Pricing discrete percentage look-back option via integral transforms of measure

We are interesting in pricing lookback option which pays

(ф (aSNNxt - St)) + , where a is a positive coefficient, (St)0

Sext 4 Ф max фSt■ k        0

We consider underlying price dynamic with known characteristic functions of the logreturn X_t,_s⁴ In (S_s/S_t) by itself, its positive Xt+s and negative X-s parts

ht,s(u) ⁴ee^, h±s(u) ⁴ E e^uX±s.

Also it is assumed that returns have independent increments. Additionally we assume the risk-free asset growing exponentially.

Price of such an option can be found by the formula

L%(t) = e^-r(T-t)Et ф (aSNNxt — S^)+ .

In the case a = 1 a percentage lookback becomes a 100%-lookback option with payoff given by

L(T) = ф Sexxt — St) .

For the numerical calculation of the price of such an option, G. Petrella and S. Kou proposed in [2] the method based on the inversion of the Laplace transform of the option value with respect to a previously achieved extrema which can be expressed in terms of characteristic functions of returns (its positive and negative parts as well). But this technique can not be applied straight forward to the case of an arbitrary positive coefficient a because of the problem we attempt to solve is essentially two-dimensional. However in this paper we show that, with some modification requiring a second Laplace transform, such an integral transform approach still can be a useful idea.

Pricing Lookback Put Option

Now we get onto the derivation of the percentage discrete lookback Put option (on maximum) price. An expression for the corresponding Call can be obtained in an analogous way.

Let us start with some transformation of the initial expression (2), assuming that a current time t lies within a time interval [t_1-1,t₁) and that the running maximum over the whole time interval [0, T] is the maximum between previously achieved extremum and the maximum of the rest SN^ax= max (SpaX, maxl<j<_NStj).

LP%(t) = e-r^1-tV(aSN" — St)₊=

= e-r(T^Et (max (aSmax, St) - St) = e r(T t)Et max ( aSPax, a max St-, St 1-1 ’ l

t

e ^r(T t)Et max I aSPaXa max Ste^Xt’^tj, Ste^{Xt/T l 1}    l<j<N

t

St ae

Pax S, S_t

exp max Xtt  , exp (XtT l≤j≤N , j                ,

— In a) J — 1

We can rewrite

max Xtt l≤j≤N ^{, j}

j-1

Xtji + max VXtk,tk. x, ’ 1    1

N-1

Xt,T

Xt,tl

+ 52 Xtk’tk+i

k=l where we use the convention that a sum is equal to 0 if the set of indices is empty. By the following simplification of the notation

Xk

tl + k-1^,tl + k ,

(iid Vk)

Yk

k

Xj,

j=i

Mk

max (0, Yi,.

.

.,Yk),

n

N

-

l,

we continue the consequence (3)

LP %

(t) =

ae

-

^r(T-t)S_t Et max

Cmax Sl-1

St

, exp (Xt,ti + Mn), exp (X_t,_tl + Yn

-

-

St

St

St

ae

-

ae

^r(T-t)Et

max

Cmax ^Sl-i

St

exp (Xt,tl + max (Mn, Yn

-

In a))

-

-

r(T—t)

E e^Zn

+ Et

max

^Sl-i

St

-

_eZn

-

,

|

+

A

{z

B

}

where we denoted Zk = X_t,_tl + max (Mk, Yk

-

ln a).

To calculate an expectation B we introduce the real-valued function f (x),

so that B can be expressed as

f (x) = E (ex

-

+^,

B = f ln

max

^Sl-i

St

.

As in [2] we consider a bilateral Laplace transform of the function f (x)

^b

f (0

∞

∞

e ^^xf (x) dx =

/ x

∞

∞

e

ξx

[ x   (ex

_J z=-^

-

ez)^z_n(z) dz dx,

where vZn (z) is the density function of the random variable Z_n.

Here we are allowed to

apply the Fubbini’s theorem and interchange integrals in (5):

^b

f (0

∞

J z = -^ L JX'

∞

e^-€x(ex

-

ez) dx vzn (z) dz =

∞ z=

-

∞

e

(^-i)z

L < -1

-

e^z

e

-

ξz

ξ

VZn^(z) ^dz =

1 p e« -1) Z=

-

∞

e

(^-iyz

VZn^{(z) dz} =

e« -1)

Ee

(^ — 1)Zn

.

It is worth noting that the inner integral in (6) converges if and only if Re £ > 1. Now we consider the characteristic function of Zn which is useful to compute both the expectation in (7) and the expectation A from (4), hzn (u) = E euZn,

so that A = hzn(1) and the expectation in (7) is equal to hzn (1 — £). Substituting the expression for Zn into (8) we obtain hzn (u) = E eu

(Xt,tl +max(Mn,Yn -In a))

= ht,tl (u) E e

u max(Mn ,Yn

—In a)

where we used the independence of increments of the process X , and ht,t_lis defined by (1). In the case a > 1 (a < 1 for the Call) the expectation in (9) becomes simply the characteristic function of the variable M_n,

E euMn and it is explained further how to calculate such an expression via Spitzer’s identity as proposed in [1]. The most interesting and understudied case is if a < 1 (a > 1 for the Call). Then we have a completely two-dimensional problem.

For the purpose to solve it we introduce the following complex-valued function

g(y; u) = E eu

max(Mn,Yn+y)

with its usual (one-sided) Laplace transform

∞ b (n; u) =

J y=0

e ^nyg(y; u) dy.

If we denote ^Yk,Mk (p,q) the joint density of the pair (Yk,Mk) then we can write g (n; u)

∞ y=0

e

—'

ηy

[/2 L

q=0

q

---------------------I

∞

Fubbini

/=o p:

---------------------I

∞

[/ y

∞

e

y=0

eu^max(q,p^+y)^Yn ,Mn (p,q) dpdq dy =

ⁿyeu^max(qp+y)  ^YnMn (p, q) dp dq =

/=o p:

euq

---------------------I

∞

[q ^Pe^—11ydy + y=0

∞

+e^up / y=q-p

e

^{(n u)y}dy ^Yn,Mn⁽p,q) dpdq =

/=o p:

euq

-

e

—'

n(q—p)

---------------------I

∞

η

+ e^up

e

(n—u)(q—p)"

η

-

u

^Yn,Mn ⁽p,q) dpdq =

/=o p:

1 e^uq₊

u

---------------------I

∞

η

П (n — u)

e^np—(,^—u)q^Y^Mn (p,q) _dpdq =

_EeuMn

η

+

u

П (n — u)

E _enYn^—(n^—u)Mn

We again remark that the second inner integral in (10) converges whenever Re η > Re u. We use the same method to calculate both expectations in (11) as proposed in [2] (and earlier in [1]).

If we denote

Xk (u) 4 E euMk, bk(u,v) 4 E euYk+vMk, ak(u) 4 h+Mk(u), ak(u,v) 4 h+,ti+k (u + v) + h-^ (-u) - 1, then the following corollary of the Spitzer’s identity holds

Due to the basic properties of the Laplace transform we consequently obtain

g(y; u) = Xn(u) + L¹

——-bn ⁽n, u ^-n) П (n — u)

(y).

And substituting all the expressions into (4) we finally get

Theorem. The price of a percentage discretely sampled lookback Put option is given by the following formulae

LP % (t)

St

ae"'^Tht,ti (1)

f m I r-1 Гbn ⁽n,^{1 —}n)] / , Л

I xn⁽¹⁾ + ^L ----7-----7Г-- ^(—In ^a)

V              L n ⁽n^— 1) J /

+

+ L^-1

+ L^-1

ht,ti^{(1 —}() e« — 1)

xn^{(1 —}^) +

^{(1 —}£) bn ⁽n,^{1 —}( ^—n) n ⁽n + ( ^—i)

(— ln a)

Cmax Sl-1

St

—1

Remark. In the formulae (13) above there are only terms dependent on the characteristics ht,t_l, h_t^±_l,tl+ksupposed to be known.