the impact of fx options on the spot market and the cost of …€¦ · overview hedging the......

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Impact on the Spot . . .

Cut-Off Time and . . .

Delta-Hedging a . . .

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The Impact of FX Options on theSpot Market and the Cost of Delayed

Currency Fixing Announcements

Uwe Wystup

MathFinance AG

mailto:[email protected]

March 15, 2006

http://www.mathfinance.de/

mailto:[email protected]

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Abstract

In Foreign Exchange Markets vanilla and barrier options are tradedfrequently. In this presentation we discuss how large barrier con-tracts traded in the OTC market can influence and explain some ofthe jump-like behavior in the spot market. We furthermore showhow clients insisting on official spot fixing sources give traders ofbarrier options a tough time hedging these options. The marketstandard is a cutoff time of 10:00 a.m. in New York for the strikeof vanillas and a knock-out event based on a continuously observedbarrier in the inter bank market. However, many clients, particu-larly from Italy, prefer the cutoff and knock-out event to be based onthe fixing published by the European Central Bank on the ReutersPage ECB37. These barrier options are called discretely monitoredbarrier options. While these options can be priced in several mod-els by various techniques, the ECB source of the fixing causes twoproblems. First of all, it is not tradable, and secondly it is publishedwith a delay of about 10 - 20 minutes. We examine here the effectof these problems on the hedge of those options and consequentlysuggest a cost based on the additional uncertainty encountered.


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1. Overview

1.1. Agenda

1. Hedging the reverse knock-out (RKO)


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1. Overview

1.1. Agenda


2. Impact on the spot market


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1. Overview

1.1. Agenda



3. Cutoffs and fixings


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1. Overview

1.1. Agenda




4. Delta-hedging a short position of a discretely mon-itored RKO


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1. Overview

1.1. Agenda




4. Delta-hedging a short position of a discretely mon-itored RKO

5. Analyze cost due to unknown spot value at knock-out or expiration


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2. Hedging the Reverse Knock-out (RKO)

2.1. Option Payoff

F (ST )e.g.= max(0, ST −K)


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2. Hedging the Reverse Knock-out (RKO)

2.1. Option Payoff

F (ST )e.g.= max(0, ST −K)

today t=0 Expiration time t=T

TSK

Payoff Profile of a Call

45°


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2.2. Option Value

The value is

v(St, t) = e−rdTIE[F (ST )]e.g.= e−rdTIE[max(0, ST −K)]

= . . . Black-Scholes formula


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2.2. Option Value

The value is

v(St, t) = e−rdTIE[F (ST )]e.g.= e−rdTIE[max(0, ST −K)]

= . . . Black-Scholes formula

vanilla

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

1.09

1.10

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22

1.23

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1.30

1.31

1.32

spot

valu

e


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2.3. Delta Hedging the Short Option

The delta is the slope of the value function

∂

∂St

v(St, t) = . . . Black-Scholes delta


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2.3. Delta Hedging the Short Option

The delta is the slope of the value function

∂

∂St

v(St, t) = . . . Black-Scholes delta

vanilla

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

1.09

1.10

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22

1.23

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1.30

1.31

1.32

spot

del

ta


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∂

∂St

v(St, t) = delta


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∂

∂St

v(St, t) = delta

• hedge a short option with value v(St, t)


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∂

∂St

v(St, t) = delta


• by buying ∂∂St

v(St, t) of the underlying


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∂

∂St

v(St, t) = delta




• for a short EUR-call USD put buy delta EUR


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∂

∂St

v(St, t) = delta




• for a short EUR-call USD put buy delta EUR

• ... a quantity between 0 and 100%


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2.4. Reverse Knock-Out (RKO) Call

is an option with a barrier in the money paying off

max(0, ST −K)II{max0≤u≤T Su<B}


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2.4. Reverse Knock-Out (RKO) Call

is an option with a barrier in the money paying off

max(0, ST −K)II{max0≤u≤T Su<B}

today t=0 Expiration time t=T

TSK

Payoff Profile of a RKOCall

45°

B


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2.5. Reverse Knock-Out Call Value

v(St, t) = e−rdTIE[max(0, ST −K)II{max0≤u≤T Su<B}

]= . . . some formula


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2.5. Reverse Knock-Out Call Value

v(St, t) = e−rdTIE[max(0, ST −K)II{max0≤u≤T Su<B}

]= . . . some formula

barrier

-0.0020

0.0000

0.0020

0.0040

0.0060

0.0080

0.0100

0.0120

0.0140

0.0160

0.01801.

09

1.10

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22

1.23

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1.30

1.31

1.32

spot

valu

e


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2.6. Reverse Knock-Out Call Delta

Hedge a short position by buying

delta =∂

∂St

v(St, t) of the underlying.


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2.6. Reverse Knock-Out Call Delta

Hedge a short position by buying

delta =∂

∂St

v(St, t) of the underlying.

barrier

-0.4000

-0.3000

-0.2000

-0.1000

0.0000

0.1000

0.2000

0.30001.

09

1.10

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22

1.23

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1.30

1.31

1.32

spot

del

ta


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Accross time this looks like

1.0

9

1.1

1

1.1

2

1.1

4

1.1

6

1.1

8

1.2

0

1.2

2

1.2

4

1.2

6

1.2

8

1.3

0

1.3

2

90

81

72

63

54

4637

2819

10

-6.0000

-5.0000

-4.0000

-3.0000

-2.0000

-1.0000

0.0000

1.0000

del

ta

time to expiration (days)

barrier


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Accross time this looks like

1.0

9

1.1

1

1.1

2

1.1

4

1.1

6

1.1

8

1.2

0

1.2

2

1.2

4

1.2

6

1.2

8

1.3

0

1.3

2

90

81

72

63

54

4637

2819

10

-6.0000

-5.0000

-4.0000

-3.0000

-2.0000

-1.0000

0.0000

1.0000

del

ta

time to expiration (days)

barrier

Delta approaches −∞ near maturity for a spot near thebarrier


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3. Impact on the Spot Market

• by valuation theory assumptions: none


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• for vanilla options: none


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• for the RKO: delta = −1000% or less is possible


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• for the RKO: delta = −1000% or less is possible

• implications ?


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3.1. How Large Barrier Contracts Affect the Mar-

ket – Example

EUR/USD RKO call strike 1.2000 and barrier 1.3000


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3.1. How Large Barrier Contracts Affect the Mar-

ket – Example

EUR/USD RKO call strike 1.2000 and barrier 1.3000

barrier value function

-0.02

0.00

0.02

0.04

0.06

1.22

1.23

1.24

1.26

1.27

1.28

1.29

1.30

spot

Buy EUR

Sell lots of EUR

Sell even more EUR

Unwind Hedge = Buy lots of EUR back


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In words:

• EUR/USD RKO call strike 1.2000 and barrier1.3000.


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In words:


• An investment bank delta-hedging a short positionwith nominal 10 Million EUR has to buy 10 Milliontimes delta EUR. As the spot goes up to the bar-rier, delta becomes smaller requiring the hedginginstitution to sell more and more EUR.


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In words:



• This can influence the market since steadily offeringEUR slows down the spot movement towards thebarrier and can in extreme cases prevent the spotfrom crossing the barrier.


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In words:



• This can influence the market since steadily offeringEUR slows down the spot movement towards thebarrier and can in extreme cases prevent the spotfrom crossing the barrier.

• Once the barrier is breached, the bank has to un-wind the delta hedge, buy lots of EUR ⇒ rate goesup fast


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3.2. Historic Examples

Barrier options crisis in the 90s.


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4. Cut-Off Time and Fixings

4.1. Cut-Off

The exact time T in ST

• Value to take at a pre-specified time at expirationdate of an option


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4.1. Cut-Off



• Source of this value must be pre-specified


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4.1. Cut-Off




• FX: OTC standard is NY cut: Traded FX spot at10:00 a.m. NY time


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4.1. Cut-Off





• Problem: not official, not transparent to public


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4.1. Cut-Off






• Advantage: tradable, transparent to FX traders


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4.1. Cut-Off







• Other sources: FED, Warshaw Cut, Tokio Cut,Bank’s own fixing


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4.1. Cut-Off







• Other sources: FED, Warshaw Cut, Tokio Cut,Bank’s own fixing

• Average of several banks

• Example: OPTREF = AVG (COMBA, DB,DREBA, HVB)


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4.2. ECB Currency Fixing

1. Many Corporate Treasurers prefer official source ofexchange rate

2. Set each business day at 2:15 p.m.

3. Published on Reuters page ECB37


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4. Not tradable


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4. Not tradable

5. Published with Delay of ∆T = 10-20 Minutes


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Reuters Page ECB37


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5. Delta-Hedging a Short Dis-cretely Monitored RKO

5.1. Model

Risk-neutral geometric Brownian motion

dSt = St[(rd − rf) dt + σ dWt]


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5. Delta-Hedging a Short Dis-cretely Monitored RKO

5.1. Model

Risk-neutral geometric Brownian motion

dSt = St[(rd − rf) dt + σ dWt]

These parameters are constant

• rd: domestic interest rate

• rf : foreign interest rate

• σ: volatility

• St: FX spot rate at time t


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5.2. Contracts

• T : maturity in years

• K: strike

• B: knock-out barrier


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5.2. Contracts


• K: strike


• fixing schedule 0 = t0 < t1 < t2 . . . , tn = T


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5.2. Contracts


• K: strike



• payoff for a discretely monitored RKO call option

V (F, T ) = (FT −K)+II{max(Ft0,...,Ftn)<B}

• Ft: fixing of the underlying exchange rate at time t


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5.2. Contracts


• K: strike






• II : the indicator function


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5.2. Contracts


• K: strike






• II : the indicator function

• payoff to hedge is

V (S, T ) = (ST −K)+II{max(St0,...,Stn)<B}


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5.3. Analysis Procedure

• simulate the spot with Monte Carlo


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• model the ECB-fixing Ft by

Ft = St + ϕ, ϕ ∈ N (µ, σ)


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Ft = St + ϕ, ϕ ∈ N (µ, σ)

• µ and σ estimated from historic data.


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Ft = St + ϕ, ϕ ∈ N (µ, σ)

• µ and σ estimated from historic data.

• difference of fixing and traded spot = normally dis-tributed random variable.


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5.3.1. Estimates

Estimated values for mean and standard-deviations ofthe quantity Spot - ECB-fixing from historic time se-riesa.

Currency pair Mean Std Dev Time horizon

EUR - USD -3.125E-6 0.0001264 23.6 - 08.8.04

USD - JPY -4.883E-3 0.0134583 22.6 - 26.8.04

USD - CHF -1.424E-5 0.0001677 11.5 - 26.8.04

EUR - GBP -1.330E-5 0.00009017 04.5 - 26.8.04

For USD-JPY take EUR-JPY / EUR-USD etc.aData provided by Commerzbank


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5.4. Error Estimation

1. introduce a bid/offer-spread δ for the spot, whichis of the size of 2 pips in the inter bank market.


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2. evaluate the payoffs for barrier options for each pathand run the simulations with the appropriate deltahedge quantities to hold.


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3. compute for each path the error encountered duethe fixing being different from the spot.


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4. average over all paths.


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5. do this for various currency pairs, parameter scenar-ios, varying the rates, volatilities, maturities, barri-ers and strikes.


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5. do this for various currency pairs, parameter scenar-ios, varying the rates, volatilities, maturities, barri-ers and strikes.

6. We expect a significant impact particularly for re-verse knock-out barrier options due to the jump ofthe payoff and hence the large delta hedge quantity.


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5.5. Hedging Error for the Discretely monitored

up-and-out Call

Two sources of error


-0.02

0.00

0.02

0.04

0.061.

25

1.26

1.28

1.29

1.30

1.32

1.33

spot

spotfixing


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up-and-out Call



-0.02

0.00

0.02

0.04

0.061.

25

1.26

1.28

1.29

1.30

1.32

1.33

spot

spotfixing


-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

fixingspot


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up-and-out Call



-0.02

0.00

0.02

0.04

0.061.

25

1.26

1.28

1.29

1.30

1.32

1.33

spot

spotfixing


-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

fixingspot

Let ∆(St) be the theoretical delta (negative near B).


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5.5.1. Spot Below Fixing: St < B and Ft ≥ B


-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

spotfixing


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-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

spotfixing

• here we unwind our hedge with delay and encounter


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-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

spotfixing


P&L = ∆(St) · (St+∆T − St),

• the seller has been short the underlying at time tand must buy it in t+ ∆T minutes to close out thehedge.


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-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

spotfixing


P&L = ∆(St) · (St+∆T − St),

• the seller has been short the underlying at time tand must buy it in t+ ∆T minutes to close out thehedge.

• he makes profit if the underlying is cheaper in t +∆T .


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5.5.2. Spot Above Fixing: St ≥ B and Ft < B


-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

fixingspot


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-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

fixingspot

• here the seller closed out the hedge at time t, thoughshe shouldn’t have done so


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-0.02

0.00

0.02

0.04

0.06

1.25

1.26

1.28

1.29

1.30

1.32

1.33

spot

fixingspot

• here the seller closed out the hedge at time t, thoughshe shouldn’t have done so

• and in t+∆T she needs to build a new hedge caus-ing

P&L = ∆(St) · (St + δ)−∆(St+∆T ) · St+∆T


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5.5.3. Calculating the Delta-Hedge Quantity

• approximation by Per Horfelt in [6]

• Assume the value of the spot is observed at timesiT/n, i = 0, . . . , n

• define


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• define

θ±∆=

rd − rf ± σ2/2

σ

√T

c∆=

ln(K/S0)

σ√

T

d∆=

ln(B/S0)

σ√

T

β∆= −ζ(1/2)/

√(2π) ≈ 0.5826


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• define

θ±∆=

rd − rf ± σ2/2

σ

√T

c∆=

ln(K/S0)

σ√

T

d∆=

ln(B/S0)

σ√

T

β∆= −ζ(1/2)/

√(2π) ≈ 0.5826

• ζ : Riemann zeta function


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• define

F+(a, b; θ)∆= N (a− θ)− e2bθN (a− 2b− θ)


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• define

F+(a, b; θ)∆= N (a− θ)− e2bθN (a− 2b− θ)

• obtain for the value of the discretely monitored up-and-out call

V (S0, 0) ≈ S0e−rfT

[F+(d, d + β/

√n; θ+)− F+(c, d + β/

√n; θ+)

]−Ke−rdT

[F+(d, d + β/

√n; θ−)− F+(c, d + β/

√n; θ−)

]


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• define

F+(a, b; θ)∆= N (a− θ)− e2bθN (a− 2b− θ)

• obtain for the value of the discretely monitored up-and-out call

V (S0, 0) ≈ S0e−rfT

[F+(d, d + β/

√n; θ+)− F+(c, d + β/

√n; θ+)

]−Ke−rdT

[F+(d, d + β/

√n; θ−)− F+(c, d + β/

√n; θ−)

]• take a finite difference approach for the computa-

tion of the theoretical delta

∆ = VS(S, t) ≈ V (S + ε, t)− V (S − ε, t)

2ε


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5.6. Analysis of EUR-USD

Spot 1.2100

Strike 1.1800

Trading days 250

domestic interest rate 2.17% (USD)

Foreign interest rate 2.27% (EUR)

Volatility 10.4%

Time to maturity 1 year

Notional 1,000,000 EUR

Consider a short position of a discretely monitored up-and-out Call


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5.6.1. Distribution of Absolute Errors in USD

The figures are the number of occurrences out of 1 mil-lion

barrier 1.2500 <1k$ <2k$ <3k$ <39k$ <40k$ <41k$ <42k$ <43k$

upside error 951744 20 1 0 0 0 0 0

downside error 48008 54 2 5 59 85 21 1

barrier 1.3000 <1k$ < 2k$ < 3k$ <89k$ < 90k$ <91k$ <92k$

upside error 974340 20 1 0 0 0 0

downside error 25475 43 0 2 40 59 20

barrier 1.4100 <1k$ <2k$ <3k$ <199k$ <200k$ <201k$ <202k$ <203k$

upside error 994854 78 0 0 0 0 0 0

downside error 4825 194 3 1 19 17 8 1


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5.6.2. Additional Hedge Cost

Hedge error with 99.9% - confidence interval

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

01.22 1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46

barrier

error

error in USDconfidence band


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5.6.3. Probability of a Miss-Hedge

Probability of mishedging

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

1.22 1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46barrier

pro

bab

ility


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5.6.4. Hedging Error / TV

Rel. hedge error with 99.9% bands

-120.00%

-100.00%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

1.22

1.24

1.26

1.28 1.3 1.3

21.3

41.3

61.3

8 1.4 1.42

1.44

1.46

barrier

pro

bab

ility

errorconfidence band


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Rel. hedge error with 99.9% bands

-4.50%

-4.00%

-3.50%

-3.00%

-2.50%

-2.00%

-1.50%

-1.00%

-0.50%

0.00%1.25 1.27 1.29 1.31 1.33 1.35 1.37 1.39 1.41 1.43 1.45

barrier

pro

bab

ility

errorconfidence band


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5.6.5. Maximum Losses

-300000

-250000

-200000

-150000

-100000

-50000

0

50000

P & L

1.22 1.25 1.28 1.31 1.34 1.37 1.4 1.43 1.46barrier

Extremal P & L for the short position

max downside errormax. upside error


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5.6.6. Maximum Losses / TV

Relative P & L

-45000%

-40000%

-35000%

-30000%

-25000%

-20000%

-15000%

-10000%

-5000%

0%

5000%1.2

41.2

61.2

81.3 1.3

21.3

41.3

61.3

81.4 1.4

21.4

41.4

6

barrier

P &

L in

US

D

max downside errormax upside error


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5.7. Summary

• other currency pairs are similar.


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5.7. Summary


• average loss is comparatively small for liquid cur-rency pairs.


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5.7. Summary



• maximum loss can be very large with smallprobability.


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5.7. Summary




• sufficient to charge a maximum of 0.1% of the TVto cover the potential average loss.


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5.7. Summary





• traders take extra premium of 10 basis points perunit of the notional of the underlying.


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5.7. Summary





• traders take extra premium of 10 basis points perunit of the notional of the underlying.

• relative errors are so small that it seems reasonablenot to pursue any further investigation with othermodels beyond Black-Scholes.


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6. Contact Information

Uwe WystupMathFinance AGSchießhohl 1965529 WaldemsGermanyPhone +49-700-MATHFINANCEMore papers are available athttp://www.mathfinance.com/wystup/papers.phpThese slides are available athttp://www.mathfinance.com/wystup/papers/fximpact_slides.pdf


http://www.mathfinance.com/wystup/papers.php

http://www.mathfinance.com/wystup/papers.php

http://www.mathfinance.com/wystup/papers/fximpact_slides.pdf

http://www.mathfinance.com/wystup/papers/fximpact_slides.pdf

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References

[1] Anagnou-Basioudis, I. and Hodges, S. (2004) Derivatives Hedgingand Volatility Errors. Warwick University Working Paper.

[2] Becker, C. and Wystup, U, (2005). On The Cost of Delayed Fix-ing Announcements in FX Options Markets. Research Report, HfB–Business School of Finance and Management, Center for PracticalQuantitative Finance.

[3] Brown, B., Lovato, J. and Russell, K. (2004)D. CDFLIB - C++- library, http://www.csit.fsu.edu/~burkardt/cpp_src/dcdflib/dcdflib.html

[4] Fusai G. and Recchioni, C. (2003). Numerical Valuation of DiscreteBarrier Options Warwick University Working Paper.

[5] Hakala, J. and Wystup, U. (2002) Foreign Exchange Risk, RiskPublications, London.

[6] Horfelt, P. (2003). Extension of the corrected barrier approximationby Broadie, Glasserman, and Kou. Finance and Stochastics, 7, 231-243.


http://www.hfb.de

http://www.hfb.de

http://www.csit.fsu.edu/~burkardt/cpp_src/dcdflib/dcdflib.html

http://www.csit.fsu.edu/~burkardt/cpp_src/dcdflib/dcdflib.html

http://www.mathfinance.de/FXRiskBook/

http://www.math.ethz.ch/~finasto/

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[7] Matsumoto, M. (2004). Homepage of Makoto Matsumoto onthe server of the university of Hiroshima: http://www.math.sci.

hiroshima-u.ac.jp/~m-mat/eindex.html

[8] Wystup, U (2000). The MathFinance Formula Catalogue. http://www.mathfinance.com


http://www.math.sci.hiroshima-u.ac.jp/~m-mat/eindex.html

http://www.math.sci.hiroshima-u.ac.jp/~m-mat/eindex.html

http://www.mathfinance.com

http://www.mathfinance.com

the impact of fx options on the spot market and the cost of …€¦ · overview hedging the......

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