fx options trading and risk management1 bs

8/13/2019 FX Options Trading and Risk Management1 BS

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FX Options Trading and RiskManagement

Paiboon Peeraparp Feb. 2010

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Risk

Uncertaintiesfor the good and worsescenariosMarket Risk

Operational Risk

Counterparty Risk

Financial Assets Stock , Bonds

Currencies

Commodities

Non-Financial Assets Weather Inflation

Earth Quake

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Today Topics

Hedging Instruments

Risk Management

Dynamic Hedging Volatilities Surface

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Instruments

Forwards Contracts to buy or sell financial assets at

predetermined price and time

Linear payout

No initial cost

Options Rights to buy or sell financial asset at

predetermined price (strike price) and time Non-Linear payout

Premium charged

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Participants

Hedgers Want to reduce risk

Speculators

Seek more risk for profit

Brokers / Dealers Commission and Trading

Regulators/ Exchanges Supervise and Control

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FX (Foreign Exchange) Market

Over the counter

Trade 24 hours

Active both spot/forwards/options Banks act as dealers

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FX Banks

Trade to accommodate clients Make profit by bid/offer spread

Absorb the risk from clients

Offer delivery service

Other Commission Fees

Trade on their own positions

Trade on their views (buy low and sell high)

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Forwards Valuation (1)

An Electronic manufacturer needs to hedge goldprice for their manufacturing in 1 years.

A dealer will need to

T= 01. borrow $ 1,000 at interest rate of 4% annually2. buy gold spot at $ 1,000

T = 1 yr

1. repay loan 10,40 (principal + interest)2. Charge this customer at $ 1,040

Valuation by replication , F = Sert

In FX and commodities market, we call F-S swap points

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We call the last construction as the arbitragepricing by replicate the cash flow of theforward.

If F is not Sert

but G G > F , sell G, borrow to buy gold spot cost = F

G < F , buy G, sell gold spot and lend to receive = F

The construction is working well for underlyingthat is economical to warehouse it.

For the others, it typically follows the meanreverting process.

Forwards Valuation (1)


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Physical / Paper Hedging

Physical Hedging Deliver goods against cash

No basis risk

Paper Hedging Cash settlement between contract rate and

market rate at maturity

Market rate reference has to be agreed on

the contracted date. Some basis risk incurred


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Option Characteristics (1)

P/L of Call Option Strike at 34.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

30.00 31.00 32.00 33.00 34.00 35.00 36.00 37.00

P/LTime value

Intrinsic Value

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C-Ke-rt> 0, we call the option is in the money

C-Ke-rt = 0, we call the option is at the money

C-Ke-rt< 0, we call the option is out of the money


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P/L Diagram

An importer needs to pay USD vs. THB for 1 year.

P/L

Rate

P/L

Rate

P/L

Rate

+ =

P/L

Rate

P/L

Rate

P/L

Rate

+ =

Underlying Option

ForwardUnderlying

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Options Details

Buyer/Seller

Put/Call

Notional Amount

European/ American

Strike

Time to Maturity

Premium

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Option Premium (1)

Normally charged in percentage ofnotional amount

Paid on spot date Depends on (S,,r,t,K) can be

represented by V= BS(S,,r,t,K) ifthe underlying follows BS model.


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Option Premium (2)

BS(S,,r,t,K)

1> 2 thenBS(S,1,r,t,K) > BS(S,2,r,t,K)t1> t2 thenBS(S,,r,t1,K) > BS(S,,r,t2,K)

r1> r2 thenBS(S,,r1,t,K) > BS(S,,r2,t,K)

In reality, the call and put are traded with the marketdemand supply.

From the equation C,P = BS(S,,r,t,K), we solve for

and call it implied volatility. The is another realized volatility is the actual realized

volatility.


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Volatilities


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Put/Call Parity

P/L

Rate

Call option for buyer

P/L

Rate

Call option for seller

P/L

Rate

Put option for seller

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P/L

Rate

+ =

F = C P

F = S-Ke-rt

C-P = S-Ke-rt

K


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Options

Path Independence Plain Vanilla

European Digital

Path Dependence Barriers

American Digital Asian

Etc.

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Combination of Options (1)

Risk Reversal

Buy Call option Sell Put option

1. View that the market is going up (Strikes are not unique).

2. Can do it as the zero cost.3. If do it conversely, the buyer of this structure view the

market is going down.

+ =

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Straddle

Butterfly Spread

Buy Call option Buy Put option

+

+

=

=

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Buy Call & Put option Sell Straddle


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Create a suitable risk and rewardprofile

Finance the premium Better spread for the banks


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Risk Reward Analysis

Combine your underlying with the options and

see how much you get and how much you lose.

Moreriskmorereturn

+

+

=

=

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Underlying

Underlying


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Structuring

Dual Currency Deposit is the most popularproduct that combine sale of option and anormal deposit .

For example, the structure give the buyer ofthis deposit at normal deposit rate + r %annually. But in case the underlying asset hasgone lower the strike, the buyer will receiveunderlying asset instead of deposit amount.

This structure will work when the interest ratesare low and volatilities are high.

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FX Option Quotation in FX market (1)

1.Quotes are in terms of BS Model impliedvolatilities rather than on option pricedirectly.

2.Quotes are provided at a fixed BS deltarather than a fixed strike.

3.However implied volatilities are nottradeable assets, we need to settle in

structures.

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FX Option Quotation in FX market (2)

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Standard Quotation in the FX markets

1 Straddle

- A straddle is the sum of call and put at the samestrike at the money forward

2 Risk Reversal (RR)

- A RR is on the long call and short put at the samedelta

3 Butterfly

- A Butterfly is the half of the sum of the long call andput and short Straddle.


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BBA FX Option Quotation

GBP/USD

Spot

Rate Option Volatility 25 Delta Risk Reversal 25 Delta Strangle

Date: 1 Month 3 Month 6 Month 1 Year 1 Month 3 Month 1 Year 1 Month 3 Month 1 Year

2-Jan-08 1.9795 9.80 9.80 9.43 9.25 -0.82 -0.79 -0.38 0.23 0.32 0.39

3-Jan-08 1.9732 9.73 9.73 9.48 9.25 -0.61 -0.59 -0.62 0.26 0.33 0.39

4-Jan-08 1.9754 9.45 9.45 9.38 9.20 -1.20 -1.22 -1.30 0.29 0.33 0.39

7-Jan-08 1.9725 9.55 9.55 9.23 9.18 -1.14 -1.18 -0.83 0.27 0.32 0.39

For 3 months (Vatm= 9.8)

VC25d

-VP25d

= -0.79

((VC25d+VP25d)/2)-Vatm= 0.32

Solve above equation

VC25d= 9.725 , VP25d= 10.045

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Volatility Smile

Volatility Smile

9.50

9.60

9.70

9.80

9.90

10.00

10.10

25d 50d 25d

Strike

ImpliedVol

Volatility Smile

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Volatility Surface (1)


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Stock Index Vol. FX Vol.

K/S

Vol.

K/S

Single Stock Vol.

K/S


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Implies volatilities are steepest for the shorterexpirations and shallower for long expiration.

Lower strike and higher strikes has highervolatilities than the ATM. implied volatilities.

Implied volatilities tend to rise fast and declineslowly.

Implied volatility is usually greater than recenthistorical volatility.


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Smile Modeling

In the BS Model the stocks volatilities are constant,independent of stock price and future time and inconsequence (S,t,K,T) =

In local volatility models, the stock realized volatilityis allowed to vary as a function of time and stockprice. we may write the evolution of stock price asdS/S = (S,t)dt + (S,t)dZ

We firstly match the (S,t) with (S,t,K,T) and thiscan be done in principle. The problem is to calibratethe (S,t) to match with the characteristic of the

pattern of the smile


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FX Option Formula

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A bank has a lot of fx optionsoutstanding in the book.

They manage overall risk by look intothe change of option price givenchange in one parameter.

Each dealer is limited by the total

amount of risk in his book.

Bank Options Hedging


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The Greek

A Taylor Expansion:

...)( 221

ScrccSctccSSrSt

),,,( trSc A call option depends on many parameters:

theta

tc

delta

Sc

vega

c

rho

rc

gamma

SSc

A dealer try to keep all parameter hedged except the onethey want to take the view.


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Dynamic Hedging (1)

Set C(S,t) be the option call price

From Taylor series expansion

Assume S = St

(S)2 = 2S2t

C(S+S,t+t) = C(S,t)+C/t t+C/S S + 2C/S2 (S)2/2 +

For a fixed t, and define = 2C/S2

Consider C(S+S,t) = C(S,t)+C/S S + (S)2/2 +

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Dynamic Hedging (2)

We like to create a hedged portfolio

Define = C/tC(S+S,t+t) = C(S,t)+t+C/S S + (S)2/2

dP&L = C(S+S,t+t) - C(S,t) - C/S S = t+ (S)2/2

Suppose r=0, the hedge portfolio has the same return as riskless

portfolio

t + (S)2/2 = 0 or t + /2 2S2t = 0 or + /2 2S2= 0

Step by step hedging

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Time OptionValue

Stock Value Cash Value Net Position

t C - C/S S (C/S S)-C 0

t+dt C+dC - C/S(S+dS)

((C/S S)-C) (1+rdt) C+dC - C/S(S+dS)+ ((C/S S)-C) (1+rdt)


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Dynamic Hedging (3)

dP&L =[C+dC - C/S (S+dS)]+ ((C/S S)-C) (1+rdt)

=dC- C/S dS r(C- C/S S)dt

Using Itos Lemma for dC we obtain

= dt+ C/S dS +1/2S22dt- C/S dS r(C-C/S S)dt=[+ 1/2S22-rC/S-rC]dt

By Black-Scholes equation with =

+ 1/2S22-rC/S-rC = 0

dP&L = 1/2 S2(2-2)dt


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Real World Hedging

A Taylor Expansion:

...)( 221

ScrccSctccSSrSt

Daily P/L = Delta P/L + Gamma P/l + Theta P/L

= C/S (S) + 1/2(S) 2+ (t)

The dealer job is to design a option book with the riskthat he feel comfortable with.

For a delta hedged position Gamma and Theta have the

opposite signs

For a long call or put, Gamma is positive and Theta isnegative.

For a short call or put, the situation is reversed.


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European Call Option Price

8 8.5 9 9.5 10 10.5 11 11.5 120

0.5

1

1.5

2

2.5

Price

S8 8.5 9 9.5 10 10.5 11 11.5 12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S

Delta

European Call Option Delta

8 8.5 9 9.5 10 10.5 11 11.5 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

S

Gamma

European Call Option Gamma European Call Option Theta

(K=10, T=0.2, r=0.05, =0.2)

8 8.5 9 9.5 10 10.5 11 11.5 12-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

S

Theta

Option Sensitivities

fx options trading and risk management1 bs

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