FX Options Trading and Risk
Management
Paiboon Peeraparp Feb. 2010
1
Risk
 Uncertainties for the good and worse
scenarios
Market Risk
Operational Risk
Counterparty Risk
 Financial Assets
 Stock , Bonds
 Currencies
 Commodities
 Non-Financial Assets
 Weather
 Inflation
 Earth Quake
2
Today Topics




Hedging Instruments
Risk Management
Dynamic Hedging
Volatilities Surface
3
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
4
Participants
 Hedgers
 Want to reduce risk
 Speculators
 Seek more risk for profit
 Brokers / Dealers
 Commission and Trading
 Regulators/ Exchanges
 Supervise and Control
5
FX (Foreign Exchange) Market




Over the counter
Trade 24 hours
Active both spot/forwards/options
Banks act as dealers
6
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)
7
Forwards Valuation (1)
An Electronic manufacturer needs to hedge gold
price for their manufacturing in 1 years.
A dealer will need to
T= 0
1.
2.
borrow $ 1,000 at interest rate of 4% annually
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
8
Forwards Valuation (1)
 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.
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
2.50
2.00
1.50
Time value
P/L
1.00
0.50
Intrinsic Value
0.00
30.00
31.00
32.00
33.00
34.00
35.00
36.00
37.00
-0.50
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
11
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.
12
P/L Diagram
 An importer needs to pay USD vs. THB for 1 year.
Underlying
P/L
Forward
P/L
P/L
+
=
Rate
Rate
Underlying
Rate
Option
P/L
P/L
P/L
+
Rate
=
Rate
Rate
13
Options Details







Buyer/Seller
Put/Call
Notional Amount
European/ American
Strike
Time to Maturity
Premium
14
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
Call option for buyer
P/L
Put option for seller
P/L
P/L
+
Rate
=
Rate
Rate
K
Call option for seller
P/L
F=C–P
Rate
F = S-Ke-rt
C-P = S-Ke-rt
18
Options
 Path Independence
 Plain Vanilla
 European Digital
 Path Dependence




Barriers
American Digital
Asian
Etc.
19
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.
20
Combination of Options (2)
 Straddle
Buy Call option
Buy Put option
+
=
 Butterfly Spread
Buy Call & Put option
Sell Straddle
+
=
21
Combination of Options (3)
 Create a suitable risk and reward
profile
 Finance the premium
 Better spread for the banks
22
Risk Reward Analysis
Combine your underlying with the options and
see how much you get and how much you lose.
Underlying
+
Underlying
+
=
=
More
risk
more
return
23
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.
25
FX Option Quotation in FX market (2)
Standard Quotation in the FX markets
1 Straddle
- A straddle is the sum of call and put at the same
strike at the money forward
2 Risk Reversal (RR)
delta
- A RR is on the long call and short put at the same
3 Butterfly
- A Butterfly is the half of the sum of the long call and
put and short Straddle.
26
BBA FX Option Quotation
GBP/USD
Spot
Rate
Date:
Option Volatility
25 Delta Risk Reversal
25 Delta Strangle
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
27
Volatility Smile
Volatility Smile
10.10
Implied Vol
10.00
9.90
9.80
Volatility Smile
9.70
9.60
9.50
25d
50d
25d
Strike
28
Volatility Surface (1)
Volatility Surface (2)
Vol.
Stock Index Vol.
FX Vol.
K/S
K/S
Single Stock Vol.
K/S
Volatility Surface (3)
•
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.
31
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
33
Bank Options Hedging
 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.
The Greek
A call option depends on many parameters: c( S ,  , r , t )
A Taylor Expansion:
c  ct t  cS S  c   cr r  12 cSS (S ) 2  ...
theta
delta
vega
rho
gamma
ct
cS
c
cr
cSS
A 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 price
From Taylor series expansion
Assume ∆S = ∑S√∆t
(∆S)2 = ∑2S2∆t
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 Γ = ∂2C/∂S2
Consider C(S+∆S,t) = C(S,t)+∂C/∂S ∆S + Γ(∆S)2/2 + …
36
Dynamic Hedging (2)
We like to create a hedged portfolio
Define 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)2/2
Suppose 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
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)
37
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 Hedging
A Taylor Expansion:
c  ct t  cS S  c   cr r  12 cSS (S ) 2  ...
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.
Option Sensitivities
(K=10, T=0.2, r=0.05, =0.2)
European Call Option Price
European Call Option Delta
1
2.5
0.9
0.8
2
0.7
0.6
Delta
Price
1.5
1
0.5
0.4
0.3
0.5
0.2
0.1
0
8
8.5
9
9.5
10
S
10.5
11
11.5
0
12
8
European Call Option Gamma
8.5
9
9.5
10
S
10.5
11
11.5
12
European Call Option Theta
0.5
0
0.45
-0.2
0.4
-0.4
0.35
-0.6
Theta
Gamma
0.3
0.25
-0.8
0.2
0.15
-1
0.1
-1.2
0.05
0
8
8.5
9
9.5
10
S
10.5
11
11.5
12
-1.4
8
8.5
9
9.5
10
S
10.5
11
11.5
12