Multi-state Lookback Options presented by Yue Kuen Kwok Hong

Multi-state Lookback Options

presented by

Yue Kuen Kwok

Hong Kong University of Science & Technology

* Joint work with Hoi Ying Michael Wong and Min Dai

Outline

1. Rollover hedging strategies

2. Lookback spread options

3. Quanto lookback options — early exercise policy of American feature

Lookback options

asset price

time

T0 t T

],[ 0 TTS

],[ 0 TTS

S[T0, T ] = maxu∈[T0,T ]

Su and S[T0, T ] = minu∈[T0,T ]

Su

Continuously monitored floating strike lookback call

cf`(ST , T ) = ST − S[T0, T ].

Continuously monitored fixed strike lookback call

cfix(ST , T ) = (S[T0, T ] − K)+, K is the strike price.

Rollover hedging strategy

How to hedge the floating strike lookback call?

• At any time, hold a European vanilla call with strike equals S[T0, t].

• To replicate the terminal payoff, replace the call with a new call whose

strike equals the newly realized minimum asset value.

Put-call parity (European options)

• A European call can be replicated fully by a forward and a European put

(same maturity, strike price/delivery price). This is a distribution free result.

• However, the forward alone gives only a sub-replication to the call.

Sub-replication and replenishing premium

• A partial replicating portfolio whose terminal value always stays equal or

below the terminal value of the derivative to be replicated is said to be a sub-

replicating portfolio (choose a portfolio whose value is readily available).

• To determine the additional premium for acquiring extra assets on top of

the sub-replication that are required to achieve the full replication.

• The total premium for these extra assets is termed the replenishing pre-

mium.

Divide the interval [O, K] into n sub-intervals,

where n∆ξ = K.

Extra replenishment required when ST falls further from (j + 1)∆ξ to j∆ξ.

This is achieved by long holding a put with strike (j +1)∆ξ and short selling a

put with strike j∆ξ. To leading order in ∆ξ, present value of this jth portfolio

= e−rτ[(j + 1)∆ξ − ST ]− [j∆ξ − ST ]P [ST ≤ ξj], where valuation is performed

under the risk neutral world.

Total premium of replenishment = e−rτ limn→∞

n−1∑

j=0

P [ST ≤ ξj]∆ξ

= e−rτ∫ K

0P [ST ≤ ξ] dξ.

This is just the value of the put option.

Alternative mathematical argument

Replenishing premium = put value = e−rτE[

(K − ST )1ST≤K]

= e−rτ∫

Ω(K − ST )1ST (ω)≤K dP (ω).

Applying the relation

(K − ST )1ST (ω)≤K =∫ K

01ST (ω)≤ξ dξ,

we obtain

put value = e−rτ∫

Ω

∫ K

01ST (ω)≤ξ dξdP (ω)

= e−rτ∫ K

0

∫

Ω1ST (ω)≤ξ dP (ω)dξ (by Fubini’s theorem)

= e−rτ∫ K

0P [ST ≤ ξ] dξ.

Non-uniqueness of the choice of sub-replicating portfolio

Suppose the sub-replicating portfolio is the null portfolio.

Replenishment is required when ST increases from K + j∆ξ to K + (j + 1)∆ξ.

This is achieved by

(i) long holding a call with strike K + j∆ξ,

(ii) short selling a call with strike K + (j + 1)∆ξ.

Total replenishment premium = e−rτ∞∑

j=1

P [ST > K + j∆ξ]∆ξ

= e−rτ∫ ∞

KP [ST > ξ] dξ.

Put-call parity relations for floating strike and fixed strike lookbacks

Terminal payoff of floating strike lookback call = cf`(ST , T ) = ST − S[T0, T ],

where S[T0, T ] = min[S[T0, t], S[t, T ]).

Here, S[t, T ] determines the occurrence of under replication.

Sub-replicating portfolio is a forward with delivery price S[T0, t].

cf`(S, t;S[T0, t]) = S − e−rτS[T0, t] + e−rτ∫ S[T0,t]

0P [S[t, T ] ≤ ξ] dξ

= S − e−rτS[T0, t] + pfix(S, t;S[T0, t]).

Strike bonus premium revisited

From cf` = forward + Pfix and call = forward + put,

strike bonus premium = cf`(S, t;S[T0, t]) − c(S, t;S[T0, t])

= pfix(S, t;S[T0, t]) − p(S, t;S[T0, t])

= e−rτ∫ S[T0,t]

0P (S(t, T ] ≤ ξ] − P [ST ≤ ξ] dξ

= e−rτ∫ S(T0,t]

0P (S[t, T ] ≤ ξ < ST ] dξ.

Remark Simplication into a single distribution is feasible since ST ≥ S[t, T ].

Financial interpretation

• Under replication occurs only when S[t, T ] < min(ST , S[T0, t]).

• To immunize under replication over [j∆ξ, (j + 1)∆ξ], the present value of

the replenishment premium is

e−rτP [S[t, T ] < ξ ≤ min(ST , S[T0, t])]∆ξ.

• Total replenishment premium

= e−rτ∫ ∞

0P [S[t, T ] < ξ ≤ min(ST , S[T0, t]) dξ

= e−rτ∫ S[T0,t]

0P [S[t, T ] < ξ ≤ ST ] dξ.

Probability distribution functions under lognormal processes

dS

S= rdt + σdZ.

Write X = lnS so that X(t) =

(

r − σ2

2

)

dt + σdZ.

Define X(t) = min0≤u≤t

X(u) and X(t) = max0≤u≤t

X(u):

P [X(t) ≥ x, X(t) ≥ x] = G(x, x, t;α)

= N

(

−x + αt

σ√

t

)

− e2αx

σ2 N

(

−x + 2x + αt

σ√

t

)

P [X(t) ≥ x] = G(x, x, t;α)

P [X(t) ≤ x, X(t) ≤ x] = G(−x,−x, t;−α)

P [X(t) ≤ x] = G(−x,−x, t;−α)

Floating strike lookback call and straddle

Try to replicate a floating strike lookback call by a straddle, the amount of

mis-replication = cf`(S, t;S[T0, t]) − [c(S, t;S[T0, t] + p(S, t;S[T0, t])]

= e−rτ∫ S[T0,t]

0P [S[t, T ] ≤ ξ < ST ] − P [ST ≤ ξ] dξ.

The mis-replication becomes zero when the asset price is lognormal and α =

r − σ2

2= 0.

Remark When full replication is not achieved, the rollover strategy of hedging

amounts to replacing the straddle with a new strike set at the newly

realized minimum asset value.

percentage of mis-replication percentage of mis-replicationasset value by a straddle by a vanilla call

α = 0.02 α = −0.02 α = 0.02 α = −0.02

60 4.9220 −6.4519 40.7589 46.774065 2.0786 −2.5823 19.6542 24.832070 0.7871 −1.8045 8.3746 10.740175 0.2874 −0.7944 3.3961 5.042480 0.1044 −0.2731 1.3554 1.444785 0.0381 −0.0884 0.5385 0.835190 0.0140 −0.0831 0.2137 0.464895 0.0052 −0.0734 0.0847 0.0846100 0.0019 −0.0670 0.0336 0.0585

Table 1 A floating strike lookback call can be partially replicated either by a

vanilla call or a straddle, all options are struck at S[T0, t]. The en-

tries show the percentage of mis-replication (ratio of mis-replication

amount to option value) by the vanilla call and the straddle at vary-

ing level of asset value and α. The other parameter values used in

the calculations are: S[T0, t] = 60, r = 3% and τ = 1.

60 70 80 90 100 110 120−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

S

delta

of r

eple

nish

ing

prem

ium

Lookback minus straddle

Lookback minus call

Lookback minus forward

The curves show the plotting of the delta of the replenishing premium against

the asset value S, corresponding to different replication strategies adopted in

the replication of a floating strike lookback call option.

Discretely monitored floating strike lookback call

At the current time, S[1, k] = min(St1, · · · , Stk).

cdisf` (ST , T ) = ST − min(St1, St2, · · · , Stn).

Choose the forward with delivery price S[1, k] as the sub-replicating instrument,

then

cdisf` (S, t;S[1, k]) = S − e−rτS[1, k] + e−rτ

∫ S[1,k]

0P [S[k + 1, n] ≤ ξ] dξ.

The distribution function P [S[k + 1, n] ≤ ξ] can be expressed as

P [S[k + 1, n] ≤ ξ] =n∑

j=k+1

E1Stj≤ξ,Stj

/Sti≤1 for all i6=j,k+1≤i≤n.

One-asset lookback spread

csp(ST , T ; k) = (S[T0, T ] − S[T0, T ] − K)+

Sub-replicating portfolio = cf`(S, t;S[T0, t]) + pf`(S, t;S[T0, t]) − Ke−rτ . Since

S[T0, T ] − S[T0, T ] − K = max(S[T0, t], S[T0, T ]) − min(S[T0, t], S[t, T ]) − K

≥ S[T0, t] − S[T0, t] − K,

so the lookback spread option is guaranteed to expire in-the-money if it is

currently in-the-money.

If the lookback spread option is currently out-of-the-money, the terminal payoff

of the sub-replicating portfolio would be less than that of the lookback spread

option if the lookback spread option expires out-of-the-money, that is,

max(S[T0, t], S[t, T ]) − min(S[T0, t], S[t, T ]) − K < 0.

Treat max(S[T0, t], S[t, T ]) as the stochastic state variable that determines full

or under replication and min(S[T0, t], S[t, T ]) + K as the effective strike price.

replenishing premium

= e−rτ∫ ∞

0P (max(S[T0, t], S[t, T ]) < ξ ≤ min(S[T0, t], S[t, T ]) + K) dξ

= e−rτ∫ S[T0,t]+K

S[T0,t]P (S[t, T ] < ξ ≤ S[t, T ] + K) dξ.

In summary,

(i) S[T0, t] − S[T0, t] − K ≥ 0

csp(S, t;S[T0, t], S[T0, t]) = cf`(S, t;S[T0, t]) + pf`(S, t;S[T0, t]) − Ke−rτ ;

(ii) S[T0, t] − S[T0, t] − K < 0

csp(S, t;S[T0, t], S[T0, t]) = cf`(S, t;S[T0, t]) + pf`(S, t;S[T0, t]) − Ke−rτ

= +e−rτ∫ S[T0,t]+K

S[T0,t]P (S[t, T ] < ξ ≤ S[t, T ] + K) dξ.

Probability distribution under lognormal asset price process

P (X ≥ x, X ≤ y) =∞∑

n=−∞e[2nα(y−x)]/σ2

[

N

(

y − αt − 2n(y − x)

σ√

t

)

− N

(

x − αt − 2n(y − x)

σ1

√t

)]

−e2αx/σ2[

N

(

y − αt − 2n(y − x) − 2x

σ√

t

)

− N

(

x − αt − 2n(y − x) − 2x

σ√

t

)]

Two-asset lookback spread option

csp(S1,T , S2,T , T ;K) = (S1[T0, T ] − S2[T0, T ] − K)+

Since S1[T0, T ]− S2[T0, T ]−K = (S1[T0, T ]− S1,T )+ (S2,T − S2[T0, T )]+ S1,T −S2,T − K, the sub-replicating portfolio would consist of long holding of one

European floating strike lookback put on asset 1, one European floating strike

lookback call on asset 2, one unit of asset 1 and short holding of one unit of

asset 2 and a riskless bond of par value K.

It is guaranteed to expire in-the-money if it is currently in-the-money; and

the sub-replicating portfolio will expire with a terminal payoff below that of the

lookback spread option if the lookback spread option expires out-of-the-money.

Formally,

csp = e−rτ∫ 0

−∞dx1

∫ ∞

0dx2csp(S1,T , S2,T , T ;K)

[

− ∂2G2

∂x1∂x2

]

P [X1(t) ≤ x1, X2(t) ≤ x2] = G2(x1, x2, t, α1, α2, ρ)

= ea1x1+a2x2+btF (r0, θ0, t)

where

F (r0, θ0, t) =2

αte−r2

02t

∞∑

n=1

sin

(

nπθ0α

)∫ α

0sin

(

nπθ

α

)

gn(θ) dθ

gn(θ) =∫ ∞

0re

−r2

2t ed1r sin(θ−α)−d2r cos(θ−α)Inπα

(

rr0t

)

dr.

α =

tan−1(

−√

1−ρ2

ρ

)

, if ρ < 0

π + tan−1(

−√

1−ρ2

ρ

)

, otherwise

θ0 =

tan−1(

−Z2

√1−ρ2

Z1−Z2ρ

)

, if − Z2

√1−ρ2

Z1−Z2ρ > 0

π + tan−1(

−Z2

√1−ρ2

Z1−Z2ρ

)

, otherwise

r0 =Z2

sin θ0, Z1 =

x1

σ1, Z2 =

x2

σ2,

a1 =α1σ2 − ρα2σ1

(1 − ρ2)σ21σ2

, a2 =α2σ1 − ρα1σ2

(1 − ρ2)σ1σ22

,

b = −α1a1 − α2a2 +1

2σ21a2

1 + ρσ1σ2a1a2 +1

2σ22a2

2,

d1 = a1σ1 + ρa2σ2, d2 = a2σ2

√

1 − ρ2.

In summary,

(i) S1[T0, t] − S2[T0, t] − K ≥ 0 (currently in-the-money)

csp(S1, S2, t;S1[T0, t], S2[T0, t])

= pf`(S1, t;S1[T0, t]) + cf`(S2, t;S2[T0, t]) + S1 − S2 − Ke−rτ ;

(ii) S1[T0, t] − S2[T0, t] − K < 0 (currently out-of-the-money)

csp(S1, S2, t;S1[T0, t], S2[T0, t])

= pf`(S1, t;S1[T0, t]) + cf`(S2, t;S2[T0, t]) + S1 − S2 − Ke−rτ

+e−rτ∫ S2[T0,t]+K

S1[T0,t]P [S1[t, T ] < ξ ≤ S2[t, T ] + K] dξ.

Semi-lookback option

V 2semi(S1, S2, T ) = (S2[T0, T ] − S1,T − K)+

Since S2[T0, T ]−S1,T−K = (S2[T0, T ]−S2,T )+S2,T−S1,T −K, the sub-replicating

portfolio is chosen to consist of long holding of one European floating strike

lookback put and one unit of forward on asset 2, and short holding of one unit

of asset 1, all instruments having the same maturity.

replenishing premium = e−rτ∫ ∞

0P [max(S2[T0, t], S2[t, T ]) < ξ ≤ S1,T + K] dξ

= e−rτ∫ ∞

S2[T0,t]P [S2[t, T ] < ξ ≤ S1,T + K] dξ.

V 2semi(S1, S2, t;S2[T0, t]) = pf`(S2, t;S2[T0, t]) + S2 − S1 − Ke−rτ

+ e−rτ∫ ∞

S2[T0,t]P [S2[t, T ] < ξ ≤ S1,T + K] dξ.

Joint probability distribution of extremum value on one asset

and terminal value of another asset

P [X1(t) ≥ x1, X1(t) ≥ x1, X2(t) ≤ x2]

= Gsemi(x1, x2, x1, t;α1, α2, ρ)

= N2

(

−x1 + α1t

σ1

√t

,x2 − α2t

σ2

√t

;−ρ

)

− e

2α1x1σ21 N2

(

−x1 + x1 + α1t

σ1

√t

,x2 − α2t

σ2

√t

;−ρ

)

P [X1(t) ≥ x1, X2(t) ≤ x2] = Gsemi(x1, x2, x1, t;α1, α2, ρ)

P [X1(t) ≤ x1, X1(t) ≤ x1, X2(t) ≤ x2] = Gsemi(−x1, x2,−x1, t;−α1, α2,−ρ)

P [X1(t) ≤ x1, X2(t) ≤ x2] = Gsemi(−x1, x2,−x1, t;−α1, α2,−ρ)

Multi-asset semi-lookback option

Let V nsemi(S1, S2, · · · , Sn, t;S1[T0, t]) denote the value of the multi-asset semi-

lookback option whose terminal payoff is given by

max(max(S2,T , · · · , Sn,T ) − S1[T0, T ], 0).

The sub-replicating portfolio is cn−1max(S2, · · · , Sn, t) + cf`(S1, t; S1[T0, t]) − S1,

where cn−1max(S2, · · · , Sn, t) denotes the value of the (n − 1)-asset maximum call

option with zero strike.

Under replication at maturity by the sub-replicating portfolio occurs when

max(S2,T , · · · , Sn,T ) < S1[T0, T ] = min(S1[T0, t], S1[t, T ]).

V nsemi(S1, S2, · · · , Sn, t;S1[T0, t])

= cn−1max(S2, · · · , Sn, t) + cf`(S1, t;S1[T0, t]) − S1

+ e−rτ∫ S1[T0,t]

0P [max(S2,T , · · · , Sn,T ) < ξ ≤ S1[t, T ]] dξ.