fx options trading and risk management 1 bs
TRANSCRIPT
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FX Options Trading and RiskManagement
Paiboon Peeraparp Feb. 2010
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Risk
Uncertainties for the good and worsescenarios
Market 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% annually
2. 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 underlying
that is economical to warehouse it. For the others, it typically follows the mean
reverting 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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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 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 then BS(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 SellPut 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
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Create a suitable risk and rewardprofile
Finance the premium Better spread for the banks
Combination of Options (3)
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Risk Reward Analysis
Combine your underlying with the options andsee 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
2RiskReversal (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
Str
ke
Im
edV
Volatility Smile
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Volatility Surface (1)
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Volatility Surface (2)
Stock Index Vol. FX Vol.
K/S
Vol.
K/S
Single Stock Vol.
K/S
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Volatility Surface (3)
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Implies volatilities are steepest for the shorterexpirations 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 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 volatility
is 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:
...)( 2
2
1 (((((!( ScrccSctccSSrSt
WW
),,,( trSc WA call option depends on many parameters:
theta
tc
delta
Sc
vega
Wc
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 + 2
C/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 portfolioDefine = C/t
C(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 risklessportfolio
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 NetPosition
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 =
+ 1/2S2
2
-rC/S-rC = 0
dP&L = 1/2 S2(2-2)dt
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Real World Hedging
A Taylor Expansion:
...)( 221 (((((!( ScrccSctcc
SSrStW
W
Daily P/L = DeltaP/L + GammaP/l + ThetaP/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 1 0 1 0 . 5 1 1 1 1 . 5 1 2 0
0. 5
1
1. 5
2
2. 5
Price
S
8 8. 5 9 9. 5 1 0 1 0 . 5 1 1 1 1 . 5 1 2 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 1 0 1 0 . 5 1 1 1 1 .5 1 2 0
0 . 0 5
0. 1
0 . 1 5
0. 2
0 . 2 5
0. 3
0 . 3 5
0. 4
0 . 4 5
0. 5
S
G
am
m
a
European Call Option Gamma European Call Option Theta
(K=10, T=0.2, r=0.05, W=0.2)
8 8. 5 9 9 . 5 1 0 1 0 . 5 1 1 1 1 . 5 1 2 -1.
- 1 . 2
-1
- 0 . 8
-0.
-0.
- 0 . 2
0
Thet
Option Sensitivities