Barriers - Advanced Options

The barrier options described so far have been European options which are knocked in or knocked out when the price of the underlying variable crosses a barrier.. An extension of this is

16 Barriers: Advanced Options

The last chapter laid out the principles of European barrier option pricing This chapter con-tinues the same analysis, applied to more complicated problems The integrals get a bit larger, but the underlying concepts remain the same Lack of space prevents each solution being given explicitly; but the reader should by now be able to specify the integrals corresponding to each problem, and then solve them using the results of Appendix A.1

These are options which knock out or in when either a barrier above or a barrier below the

starting stock price is crossed The analysis is completely parallel to what we have seen for a single barrier option (Ikeda and Kunitomo, 1992)

(i) In the notation of this chapter, F0(x T , T ) is the normal distribution function for a particle starting at x0= 0 The function is given explicitly in Section 15.1(i) Fnon-absis the

probabil-ity distribution function for particles starting at x0= 0 which have not crossed either barrier before time T Two expressions have been derived for this function, which are given by

equa-tions (A8.9) and (A8.10) of the Appendix They are both infinite series although there is no correspondence between individual terms of the two series:

1 Fnon-abs (x T , T ) = 1

σ√2πT

+∞

n=−∞

 exp

+mun

σ2

 exp



2σ2T (x T − mT − u n)2



− exp

+mvn σ2

 exp



2σ2T (x T − mT − v n)2



2 Fnon-abs (x T , T ) = exp

mxT

σ2

∞

n=1



ane−b n T

sinnπ

L (x T + b)



L = a + b; un = 2Ln; vn = 2(Ln − b)

an = 2

Lsin

nπb

2

µ σ

2 +

nπσ

L

2

(ii) The reasoning of Appendix A.8(iv) and (v) demonstrates that the distribution functions of

particles which start at x0= 0, then cross either the barrier at −b or +a, and then return to the

region−b to +a can be written

Freturn = F0− Fnon-abs

-b X a

return

0

F

0

S

Figure 16.1 Double barrier-and-in call

The total probability distribution function for all

those particles that cross one of the barriers can now

be written

Fcrossers=







F0 (x T , T ) xT < −b Freturn (x T , T ) −b < x T < a

F0(x T , T ) a < xT

(iii) As an example, we will look at the knock-in call

op-tion shown in Figure 16.1 We do not really need to

worry about whether we move up or down to the barrier:

Cki = e−rT +∞

0

(S T − X)+Fcrossers dS

T = e−rT +∞

X (S T − X)FcrossersdS T

= e−rT a

X

(S T − X)FreturndS T+ e−rT ∞

a (S T − X)F0dS T (16.1) The second integral is completely standard The first depends on which form of series is used

for Fnon-abs The sine series is completely straightforward to integrate while the other alternative

is handled using the procedures of Section 15.1

(iv) The question of which of the two series to use and how many terms to retain is best handled pragmatically Set up both series and see how fast convergence takes place in each case Both series dampen off regularly, so it is for us to choose how accurate the answer needs to be We should expect to perform the calculation with one series or the other within four to six terms, and often less

The barrier options described so far have been European options which are knocked in or knocked out when the price of the underlying variable crosses a barrier An extension of this is

a European option which knocks in when the price of a commodity other than the underlying stock crosses a barrier For example, an up-and-in call on a stock which knocks in when a foreign exchange rate crosses a barrier These options are called outside barrier options, as distinct from inside barrier options, where the barrier commodity and the commodity underlying the European option are the same The reason for the terminology is anybody’s guess (Heynen and Kat, 1994a)

We could repeat most of the material presented so far in this chapter, adapted for outside barriers rather than inside barriers However, these options are relatively rare so we will simply describe a single-outside-barrier up-and-in call option; the reader should be able to generalize this quite easily to any of the other options in this category

(i) Outside Barrier, Up and In: The general principle remains as before; it is merely the form of

some of the distributions that is different The price of the option is the present value of the risk-neutral expectation of the payoff (Figure 16.2):

C uoutside−i = e−rT +∞

S T=0

 +∞

Q T=0(S T − X)+F

jo intdS T dQ T (16.2)

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16.2 OUTSIDE BARRIER OPTIONS

0

return

F

0

F

0

Figure 16.2 Outside barrier, up-and-in call

where S T is the maturity value of the stock

un-derlying the call option and Q T is the maturity

value of the barrier commodity The form of this

is the same as for inside barriers; but we need

to find an expression for Fjo int which is the joint

probability distribution for the two price

vari-ables The large topic of derivatives which

de-pend on the prices of two underlying assets is

at-tacked in Chapter 12 The material of that chapter

and of Appendix A.1 is used to solve equation

(16.2)

(ii) Separate Distributions of the Two Variables: As

before, we transform to the logs of prices: x T =

ln(S T S0 ); y T = ln(Q T /Q0 ) The distribution of x T

is normal and the variable z T = [ln(S T /S0)− mT ]/

σ√T is a standard normal variate (mean 0, variance 1); σ is the volatility of the stock and

m − r − q −1

2σ2

The variate y T has a more complex distribution As explained in Section 13.1(i), y T is

distributed as Fcrossers(y T , T ) which has different forms above and below the barrier at QT = K

or y T = ln(K/Q0)= b.

b < yT : Fcrossers= F0(y T , T ) which is the distribution at time T of a particle which started at

y0 = 0 and has drift m Q = r − q Q−1

2σ2

Qand varianceσ2

Q The variable

wT =ln(Q T /Q0)− m QT

σQ√T

is a standard normal variate

yT < b: Fcrossers = Freturn= AF0(y T − 2b, T ) where A = exp(2m Qb/σ2

Q)= (K/Q0)2m Q /σ2

Q and F0(y T − 2b, T ) is the distribution function for a particle which started at y0= 2b and has drift m Q The variable

w

T =ln(Q T /Q0)− m Q T − 2b

σQ√T

is therefore a standard normal variate

(iii) Equation (16.2) may be rewritten

C uoutside−i = e−rT +∞

S T =X

 K

Q T=0A(ST − X)F1 jo intdQ T dS T

+

 +∞

S T =X

 +∞

Q T =K (S T − X)F2 jo intdQ T dS T



and transforming to the variables Z T, wT andw

T, this last equation can be written more

191

precisely as

C uoutside−i = e−rT



A

 +∞

Z X

 W

K

−∞ (S0e

mT +σ√T z T − X)n2(z T , w

T;ρ) dzTdw

T

+

 +∞

Z X

 +∞

W K (S0emT +σ

√

T z T − X)n2(z T , wT;ρ) dzTdwT



Z X =ln(X /S0)− mT

K = ln(K /Q0)− m Q T − 2b

σQ√T ; W K =ln(K /Q0)− m QT

σQ√T

n2(z T , w

T;ρ) is the standard bivariate normal distribution describing the joint distribution of the two standard normal variates z t andw

t, which have correlation ρ n2 (z T , wT;ρ) is the

standard bivariate normal distribution describing the joint distribution of the two standard

normal variates z t andwt, which have correlationρ.

Note that the correlations between z tandw

t are the same as between z tandwt;w

T andwT essentially refer to the same random variable Q T, and differ only in their means, which does not affect the correlations

Using the results of equations (A1.20) and (A1.21), this last integral is evaluated as follows:

C uoutside−i = AS0e−qTN[(σ√T − Z X)]− X e −rTN[−ZX]

−S0e−qTN2[−(σ√T − Z X),−(ρ σ√T − W

X);ρ]−X e −rTN

2[−ZX ,−W

K;ρ]

+S0e−qTN2[−(σ√T − Z X),−(ρ σ√T − W

X);ρ] − X e −rTN

2[−ZX ,−WK;ρ]

(16.3)

In the foregoing it was always assumed that a barrier is permanent However, the barrier could

be switched on and off throughout the life of the option Such a pricing problem is usually handled numerically, but the simplest case can be solved analytically using the techniques of the last section (Heynen and Kat, 1994b)

This is an option on a single underlying stock at two different times, as described in

Chapter 14 The specific case we consider is an up-and-in call of maturity T, which knocks in

if the barrier is crossed before timeτ, i.e the barrier is switched off at time τ Its value can be

written analytically as

C upartial−i = e−rT +∞

S T=0

 +∞

S τ=0(S T − X)+F

jo intdS τ dS T

Fjo int is the joint probability distribution of two random variables S τ and S T , where S τis subject

to an absorbing barrier This problem is almost precisely the same as the outside barrier option problem solved in the last section The formula given in equation (16.3) can therefore be applied directly, with the following modifications:

r Q0→ S0,σQ → σ and m Q → m.

rT → τ in the formulas for w

K andwK

rThe correlation between S τ and S T is shown in Appendix A.1(vi) to beρ =√

τ/T

192

16.4 LOOKBACK OPTIONS

These are probably the most discussed and least used of the standard exotic options The problem is that on the one hand they have immense intuitive appeal and pricing presents some interesting intellectual challenges; but on the other hand they are so expensive that no-one wants to buy them However, this book would not be complete without an explanation of how

to price them (Goldman et al., 1979).

0

S

max

S

min

S

T

Figure 16.3 Notation for lookbacks

(i) Floating Strike Lookbacks: Lookback options are quoted in two ways The most common

way is with a floating strike, where the payoffs are defined as follows:

Payoff of Cfl str= (S T − Smin)

Payoff of Pfl str= (Smax− S T)

The lookback call gives the holder the right to buy stock at maturity at the lowest price achieved

by the stock over the life of the option Similarly, the lookback put allows the holder to sell

stock at the highest price achieved

The form of the payoff is unusual in that it does not involve an expression of the form max[0, ], since (ST − Smin) can never be negative; it has therefore been suggested that this

is not really an option at all, although this is largely a matter of semantics However, it does make the pricing formula straightforward to write out: risk neutrality gives

Cfl str= e−rT {ES T − ESmin }

Pfl str= e−rT {vmax− F 0T}

where F 0T is the forward price

(ii) Fixed Strike Lookbacks: As the name implies, these options have a fixed strike X Referring

to Figure 16.3, the payoffs of the fixed strike call and put are given by

Payoff of Cfix str= max[0, Smax− X]

Payoff of Pfix str = max[0, X − Smin] These are sometimes referred to as lookforward options They give the option holder the right

to exercise not at the final stock price, but at the most advantageous price over the life of the option The payoffs look more like normal option payoffs, containg the familiar “max” function However, in practice, the payoff can be further simplified, since the options are usually

193

quoted at-the-money, i.e with X = S0 This implies that X ≤ Smaxor Smin≤ X, so that

Cfix str= e−rT {ESmax − X}

Pfix str= e−rT {X − vmin}

(iii) Distributions of Maximum and Minimum: The prices of both floating and fixed strike lookback

options depend on the quantitiesvminandvmax, which are defined in the last two subsections

It is shown in Appendix A.8(viii) that the distribution functions for xmax= ln(Smax/S0) and

xmin = ln(Smin/S0) are

Fmax (xmax, T ) = 2

σ√2πT exp



2σ2T (xmax− mT )2



−2m

σ2 exp

+2mxmax

σ2

 N



σ√T (xmax+ mT )

 (16.6)

Fmin (xmin, T ) = 2

σ√2πT exp



2σ2T (xmin− mT )2



+2m σ2 exp

+2mxmax

σ2

 N



σ√T (xmin+ mT )



t

0

S

max

S

H ; previous max

path A

path B

max

path A

path B

t = 0

S

Figure 16.4 Previous maximum

(iv) When we derive the formula for the price of an option, we do not usually have to concern ourselves with what happened in the past: if a call option was issued for an original maturity

of 3 months, its price after 2 months is exactly the same as the price of a newly issued 1-month option However, the pricing of a lookback is a little more difficult: after 2 months, the maximum

or minimum value of S tfor the whole period may already have been achieved

Let us assume that a previous maximum H has been established and we wish to find the

value ofvmax at time t= 0 Consider the two paths shown in Figure 16.4: path A establishes a

new maximum at Smaxwhile path B does not make it so that the established maximum remains

at H This generalized definition, accommodating a previous maximum, is expressed in the

general definition

vmax = Emax[H, Smax] = H PSmax< H + ESmaxH < Smax

194

16.5 BARRIER OPTIONS AND TREES

or

vmax = H

 H

0

Fmax dSmax+

 ∞

H Smax Fmax dSmax

There is an analogous expression forvmin in terms of a previously established minimum L.

(v) Expressions forvmax andvmin can be obtained by using equations (16.6) for Fmaxand Fmin,

making the substitution Smax= S0exmax The resulting integrals are performed using the results

of Appendix A.1(v), item (E); the algebra is straightforward but very tedious The expressions can be combined to give the generalized formula

vmax / min = H N[ψ Z K]− σ2

2(r − q)exp

2mb

σ2

 N[ψ Z

K]



+F 0T 1+ σ2

2(r − q)N[−ψ(ZK − σ√T )]



(16.7)

b = ln K/S0; Z K = (ln K /S0 − mT )

K = Z K − 2b for max: K = H, ψ = +1; for min: K = L, ψ = −1

(vi) Strike Bonus: Using equation (16.7) to obtain an expression for vmin, substituting this into equation (16.4) and rationalizing gives (Garman, 1989)

Cfl str = e−rT {F 0TN[−(ZL − σ2T )] − L N[−Z L]}

+ σ2S0

2(r − q) e−rTexp



2(r − q)b

σ2

 N[−Z

L]+ e−qT N[Z

L − σ2T ]

 (16.8)

The first term is simply the Black Scholes formula for a call option with strike X = L, the previously achieved minimum At t = 0 we consider two possibilities:

rNo new minimums are formed below L before the maturity of the option The payoff of Cfl str

is then equal to the payoff of C(S0, L, T ), i.e max[0, ST− L].

rA new minimum is established at Sminwhich is below L The payoff of the option is then

max[0, ST − Smin]

Comparing these two possible outcomes, it is clear that the second term in equation (16.8)

prices an option to reset the strike price of a call option from L down to the lowest value achieved by S t before maturity This option is called the strike bonus.

(i) Binomial Model (Jarrow–Rudd): Binomial and trinomial trees are a standard way of solving

barrier option pricing problems which are not soluble analytically, such as American barrier options However, these methods do display some special features which will be illustrated

with the example of a European knock-out call C u −o (X < K ) The example uses S0= 100,

X = 110, K = 150, r = 10%, q = 4%, σ = 20%, T = 1 year This is similar to the example

195

examined in some detail in Section 7.3(v) and again in Section 8.6, except for the existence

of a knock-out barrier In the previous investigations we examined the variation of the values obtained from the binomial model, as a function of the number of time steps

In order to accommodate the knock-out feature in a binomial tree, we simply set the option value equal to zero at each node for which the stock price is outside the barrier Consider the above knock-out option, priced on a three-step binomial tree Using the Jarrow–Rudd discretization, there is no node with a stock price higher than 150; therefore there is no node at which we would set the stock price equal to zero We are therefore unable to price this option –

or alternatively put, this model gives the same value for a barrier at K = 150 and for K = ∞.

K=150

2.21

112.24

100.00 100.00

125.98

141.40

89.09

79.38

112.24

89.09

70.72

31.40

2.21

0

0

17.92

1.21

0

10.19

2.41 5.79

100.00

112.24

141.40

89.09

100.00

79.38

112.24

89.09

70.72 0

0.96 1.21

0

1.03

.65 84

K=130

K=140

125.98

0

0

Figure 16.5 Knock-out barriers in binomial trees

If we now look at the same model with the barrier at K = 140 as in the second diagram of Figure 16.5, we see how the tree is modified, giving a very different value for the option But

the tree would give exactly the same answer for K = 130; the barrier would have to be below

K = 125.98 for the tree to be modified any further In general, the value of a knock-out option

is a step function of the barrier level, with a jump each time the barrier crosses a line of nodes

(ii) Price vs Number of Steps: Figure 16.6 shows the value of our knock-out call option plotted

against the number of steps in the binomial tree; the analytical value of this option is 3.77 It is instructive to compare this graph with Figure 7.11 for a similar option but without a knock-out barrier The European option shows the characteristic oscillations which are gradually damped away; by the time we reach about 300/400 steps (not shown), the answer obtained is stable

enough for commercial purposes The knock-out option on the other hand shows three different features of interest (Boyle and Lau, 1994):

196

16.5 BARRIER OPTIONS AND TREES

3.50

3.70

3.90

4.10

4.30

4.50

4.70

4.90

Number of Steps

Option Price

(Stock = 100)

3.77 ± 1%

Figure 16.6 Up-and-out knock-out call price vs number of binomial steps (Jarrow–Rudd discretization)

rThe graph continues its relentless sawtooth pattern long after the 1500 steps which we have

shown Convergence to the analytical answer is difficult to achieve

rHowever, within about 150 steps, we find the bottom of the zig-zags within 1% of the

analytical answer; certainly, within 300/400 steps, the envelope of low points gives an

almost perfect answer

rThe answers converge to the theoretical answer from above, i.e apart from a few outliers,

the tree always gives answers greater than the analytical value

We start by turning our attention to the first two features

Row N

x x x x x x

o o o o o o

o o o o o o

x x x x x x

o o o o o o

x x x x x x

K

1 extra binomial step

Figure 16.7 Effect of increasing number of binomial steps

In subsection (i) we saw that a knock-out

op-tion value calculated with a binomial tree is a

step function of the barrier level This same

ef-fect causes oscillations in the calculated value of

a knock-out option plotted against the number

of steps If we use the Jarrow–Rudd

discretiza-tion with N time steps, each propordiscretiza-tional up-jump

is given by u= eσ√δt = eσ√T /N Therefore as N

increases, the spacing between adjacent rows of

nodes decreases; the rows of nodes become

pro-gressively compressed together and at a certain

point an entire row of nodes crosses the barrier

At this point there is a jump in the value of the

option calculated by the tree

(iii) As the number of steps N is increased, we would

expect the most accurate binomial calculation to

occur when a row is just above the knock-out barrier; the option value at these nodes is put at

197

zero Increasing the number of steps by one would then push the row down through the barrier and change this row of zeros into positive numbers (see Figure 16.7)

Let Ncbe a critical number of steps such that row n of nodes in the binomial tree lies just above K, i.e S0en σ√T /Ncis greater than K; but S0en σ√T /(Nc +1)is less than K We find N

cfrom

S0en σ√T /Nc < K < S0en σ√T /(Nc +1) or N

c= round down

n2σ2T (ln K /S0)2



where “round down” means round down to the nearest integer Note: It is important to be

accurate at this point since Ncwill give a best answer while Nc+ 1 gives a worst answer.

Figure 16.8 is just a blown up detail taken from Figure 16.6 Use of the formula just given gives the following results:

28 190

29 204

30 218

which correspond precisely to the jumps in the diagram The rippling effect of the option values between jumps is the residual effect of the oscillations always observed in binomial calculations

Number of Steps Calculated Price

3.7 3.8 3.9 4.0 4.1

184 186 188 190 192 194 196 198 200 202 204 206 208 210 212 214 216 218 220 222 224

Figure 16.8 Knock-out option price vs steps

(iv) Alternative Discretizations: The above sawtooth effect is particularly pronounced when we

use the Jarrow–Rudd discretization, since all nodes lie along horizontal levels: entire rows

of nodes then cross the barrier at once as the number of time steps is increased If we use a discretization which does not have horizontal rows of nodes, then only a few grid points cross the barrier each time the number of steps is increased Figure 16.9 is analogous to Figure 16.6, but using the Cox–Ross–Rubinstein discretization It continues to display the sawtooth effect, but with much reduced amplitude; the envelope of the low points no longer coincides with the

198