Ensuring Efficient Hedging of Barrier Options - CiteSeerX

Ensuring Efficient Hedging of Barrier Options Uwe Wystup Commerzbank Treasury and Financial Products Mainzer Landstr. 153 60261 Frankfurt am Main GERMANY phone: +49 - 69 - 136 - 41067 [email protected] http://www.mathfinance.de February 5, 2002

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Uwe Wystup - http://www.mathfinance.de

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Contents 1 Introduction 1.1 Digital options . . . . . . . . . . . . . . . . . . . . . 1.1.1 payoff . . . . . . . . . . . . . . . . . . . . . . 1.1.2 valuation in the Black-Scholes model . . . . . 1.1.3 Greeks . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Foreign-domestic symmetry . . . . . . . . . . 1.1.5 Relationship between cash, asset and vanilla . 1.1.6 Static hedge using vertical spreads . . . . . . 1.2 One-touch options . . . . . . . . . . . . . . . . . . . 1.2.1 payoff . . . . . . . . . . . . . . . . . . . . . . 1.2.2 other names . . . . . . . . . . . . . . . . . . . 1.2.3 rebates for knock-in options . . . . . . . . . . 1.2.4 payment date . . . . . . . . . . . . . . . . . . 1.2.5 Pricing . . . . . . . . . . . . . . . . . . . . . 1.2.6 Greeks . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Knock-out probability . . . . . . . . . . . . . 1.2.8 Properties of the first hitting time τB . . . . 1.2.9 Derivation of the value function . . . . . . . . 1.2.10 deviation of Market prices from the TV . . . 1.3 Double no-touch options . . . . . . . . . . . . . . . . 1.3.1 payoff . . . . . . . . . . . . . . . . . . . . . . 1.3.2 incentive . . . . . . . . . . . . . . . . . . . . . 1.3.3 Pricing . . . . . . . . . . . . . . . . . . . . . 1.4 Single Barrier Options . . . . . . . . . . . . . . . . . 1.4.1 value . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Greeks . . . . . . . . . . . . . . . . . . . . . . 1.5 Double barrier options . . . . . . . . . . . . . . . . . 1.5.1 Pricing . . . . . . . . . . . . . . . . . . . . . 1.5.2 Difference to two barrier options . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 3 3 5 5 5 6 6 6 6 6 6 7 8 8 9 10 10 10 10 10 11 13 14 16 17 17

2 Difficulties created by the characteristics of barrier options 2.1 Infinite deltas and gammas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 19

3 Exotic barrier options to minimise such 3.1 Rebates . . . . . . . . . . . . . . . . . . 3.2 Step options . . . . . . . . . . . . . . . . 3.3 Parisian options . . . . . . . . . . . . . .

difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 20 21 21

4 Applying static hedging to barrier options 4.1 Coping with the effects of liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Vega hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 23 25

5 Applying dynamic hedging to barrier options

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6 Over-hedging strategies 6.1 Constrained portfolios such as limited leverage . . . . . . 6.1.1 Model formulation and survey of super- replication 6.2 Application to reverse knock-out barrier options . . . . . 6.2.1 Constrained in-the-money knock-out call . . . . . . 6.2.2 Analytical Solutions . . . . . . . . . . . . . . . . .

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6.2.3 6.2.4 6.2.5 6.2.6 6.2.7 6.2.8 6.2.9 6.2.10 6.2.11 6.2.12 6.2.13

Digital Options . . . . . . . . . . . . . Reverse Barriers . . . . . . . . . . . . The Up-and-Out Call . . . . . . . . . Joint Formulae for all Reverse Barriers Comparative Statics . . . . . . . . . . One-Touch Digitals . . . . . . . . . . . Numerical Solutions . . . . . . . . . . Range Binaries . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . Interpretation as transaction cost . . . Interpretation as moving the barrier .

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4

Introduction

Theoretical Black-Scholes prices can differ from market prices substantially if the derivative has 1. kinks 2. jumps 3. barries because of the hedging difficulties created by their large delta and gamma values. We consider barrier options as a typical example of first-generation exotics. One problem for practitioners is to quantify the price of difficult hedging systematically. Adjusting the volatility is inconsistent since option values are not guaranteed to be montone in the volatility.

1.1 1.1.1

Digital options payoff v(T ) = I{φST ≥φK} cash-or-nothing, w(T ) = ST I{φST ≥φK} asset-or-nothing.

(1) (2)

In the cash-or-nothing case the payment of the fixed amount is in domestic currency, whereas in the assetor-nothing case the payment is in foreign currency. We use the abbreviations 1.1.2

valuation in the Black-Scholes model F

, E[ST |St = x] = xe(rd −rf )τ (forward price of the underlying) ,

d±

,

d˜±

,

x K

F K

σ2 2 τ

+ σθ± τ ln ± √ √ = σ τ σ τ x − σθ± τ ln K √ , σ τ ln

,

(3) (4) (5)

and obtain for the value functions v(x, K, T, t, σ, rd , rf , φ) = e−rd τ N (φd− ), w(x, K, T, t, σ, rd , rf , φ) = xe−rf τ N (φd+ ). 1.1.3

(6) (7)

Greeks

(Spot) Delta ∂v ∂x ∂w ∂x

n(d− ) √ xσ τ n(d+ ) = φe−rf τ √ + e−rf τ N (φd+ ) σ τ = φe−rd τ

(8) (9)

Gamma ∂2v ∂x2 ∂2w ∂x2

n(d− )d+ x2 σ 2 τ n(d + )d− = −φe−rf τ xσ 2 τ = −φe−rd τ

(10) (11)

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Theta ∂v ∂t ∂w ∂t

φn(d− )d˜− rd N (φd− ) + 2τ

−rd τ

= e

−rf τ

= xe

!

φn(d+ )d˜+ rf N (φd+ ) + 2τ

(12) ! (13)

Vega ∂v ∂σ ∂w ∂σ

d+ σ d− −rf τ = −φxe n(d+ ) σ

= −φe−rd τ n(d− )

(14) (15)

Volga ∂2v n(d− ) = −φe−rd τ (d− d2+ − d− − d+ ) ∂σ 2 σ2 n(d+ ) ∂2w = −φxe−rf τ (d+ d2− − d+ − d− ) ∂σ 2 σ2

(16) (17)

Rho ∂v ∂rd ∂v ∂rf ∂w ∂rd ∂w ∂rf

 √ φ τ n(d− ) −τ N (φd− ) + σ  √  φ τ n(d− ) = e−rd τ − σ  √  φ τ n(d+ ) −rf τ = xe σ   √ φ τ n(d+ ) −rf τ = −xe τ N (φd+ ) + σ = e−rd τ



(18) (19) (20) (21)

Dual Delta ∂v ∂K ∂w ∂K

φn(d− ) √ Kσ τ φn(d− ) = −e−rd τ √ σ τ = −e−rd τ

(22) (23)

Dual Gamma ∂2v ∂K 2 ∂2w ∂K 2

n(d− ) √ (σ τ − d− ) K 2 σ2 τ n(d− )d− = −φe−rd τ Kσ 2 τ = φe−rd τ

(24) (25)

Dual Theta ∂v ∂T

= −vt

(26)

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1.1.4

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Foreign-domestic symmetry

One can directly verify the relationship 1 1 1 v(x, K, T, t, σ, rd , rf , φ) = w( , , T, t, σ, rf , rd , −φ). x x K

(27)

The reason is that the value of an option can be computed both in a domestic as well as in a foreign scenario. We consider the example of St modeling the exchange rate of EUR/USD. In New York, the cash-or-nothing digital call option costs v(x, K, T, t, σ, rusd , reur , 1) USD and hence v(x, K, T, t, σ, rusd , reur , 1)/x EUR. If it ends in the money, the holder receives 1 USD. For a Frankfurt-based holder of the same option, receiving one USD means receiving asset-or-nothing, where he uses reciprocal values for spot and strike and for him domestic currency is the one that’s foreign to the New Yorker and vice versa. Since St and S1t have the same volatility, the New York value and the Frankfurt value must agree, which leads to (27).

1.1.5

Relationship between cash, asset and vanilla

The simple equation of payoffs φ(w(T ) − Kv(T )) = [φ(ST − K)]+

(28)

leads to the formula vanilla(x, K, T, t, σ, rd , rf , φ) = φ[w(x, K, T, t, σ, rd , rf , φ) − Kv(x, K, T, t, σ, rd , rf , φ)]. 1.1.6

(29)

Static hedge using vertical spreads

The mathematical derivative of the positive part function I{φSt ≥φK} = lim ↓0

 1  (φ(ST − (K − φ)))+ − (φ(ST − (K + φ)))+ 2

(30)

leads to an approximate static hedge (and hence price) v(x, K, T, t, σ, rd , rf , φ) ≈ 1 [vanilla(x, K − φ, T, t, σ, rd , rf , φ) − vanilla(x, K + φ, T, t, σ, rd , rf , φ)] 2

(31)

for small  > 0. In practice, arbitrarily small  corresponds to arbitrarily large nominal amounts of the vanilla options and can thus not be chosen arbitrarily small. Furthermore, there will be different volatilities for the bid and ask price of the vanilla options, which lead to a more realistic pricing for digital options using this approximation. Greeks in the static hedge Static hedges normally perform well hedging the actual model variable risk like delta, gamma and theta. In this static hedge even the model parameter uncertainty vega is hedged. The hedge vega is given by √

τ xe−rf τ

n(dK−φ ) − n(dK+φ ) + + , 2 2 F ln K ± σ2 τ √ dK , . ± σ τ

(32) (33)

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Replacing the difference quotient by its derivative at K we obtain n(dK−φ ) − n(dK+φ ) + + 2 √ −1 √ ≈ φ τ xe−rf τ · n(d+ )d+ Kσ τ d+ = −φe−rd τ n(d− ) , σ which is the vega of the digital option. √

1.2 1.2.1

τ xe−rf τ

(34) (35) (36)

One-touch options payoff RI{τB ≤T } ,

(37)

τB , inf{t ≥ 0 : ηSt ≤ ηB}.

(38)

This type of option pays a domestic cash amount R if a barrier B is hit any time before expiry. We use the binary variable η to describe whether B is a lower barrier (η = 1) or an upper barrier (η = −1). The stopping time τB is called the first hitting time. 1.2.2

other names

• rebate portion of a knock-out barrier option • American cash-or-nothing digital option • one-touch-digital • hit options 1.2.3

rebates for knock-in options

The modified payoff RI{τB ≥T } describes a rebate which is paid if a knock-in-option has not knocked in by the time it expires and can be valued similarly simply by exploiting the identity RI{τB ≤T } + RI{τB ≥T } = R. 1.2.4

(39)

payment date

We will further distinguish whether the rebate is paid at hit (ω = 0) or at end (ω = 1) and use the abbreviations q 2 + 2(1 − ω)r , ϑ− , θ− (40) d e± 1.2.5

,

± log

x B

− σϑ− τ √ . σ τ

(41)

Pricing

The value of the one-touch option turns out to be   − −   θ− +ϑ   θ− −ϑ σ σ B B v(t, x) = Re−ωrd τ  N (−ηe+ ) + N (ηe− ) . x x Note that ϑ− = |θ− | for rebates paid at end (ω = 1).

(42)

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1.2.6

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Greeks

Delta  −     θ− +ϑ σ Re−ωrd τ  B η vx (t, x) = − (θ− + ϑ− )N (−ηe+ ) + √ n(e+ )  x σx τ  −     θ− −ϑ σ η B + (θ− − ϑ− )N (ηe− ) + √ n(e− )  x τ

(43)

Theta vt (t, x)

  − −   θ− +ϑ   θ− −ϑ σ σ ηRe−ωrd τ  B B = ωrd v(t, x) + n(e+ )e− − n(e− )e+  2τ x x ηRe−ωrd τ = ωrd v(t, x) + στ (3/2)



B x

−  θ− +ϑ σ

 n(e+ ) log

B x

 .

(44)

The computation exploits the identities (61), (62) and (63) derived below. Gamma Gamma can be obtained using vxx =

vxx (t, x)

=

2 σ 2 x2 [rd v

2Re−ωrd τ · 2 2  σ x θ +ϑ  B  − σ −  x

− vt − (rd − rf )xvx ] and turns out to be

(45)   θ− + ϑ− N (−ηe+ ) rd (1 − ω) + (rd − rf ) σ

−  θ− −ϑ   σ B θ− − ϑ− + N (ηe− ) rd (1 − ω) + (rd − rf ) σ x −     θ− +ϑ σ r d − rf e− B √ + n(e+ ) − +η τ x σ τ  −   θ− −ϑ   σ B e+ rd − rf  √ +η n(e− ) + . x τ σ τ 



Vega To compute vega we use the identities ∂θ− ∂σ ∂ϑ− ∂σ ∂e± ∂σ ∂ θ− ± ϑ− A± , ∂σ σ

θ+ , σ θ− θ+ = − , σϑ−

= −

log B θ θ √ √x + − + τ , 2 σϑ− σ τ    1 θ− θ+ = − 2 θ+ + θ− ± + ϑ− , σ ϑ− = ±

(46) (47) (48) (49)

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and obtain vσ (t, x)

1.2.7

= Re−ωrd τ ·  θ +ϑ     B  − σ −  B ∂e+ N (−ηe+ )A+ log − ηn(e+ )  x x ∂σ  −    θ− −ϑ    σ B B ∂e− + N (ηe− )A− log + ηn(e− ) . x x ∂σ 

(50)

Knock-out probability

The risk-neutral probability of knocking out is given by   1 P[τB ≤ T ] = E I{τB ≤T } = erd T v(0, S0 ). R 1.2.8

(51)

Properties of the first hitting time τB

As derived, e.g., in [26], the first hitting time τ˜ , inf{t ≥ 0 : θt + W (t) = x}

(52)

of a Brownian motion with drift θ and hit level x > 0 has the density   x (x − θt)2 P[˜ τ ∈ dt] = √ exp − dt, t > 0, 2t t 2πt

(53)

the cumulative distribution function  P[˜ τ ≤ t] = N

θt − x √ t



2θx

+e

 N

−θt − x √ t

 , t > 0,

(54)

the Laplace-transform n o p Ee−α˜τ = exp xθ − x 2α + θ2 , α > 0, x > 0,

(55)

and the property  P[˜ τ < ∞] =

1 e2θx

if θ ≥ 0 . if θ < 0

(56)

For upper barriers B > S0 we can now rewrite the first passage time τB as τB

= =

inf{t ≥ 0 : St = B}    1 B inf t ≥ 0 : Wt + θ− t = log . σ S0

(57)

The density of τB is hence P[τ˜B ∈ dt] =

1 σ

       ( σ1 log SB − θ− t)2  log SB0 0 √ exp − , t > 0.   2t t 2πt

(58)

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1.2.9

10

Derivation of the value function

Using the density (58) the value of the paid-at-end (ω = 1) upper rebate (η = −1) option can be written as   v(T, S0 ) = Re−rd T E I{τB ≤T } (59)       Z T 1 log B  ( σ1 log SB − θ− t)2  σ S0 0 −rd T √ = Re exp − dt.   2t t 2πt 0 To evaluate this integral, we introduce the notation e± (t) ,

± log

S0 B

− σθ− t √ σ t

(60)

and list the properties   2 1 B e− (t) − e+ (t) = √ log , S0 tσ  − 2θσ− B n(e+ (t)) = n(e− (t)), S0 ∂e± (t) e∓ (t) = . ∂t 2t

(61) (62) (63)

We evaluate the integral in (59) by rewriting the integrand in such a way that the coefficients of the exponentials are the inner derivatives of the exponentials using properties (61),(62) and (63),       B 1 2 Z T 1 log B  log − θ t) ( − σ S0 σ S0 √ dt exp −   2t t 2πt 0  Z T 1 B 1 = log n(e− (t)) dt (3/2) σ S0 t 0 Z T 1 = n(e− (t))[e− (t) − e+ (t)] dt 2t 0   2θσ− Z T e+ (t) B e− (t) dt = − n(e− (t)) + n(e+ (t)) 2t S0 2t 0   2θσ− B = N (e+ (T )) + N (−e− (T )). (64) S0 The computation for lower barriers (η = 1) is similar.

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1.2.10

11

deviation of Market prices from the TV

Figure 1: Comparison of value adjustments of one-touch options in Heston’s model

1.3 1.3.1

Double no-touch options payoff I{L≤min[0,Te ] St B K≤B K>B K≤B K>B K≤B K>B K≤B K>B K≤B K>B K≤B

combination A1 A2 − A3 + A4 A1 − A2 + A4 A3 A3 A1 − A2 + A4 A2 − A3 + A4 A1 0 A1 − A2 + A3 − A4 A2 − A4 A1 − A3 A1 − A3 A2 − A4 A1 − A2 + A3 − A4 0

Greeks

delta A1 A2 A3

A4

= φf N (φX) √ = φf N (φx1 ) + f n(x1 )z/σ τ " #  2λ−2  2 √ −2µ B B = φ 2 xf N (ηy) − KdN (η(y − σ τ )) σ x x x  2λ B −φ f N (ηy) x #  2λ−2 "  2 √ B B = φ xf N (ηy1 ) − KdN (η(y1 − σ τ )) x x

(93)

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gamma A1 A2 C3 B3 A3 C4 B4 A4

√ = f n(X)/(xσ τ ) √ √ = f n(x1 )/(xσ τ )(1 − zx1 /σ τ ) " #  2λ−2  2 √ B B = φ xf N (ηy) − KdN (η(y − σ τ )) x x  2λ −2µ B = C3 − φ f N (ηy) 2 σ x x  √  2µ = (C3 /x − B3 ) + φf B 2λ /x2λ+1 2λN (ηy) + ηn(y)/σ τ σ2 x #  2λ−2 "  2 √ B B N (ηy1 ) − KdN (η(y1 − σ τ )) = φ xf x x  2λ  2λ √ −2µ B B = C4 − φ f N (ηy1 ) − φηf n(y1 )z/σ τ 2 σ x x x  √  2µ (C4 /x − B4 ) + φf B 2λ /x2λ+1 2λN (ηy1 ) + ηn(y1 )/σ τ ) = σ2 x  2λ √ √ B +φηf zn(y1 ) /(xσ τ )(2λ − y1 /σ τ ) x

(94)

theta A1 A2

A3

A4

√ √ 1 = − σxf n(X)/ τ + φxf N (φX)rf − φKdN (φ(X − σ τ ))rd 2 √ √ 1 = − σxf n(x1 )K/(B τ ) + φxf N (φx1 )rf − φKdN (φ(x1 − σ τ ))rd 2 −xf n(x1 )zy1 /(2τ )  2λ B 1 √ xf ηn(y) σ/ τ = −φ x 2 #  2λ−2 "  2 √ B B +φ rf xf N (ηy) − rd KdN (η(y − σ τ )) x x  2λ   √ B 1 = −φ xf ηn(y1 ) x1 /(2τ )z + σK/( τ B) x 2 #  2λ−2 "  2 √ B B +φ rf xf N (ηy1 ) − rd KdN (η(y1 − σ τ )) x x

(95)

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vega A1 A2 B3 A3 B4 A4

√ = xf n(X) τ √ = xf n(x1 )( τ − x1 z/σ) #  2λ−2 "  2 √ B B = φ xf N (ηy) − KdN (η(y − σ τ )) x x    2λ √ −4 B B = log (rd − rf )B3 + φ xf ηn(y) τ 3 σ x x #  2λ−2 "  2 √ B B = φ xf N (ηy1 ) − KdN (η(y1 − σ τ )) x x   −4 B = log (rd − rf )B4 σ3 x   2λ  √ B K√ +φ xf ηn(y1 ) ( τ − y1 /σ)z + τ x B

(96)

(97)

joint density for final time value and running extremum As derived, e.g., in [26], joint density f (x, y) for a Brownian motion with drift and its running extremum (η = +1 for a maximum and η = −1 for a minimum)   W (T ) + θ− T, η min [η(W (t) + θ− t)] (98) 0≤t≤T

1

2

f (x, y) = −ηeθ− x− 2 θ− T

derivation of the value function the following integral

  2(2y − x) (2y − x)2 √ exp − , 2T T 2πT ηy ≤ min(0, ηx).

Using the density (99) the value of a barrier option can be written as

barrier(S0 , σ, rd , rf , K, B, T )   = e−rd T E [φ(ST − K)]+ I{ηSt >ηB,0≤t≤T } Z x=−∞ Z + −rd T = e [φ(S0 eσx − K)] I{ηy>η 1 log x=−∞

(99)

ηy≤min(0,ηx)

σ

(100)

B S0

} f (x, y) dy dx.

Further details on how to evaluate this integral can be found in [26].

1.5

Double barrier options

payoff +

(φ (STe − K)) I{L