of this type of rebate is more complex since the lump sum is paid as soon as the knock-out occurs; we cannot then calculate

the present value just by discounting back over the period T

In Appendix A.8(vii) it is seen that the first passage timeτ (time to first crossing) is a random

variable with a well-defined probability distribution function

gabs(τ) = ψb

σ√2πτ3exp



2σ2τ (b − mτ)2



By definition, we can write

Pcros sin g= exp

2mb

σ2

 N



−ψ (b + mT )

σ√T

 + N



−ψ (b − mT )

σ√T



(15.2)

The value of a knock-out rebate of $1 is given by the following integral:

Rfirst passage=

 T

0

e−rτ gabs(τ) dτ

On the face of it, this looks like a very difficult integral to solve: but a little trick helps; completing the square in the exponential gives

e−rτ gabs(τ) = exp



−b(γ − m) σ2 ψb

σ√2πτ3exp



2σ2τ (b + γ τ)2



= exp



−b(γ − m)

σ2



habs(τ)

whereγ =√m2+ 2rσ2and habs(τ) is the same as gabs(τ), but with the replacement m → γ

Using the result of equation (15.2), we can write

Rfirst passage =

 T

0

e−rτ gabs(τ) dτ = exp



−b(γ − m)

σ2

  T

0

habs(τ) dτ

= exp



−b(γ − m) σ2 exp



2γ b

σ2

 N



−ψ (b − γ T )

σ√T

 + N



−ψ (b + γ T )

σ√T

 (15.3)

(i) Recap of Straight Binaries: Referring back to Section 11.4(iv), a gap option can be written as

(Reiner and Rubinstein, 1991b)

fgap= φ{S0[BS]1− R[BS]2} = fasset− fcash where [BS]1and [BS]2are the first and second terms in the Black Scholes formula R is a cash sum which may or may not be equal to the strike price X ; if it is, we just have the formula for

a put or a call option.φ(= ±1) differentiates between puts and calls fasset and fcashare the

prices of asset-or-nothing and cash-or-nothing options with strike X

(ii) Barrier options may be decomposed into digital options in just the same way This is best illustrated by way of an example Returning to the example of Section 15.1(iii), the formula for the down-and-in call can be decomposed as described in the last subsection:

C d −i (X < K ) = {S0[BSC]1− X[BS C]2} − {S0[GC]1− X[G C]2} + {S0[JC]1− X[J C]2}

0 F

return F

$1

Figure 15.3 Digital knock in:

down-and-in; cash or nothing

Collect together the terms in −X; its coefficient

[BSC]2− [GC]2+ [JC]2 is the price of an option

with the following payoff at time T (Figure 15.3):

• $1, if the barrier has been crossed and X < S T;

• 0 otherwise

Similarly, the terms in S0give the price of an option

with the following payoff at time T (Figure 15.4):

rS T , if the barrier has been crossed and X < S T;

r0 otherwise.

0

F

return

F

0

Figure 15.4 Digital knock in: down-and-in; asset or nothing

These last two examples are of course, for specific

configurations of S0, X and K Formulas for other

configurations can be obtained from Tables 15.1 and

15.2

(iii) One Touch Options (Immediate Payment): The

bi-nary options of the last subsection give a positive

payoff if two conditions are met: the barrier is

crossed and the option expires in-the-money One

touch options are closely related but do not have

the second condition They also pay out as soon as the barrier has been crossed

The one-touch cash-immediately option with payout R is clearly just the same as the

knock-out rebate and is priced by equation (15.3)

The one-touch asset-immediately option is priced in just the same way: at timeτ when the barrier is crossed, S τ is equal to K ; but S τis the payout, so we price this option as a knock-out

rebate in which the lump sum payment is equal to K.

(iv) One Touch Options (Payout at Expiry): These are simple adaptations of previously obtained

formulas:

Cash at expiry: use e −rT R Pcros sin g

Asset at expiry uses the appropriate digital barrier option, putting the strike price equal

to zero

(i) American Capped Calls (Exploding Calls): These are American call options in which the

payout is capped at a certain certain amount (K − X), irrespective of when the option is

exercised

A European capped call is the same as a call spread If we buy a call with strike X and sell

a call with a higher strike K , the maximum payoff of the combination at maturity is (K − X).

15.6 COMMON APPLICATIONS However, this structure does not carry over to American options because each option holder can choose when to exercise: the person to whom we have sold the call may not wish to exercise when we do

The American capped call can instead be priced as an up-and-out call, (X < K ) with a rebate

of (K − X) paid at knock out A similar approach is used to price an American capped put.

0

S K 1 K 2 K 3

(ii) Ladders: When investors buy European call options,

it is not uncommon for them to watch the price of

the underlying stock soar, and with it the value of the

option – only to see both plunge out-of-the-money at maturity Ladder options have payoffs which capture the effects of such movements

The simplest form of such a scheme would be a series of one-touch cash-immediately options The payoffs would be

r K1− S0received as soon as the stock price reaches K1;

r K2− K1received as soon as the stock price reaches K2; etc.

X K 1 K 2

(iii) Fixed Strike Ladders: The simple ladder of the last

subsection does not really display the features of a call

option There are two commonly used structures which

are fundamentally call options but which at the same

time capture large up-swings in the stock price (Street,

1992) The fixed strike ladder has the following payoff (we assume for simplicity that the call

option is at-the-money, i.e S0= X):

rIf S t never reaches K1, we just have a plain call option with strike X ;

rIf S t gets as far as K1before maturity, the call payoff has a minimum of K1− X;

rIf S t gets as far as K2, the minimum payoff is K2− X; etc.

X

1

knock in at K

2

knock in at K

K 1 K 2 X

knock in at K 1 knock in at K 2

knock in at K 1

knock in at K 2

Figure 15.5 Construction of fixed strike ladder

The combination of options which gives this

pay-off is summarized below The analysis is easiest

to follow by referring to Figure 15.5

• C(X) Buy a European call option, strike X.

If S t never rises above K1, this gives the payoff

needed

• P u −i (K1, K1)− P u −i (X , K1) Buy a knock-in

put, strike K1and sell a knock-in put, strike X

If at some point S t crosses K1 (but not K2),

there are two possibilities: if the final stock

price S T is between K1and K2, the two knocked-in puts are out-of-the-money so the payoff

comes just from the original call option: S T − X For S T anywhere below K1, the payoff is

(K1− X).

• P u −i (K2, K2)− P u −i (K1, K2) As in the last step, we have a long put with strike K2and a

short put with strike K1, both of which knock in at K2 We use precisely the same reasoning

as for the last step: if at some point S t crosses K2, there are two possibilities: we have a call

option payoff for K2 < S T and a payoff (K2− X) for all S T < K2, etc

1 K 2

(iv) Floating Strike Ladders: This structure captures large

downward swings in the stock price, by changing the

strike price to lower values as barriers are crossed The

payoff is as follows:

rIf S t never reaches K1, we just have the call option

with strike X

rIf S t reaches K1(but not K2) before maturity, the call option with strike X is replaced by a

call with strike K1

rIf S t reaches K2, the call option with strike K1is replaced by a call with strike K2, etc.

The structure of the barrier options needed to produce this payoff is simpler to follow than in the last subsection (see Figure 15.6)

K 1

K 2

X

1

knock in at K

2

knock in at K

2

knock in at K 1 knock in at K 2

knock in at K 2 knock in at K 1

Figure 15.6 Construction of floating strike ladder

• C(X) Buy a European call option with strike X.

If S t never falls as far as K1, this gives the payoff

we need

• C d −i (K1, K1)− C d −i (X , K1) Buy a knock-in

call option with strike K1and sell a knock-in call

with strike X ; both knock in at K1 The sold

option cancels the original call option of the first

step above, and we are left with a new call,

strike K1

• C d −i (K2, K2)− C d −i (K1, K2) Again, the second of these cancels the call option left from

the previous step The net result is that if these two options knock in (S t crosses the K2 barrier), we are left with a call option, strike K2, etc

10.00 20.00

90.00 100.00 110.00 120.00 130.00

Figure 15.7 Up-and-out call option

By their nature, barrier options display a sudden increase or decrease in value as the stock price crosses a barrier We have already seen in the discussion of digital options in Section 11.4(v) that sudden changes in option value

for small changes in the price of

the underlying stock can cause

problems in hedging

(i) Figure 15.7 shows the value of

an up-and-out call option

plot-ted against the stock price Far

from the barrier, the value of the

option coincides with that of the

corresponding European call

op-tion In this region the

probabil-ity of a knock-out is remote; but

as the barrier is approached, the

value of the knock-out option declines sharply This creates a very pointed peak in the value of the option; put another way, the negative gamma of the option becomes very