FX Options Trading and Risk Management1 BS

FX Options Trading and Risk Management

Paiboon Peeraparp Feb. 2010

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Risk Uncertainties for the good and worse

scenariosMarket RiskOperational RiskCounterparty 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

gold price for their manufacturing in 1 years.

A dealer will need toT= 0 1. borrow $ 1,000 at interest rate of 4% annually2. buy gold spot at $ 1,000T = 1 yr1. repay loan 10,40 (principal + interest) 2. Charge this customer at $ 1,040Valuation 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 arbitrage pricing by replicate the cash flow of the forward.

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

underlying that is economical to warehouse it.

For the others, it typically follows the mean reverting process.

Forwards Valuation (1)

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

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 moneyC-Ke-rt = 0, we call the option is at the moneyC-Ke-rt < 0, we call the option is out of the money

Option Characteristics (2)For a plain vanilla option

An option buyer needs to pay a premium. An option buyer has unlimited gain. An option seller has earned the premium

but face unlimited risk. This is the zero-sum game.

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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 of notional amount

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

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

Option Premium (2)

BS(S,σ,r,t,K)σ1> σ2 then BS(S,σ1,r,t,K) > BS(S,σ2,r,t,K) t1> t2 then BS(S,σ,r,t1,K) > BS(S,σ,r,t2,K)r1> r2 then BS(S,σ,r1,t,K) > BS(S,σ,r2,t,K)

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

From the equation C,P = BS(S,σ,r,t,K), we solve for σ and call it implied volatility.

The is another realized volatility is the ∑actual realized volatility.

Volatilities

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 – PF = S-Ke-rt

C-P = S-Ke-rt

K

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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Combination of Options (2) Straddle

Butterfly Spread

Buy Call option Buy Put option

+

+

=

=

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Buy Call & Put option Sell Straddle

Create a suitable risk and reward profile

Finance the premium Better spread for the banks

Combination of Options (3)

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Risk Reward AnalysisCombine your underlying with the options and see how much you get and how much you lose.

More risk more return

+

+

=

=

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Underlying

Underlying

Structuring

Dual Currency Deposit is the most popular product that combine sale of option and a normal deposit .

For example, the structure give the buyer of this deposit at normal deposit rate + r % annually. But in case the underlying asset has gone lower the strike, the buyer will receive underlying asset instead of deposit amount.

This structure will work when the interest rates are low and volatilities are high. 24

FX Option Quotation in FX market (1)

1.Quotes are in terms of BS Model implied volatilities rather than on option price directly.

2.Quotes are provided at a fixed BS delta rather than a fixed strike.

3.However implied volatilities are not tradeable 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 markets1 Straddle

- A straddle is the sum of call and put at the same strike at the money forward2 Risk Reversal (RR)

- A RR is on the long call and short put at the same delta3 Butterfly

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

BBA FX Option Quotation

GBP/USDSpot Rate Option Volatility

25 Delta Risk Reversal 25 Delta Strangle

Date:  1

Month3

Month6

Month1

Year1

Month3

Month1

Year1

Month3

Month1

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 = 0.32Solve above equationVC25d = 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 25dStrike

Implied Vo

l

Volatility Sm ile

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Volatility Surface (1)

Volatility Surface (2)

Stock Index Vol.

FX Vol.

K/S

Vol.

K/S

Single Stock Vol.

K/S

Volatility Surface (3)

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• Implies volatilities are steepest for the shorter expirations and shallower for long expiration.

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

• Implied volatilities tend to rise fast and decline slowly.

• Implied volatility is usually greater than recent historical volatility.

Smile Modeling

In the BS Model the stock’s volatilities are constant, independent of stock price and future time and in consequence (S,t,K,T) = ∑ σ

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

We firstly match the σ(S,t) with (S,t,K,T) ∑and this can be done in principle. The problem is to calibrate the σ(S,t) to match with the characteristic of the pattern of the smile

FX Option Formula

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A bank has a lot of fx options outstanding in the book.

They manage overall risk by look into the change of option price given change in one parameter.

Each dealer is limited by the total amount of risk in his book.

Bank Options Hedging

The Greek

A Taylor Expansion:

...)( 221 ScrccSctcc SSrSt

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

thetatc

delta

Scvega

c rho

rcgamma

SScA dealer try to keep all parameter hedged except the one they want to take the view.

Dynamic Hedging (1)Set C(S,t) be the option call priceFrom Taylor series expansion

Assume ∆S = ∑S√∆t (∆S)2 = ∑2S2∆tC(S+∆S,t+∆t) = C(S,t)+∂C/∂t ∆t+∂C/∂S ∆S + ∂2C/∂S2 (∆S)2/2

+ …For a fixed t, and define Γ = ∂2C/∂S2

Consider C(S+∆S,t) = C(S,t)+∂C/∂S ∆S + Γ(∆S)2/2 + …

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Dynamic Hedging (2)We like to create a hedged portfolioDefine θ = ∂C/∂t C(S+∆S,t+∆t) = C(S,t)+θ∆t+∂C/∂S ∆S + Γ(∆S)2/2

dP&L = C(S+∆S,t+∆t) - C(S,t) - ∂C/∂S ∆S = θ∆t+ Γ(∆S)2/2Suppose r=0, the hedge portfolio has the same return as riskless portfolio θ∆t + Γ(∆S)2/2 = 0 or θ∆t + Γ/2 ∑2S2∆t = 0 or θ + Γ/2 ∑2S2 = 0

Step by step hedging

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Time Option Value

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)

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 Ito’s Lemma for dC we obtain

= θdt+ ∂C/∂S dS +1/2ΓS2σ2dt- ∂C/∂S dS –r(C-∂C/∂S S)dt = [θ+ 1/2ΓS2σ2-r∂C/∂S-rC]dt

By Black-Scholes equation with σ = ∑

θ+ 1/2ΓS2∑2-r∂C/∂S-rC = 0

dP&L = 1/2 ΓS2(σ2-∑2)dt

Real World HedgingA Taylor Expansion:

...)( 221 ScrccSctcc SSrSt

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

= ∂C/∂S (∆S) + 1/2Γ (∆S) 2 + θ (Δt)•The dealer job is to design a option book with the risk that 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 is negative.•For a short call or put, the situation is reversed.

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