Chapter 6 fx options

Managing Market Risk Under The Basel III Framework
Copyright © 2016 CapitaLogic Limited
Chapter 6
Foreign Exchange
Options
Managing Market Risk Under The Basel III Framework
The Presentation Slides
Website : https://sites.google.com/site/quanrisk
E-mail : quanrisk@gmail.com

Copyright © 2016 CapitaLogic Limited 2
Declaration
Copyright © 2016 CapitaLogic Limited.
All rights reserved. No part of this presentation file may be
reproduced, in any form or by any means, without written
permission from CapitaLogic Limited.
Authored by Dr. LAM Yat-fai (林日林日林日林日辉辉辉辉),
Principal, Structured Products Analytics, CapitaLogic Limited,
Adjunct Professor of Finance, City University of Hong Kong,
Doctor of Business Administration (Finance),
CFA, CAIA, FRM, PRM.

BIS statistics on FX derivatives
Notional amount (USD bn)
75,87974,519Total
14,60013,558Options
24,20424.724Swaps
http://stats.bis.org/statx/srs/table/d6?p=20151&c=
37,07637,238Forwards
20142015FX derivatives

Currency mis-match problem
FX call options
FX put options
FX options market
Outline

A currency mis-match problem
A trading firm
Buys goods from United Kingdom
Pays cost in GBP
Sells goods to United States
Receives income in USD
Both cost in GBP and income in USD in three months are
well estimated
How to ensure that the revenue in USD which will be used to
buy GBP, is sufficient to pay the cost in GBP, in case the FX
rate of GBP goes up in three month?
How to avoid buying expensive GBP at the strike rate in
case the FX rate of GBP goes down in three months?

FX call option
An agreement between an investor and a bank
on a foreign currency such that
The investor has the right but not obligation to
buy from the bank the foreign currency at an
agreed strike rate K at maturity T
The bank must sell to the investor the foreign
currency at an agreed strike rate K at maturity T
ONLY upon the request from the investor

At maturity
ST above K
The investor pays to the
bank K units of domestic
currencies
The bank
supplements ST - K
buys one unit of foreign
currency at ST
delivers to the investor one
unit of foreign currency
ST below K
The investor ignores the
FX call option
The bank does nothing
The investor may buy the
foreign currency with ST
from the bank or any other
banks

Payoff
The benefit to the investor resulting from a
FX call option at maturity
Single FX call option
Payoff = Max[ST - K, 0]
FX call options contract
An agreement to enter many identical FX call
options in one transaction
Contract payoff = Quantity × Max[ST - K, 0]

Payoff diagram
0
0.01
0.02
0.03
0.04
1.52 1.54 1.56 1.58 1.6
FX rate at maturity (USD per GBP)
Payoff(USD)
Payoff

Long position and short position
Long position
An investor has the right but not obligation to buy from
the bank the foreign currency at strike rate K at maturity T
Short position
The bank has the obligation but not right to sell to the
investor a foreign currency at strike rate K at maturity T
An investor can choose to enter a short position
The bank then enters the long position

FX call options
FX put options
FX options market
Outline

Analysis of FX call options
Functional purposes
Hedging
Investment
Cash flows
Outflows
Inflows
Valuation
When there is market price,
Value
= Quantity × Market price
When there is no market
price, how much does it
worth?
Market risk
VaR
Worst case loss

FX rate insurance
Investor
An insurance to ensure that an investor can buy a
foreign currency from a bank at the strike rate at
maturity in case the investor cannot buy a foreign
currency at or below the strike rate from other
institutions
Bank
Earn insurance premium by providing the
insurance service

Functional purposes
Long position
In case the FX rate goes up
To hedge the value of a
future outflow in a foreign
currency
To speculate on the up trend
of a FX rate with small
upfront cash outflow
To hedge the short position
in a foreign currency
Short position
To earn an insurance
premium

Investment in FX rate
with FX call option
Long position
Short position
[ ]
T
T
T
T
Profit = S - K - Premium > 0
if S > K + Premium
Profit = Min Premium, Premium + K - S > 0
if S < Premium + K

with FX call option
When an investor expects that the FX rate will
go up
Long a FX call option
go down
Short a FX call option

Modelling FX rate
K: Strike rate
T: Maturity in years
S0: Current FX rate
rd: Annualized risk-free rate of domestic currency
rf: Annualized risk-free rate of foreign currency
σ :Annualized volatility of foreign currency
ST: FX rate in T years
c: Value of FX call option

Modelling FX rate
Drift
Volatility
Log-normal FX rate model
[ ]
t
t-1
2
T d f
S
µ = ln
S
σ = Expected annualzied standard deviation of drifts
σ
S = exp r - r + T + σ T × Normal 0,1
2
 
 
 
   
  
   

Cash flows – physical settlement
Receive premiumPay premium
Cash flow at
origination
Foreign currencyK
Outflow at
maturity if ST > K
Foreign currency
Long
K
Inflow at
maturity if ST > K
ShortPosition

Cash flows – Cash settlement
K - STST - K
Cash flow at
maturity if ST > K
Premium- Premium
Cash flow at
origination
Long ShortPosition

Valuation – long position
Black-Scholes formula
( ) ( ) ( ) ( )
1
2
0 f 1 d 2
0
d d
2
d
1 1 -
2
d
2 1 2 -
c = S exp -r T Φ d - Kexp -r T Φ d
2S σ
ln + r - r + T
K 2 1 t
where d = Φ(d ) = exp - dt
2σ T 2π
1 t
d = d - σ T Φ(d ) = exp - dt
22π
∞
∞
           
 
 
 
 
 
∫
∫
Example 6.2

User defined VBA function
for FX options valuation
BSValue(Option type,
Current FX rate,
Volatility,
Domestic risk-free rate,
Foreign risk-free rate,
Strike rate,
Maturity)

Value of FX call option
Option value
Intrinsic value
Caused by payoff
Time value
Caused by time to maturity
( ) ( ) ( ) ( )
( ) ( ) ( ){ }
0 f 1 d 2
0 0 f 1 d 2
= S - K + S exp(-r T)Φ d - 1 - K exp(-r T)Φ d - 1      
[ ]
( ) ( )
0
T
0 f 1 d 2
Intrinsic value = S - K
= Max S - K,0 at maturity
Time value = S exp(-r T)Φ d - 1 - K exp(-r T)Φ d - 1
= 0 at maturity
      

Payoff diagram
0.00
0.01
0.02
0.03
0.04
1.52 1.54 1.56 1.58 1.6
Current FX rate (USD per GBP)
(USD)
Value Payoff Replicating portfoliio
Intrinsic value
Time value

Moneyness
<< K
< K
= K
> K
>> K
FX rate (S0)
Out-of-the-money
S0 - Kexp(-rfT)
Deeply
in-the-money
0
Deeply
out-of-the-money
At-the-money
Black-Scholes
formula
In-the-money
ValueMoneyness

Moneyness
0.00
0.02
0.04
0.06
0.08
0.10
1.46 1.51 1.56 1.61 1.66
(USD)
Value Payoff

Delta
The sensitivity of a FX call option to its underlying FX rate
A ratio between
Change in FX call value
A very small change in FX rate
Approaching
1 when deeply in-the money
0 when deeply out-of-the money
[ ] [ ]
( ) ( )0 0
f 1
h
c S + h - c S - h
Delta = lim = exp -r T Φ d
2h→∞

Replicating portfolio
A portfolio comprising
A long position in exp(-rfT)Φ(d1) units of foreign currency
A short position in exp(-rdT) Φ(d2) units of domestic currency
Replication
A FX call option is equivalent to long Delta units of foreign currency
and short a risk-free security
Effective only when the current FX rate remains steady
To be adjusted when the current FX rate changes
( ) ( ) ( ) ( )0 f 1 d 2c = S exp -r T Φ d - Kexp -r T Φ d
Example 6.3

Hedge the risk of a foreign currency
Long position in
exp(rfT)/Φ(d1)
quantity of FX
call option
Short position in
exp(rfT)/Φ(d1)
quantity of FX
call option
Option hedge
ShortLongFX position
( )
( )
( )
( )
( )
f 2
0 f d
1 1
exp r T Φ d
S = c × + Kexp r -r T ×
Φ d Φ d
  

Hedged worst case loss
Value0
0
Worst case value
Expected value
Value-at-risk
1 - q%
q%
T days
ValueT
Unrealized
loss
Worst case loss
Acquisition cost
Hedging
Hedged
worst case loss

FX rate risk of a FX call option
Foreign currency has FX rate risk
Domestic currency has no risk
Equivalent to Delta units of foreign currency
within a short holding period
Example 6.4
Example 6.5
( ) ( ) ( ) ( )0 f 1 d 2c = S exp -r T Φ d - Kexp -r T Φ d
Example 6.6

FX rate risk factors for foreign
currency and FX call options portfolio
FX rate risk
Value
Quantity
Holding period
dispersion
FX rate
Standard
deviation
Holding period
Diversification
effect
Concentration of
foreign currencies
% change
dependency
Volatility

Hedge the risk of FX call option
Long position in a
FX forward
Short position in a
FX forward
Forward
hedge
Long position in
another FX call
Short position in
another FX call
Option
hedge
Long position in a
foreign currency
Short position in a
foreign currency
Currency
hedge
ShortLong
Forward
position

FX call options
FX put options
FX options market
Outline

A reverse currency mis-match problem
A trading firm
Buys goods from United States
Pays cost in USD
Sells goods to United Kingdom
Receive income in GBP
Both cost in USD and income in GBP in three months are
well estimated
How to ensure that the income in GBP which will be sold and
become USD, is sufficient to pay the cost in USD, in case the
FX rate of GBP goes down in three month?
How to avoid selling cheap GBP at the strike rate in case
the FX rate of GBP goes up in three months?

FX put option
on a foreign currency such that
The investor has the right but not obligation to
sell to the bank the foreign currency at an agreed
strike rate K at maturity T
The bank must buy from the investor the foreign
currency at an agreed strike rate K at maturity T
ONLY upon the request from the investor

At maturity
ST below K
The investor pays to the
bank one unit of foreign
currency
The bank
sells one unit of foreign
currency at ST
supplements K - ST
delivers to the investor K
units of domestic currency
ST above K
The investor ignores the
FX put option
The bank does nothing
The investor may sell the
foreign currency with ST to
the bank or any other banks

Payoff
The benefit to the investor resulting from a
FX put option at maturity
Single FX put option
Payoff = Max[K - ST, 0]
FX put options contract
An agreement to enter many identical FX put
options in one transaction
Contract payoff = Quantity × Max[K - ST, 0]

Payoff diagram
0
0.01
0.02
0.03
0.04
1.52 1.54 1.56 1.58 1.6
FX rate at maturity (USD per GBP)
Payoff(USD)
Payoff

Long position and short position
Long position
An investor has the right but not obligation to sell to the
bank the foreign currency at strike rate K at maturity T
Short position
The bank has the obligation but not right to buy from the
investor a foreign currency at strike rate K at maturity T
An investor can choose to enter a short position
The bank then enters the long position

FX rate insurance
Investor
An insurance to ensure that an investor can sell a
foreign currency to a bank at the strike rate at
maturity in case the investor cannot sell a foreign
currency at or above the strike rate to other
institutions
Bank
Earn insurance premium by providing the
insurance service

Functional purposes
Long position
In case the FX rate goes
down
To hedge the value of a
future inflow in a foreign
currency
To speculate on the down
trend of a FX rate with small
upfront cash outflow
To hedge the long position
in a foreign currency
Short position
To earn an insurance
premium

with FX call option
Long position
Short position
[ ]
T
T
T
T
Profit = K - S - Premium > 0
if S < K - Premium
Profit = Min Premium, Premium + S - K > 0
if S > K - Premium

with FX put option
go down
Long a FX put option
go up
Short a FX put option

Modelling FX rate
K: Strike rate
T: Maturity in years
S0: Current FX rate
rd: Annualized risk-free rate of domestic currency
rf: Annualized risk-free rate of foreign currency
σ :Annualized volatility of foreign currency
ST: FX rate in T years
p: Value of FX put option

Cash flows – physical settlement
Receive premiumPay premium
Cash flow at
origination
KForeign currency
Outflow at
maturity if ST < K
K
Long
Foreign currency
Inflow at
maturity if ST < K
ShortPosition

Cash flows – Cash settlement
ST - KK - ST
Cash flow at
maturity if ST < K
Premium- Premium
Cash flow at
origination
Long ShortPosition

Valuation – long position
Black-Scholes formula
( ) ( ) ( ) ( )
1
2
d 2 0 f 1
0
d d
2
d
1 1 -
2
d
2 1 2
-
p = Kexp -r T Φ -d - S exp -r T Φ -d
2S σ
ln + r - r + T
K 2 1 t
where d = Φ(-d ) = 1 - exp - dt
2σ T 2π
1 t
d = d - σ T Φ(-d ) = 1 - exp - dt
22π
∞
∞
           
 
 
 
 
 
∫
∫
Example 6.2

Value of FX put option
Option value
Intrinsic value
Caused by payoff
Time value
Caused by time to maturity
( ) ( ) ( ) ( )
( ) ( ) ( ){ }
d 2 0 f 1
0 d 2 0 f 1
p = Kexp -r T Φ -d - S exp -r T Φ -d
= K - S + K exp(-r T)Φ -d - 1 - S exp(-r T)Φ -d - 1      
[ ]
( ) ( )
0
T
d 2 0 f 1
Intrinsic value = K - S
= Max K - S , 0 at maturity
Time value = K exp(-r T)Φ -d - 1 - S exp(-r T)Φ -d - 1
= 0 at maturity
      

Payoff diagram
0.00
0.01
0.02
0.03
0.04
1.52 1.54 1.56 1.58 1.6
(USD)
Value Payoff Replicating portfoliio
Intrinsic value
Time value

Moneyness
>> K
> K
= K
< K
<< K
FX rate (S0)
Out-of-the-money
Kexp(-rfT) - S0
Deeply
in-the-money
0
Deeply
out-of-the-money
At-the-money
Black-Scholes
formula
In-the-money
ValueMoneyness

Moneyness
0.00
0.02
0.04
0.06
0.08
0.10
1.46 1.51 1.56 1.61 1.66
(USD)
Value Payoff

Delta
The sensitivity of a FX put option to its underlying FX rate
A ratio between
Change in FX put value
A very small change in FX rate
Approaching
1 when deeply in-the money
0 when deeply out-of-the money
[ ] [ ]
( ) ( )0 0
f 1
h
p S + h - p S - h
Delta = lim = - exp -r T Φ -d
2h→∞

Replicating portfolio
A portfolio comprising
A long position in exp(-rdT) Φ(-d2) units of domestic currency
A short position in exp(-rfT)Φ(-d1) units of foreign currency
Replication
A FX put option is equivalent to long and risk-free security and short
Delta units of foreign currency
Effective only when the current FX rate remains steady
To be adjusted when the current FX rate changes
( ) ( ) ( ) ( )d 2 0 f 1p = Kexp -r T Φ -d - S exp -r T Φ -d

Hedge the risk of a foreign currency
Short position in
exp(rfT)/Φ(-d1)
quantity of FX put
option
Long position in
exp(rfT)/Φ(-d1)
quantity of FX put
option
Option hedge
ShortLongFX position
( )
( )
( )
( )
( )
f 2
0 f d
1 1
exp r T Φ -d
S = - p × + Kexp r -r T ×
Φ -d Φ -d
  

FX rate risk of a FX put option
Foreign currency has FX rate risk
Domestic currency has no risk
Equivalent to Delta units of foreign currency
within a short holding period
( ) ( ) ( ) ( )d 2 0 f 1p = Kexp -r T Φ -d - S exp -r T Φ -d

Hedge the risk of FX call option
Short position in a
FX forward
Long position in a
FX forward
Forward
hedge
Short position in
another FX call
Long position in
another FX call
Option
hedge
Short position in a
foreign currency
Long position in a
foreign currency
Currency
hedge
ShortLong
Forward
position

Put-call parity
At maturity
Valuation day
[ ]
[ ]
[ ] [ ]
[ ] [ ]
( ) ( )
( ) ( )
Call T
Put T
Call Put T T
T T
T
T f d
d T f
Payoff = Max S - K, 0
Payoff = Max K - S , 0
Payoff - Payoff = Max S - K, 0 - Max K - S , 0
= Max S - K, 0 + Min S - K, 0
= S - K
c - p = S exp -r T - Kexp -r T
c + Kexp -r T = p + S exp -r T

FX call options
FX put options
FX options market
Outline

Hypothetical sale of a FX call option
A long position at time 0
Value of a long position at time τ
Sell the long position at time τ
At maturity T
( ) ( ) ( ) ( )
[ ] ( ) [ ] ( )
0 f 1 d 2
τ τ f 1τ d 2τ
τ
c = S exp -r (T-τ) Φ d - Kexp -r (T-τ) Φ d
Profit = c - c
0

A long-short portfolio
A long position at time 0
Value of a long position at time τ
Enter a short position at time τ
At maturity T
( ) ( ) ( ) ( )
[ ] ( ) [ ] ( )
[ ] [ ]
0 f 1 d 2
τ τ f 1τ d 2τ
τ
T T
c = S exp -r (T-τ) Φ d - Kexp -r (T-τ) Φ d
Profit = c - c
Payoff = Max S - K, 0 - Max S - K, 0 0=

Reverse position
Long position
Entering a short position with the same strike rate K
maturity day
At maturity
Net payoff = Max[ST - K, 0] – Max[ST - K, 0] = 0
Equivalent to close a long position
Short position
Entering a long position with the same maturity day
At maturity
Net payoff = - Max[ST - K, 0] + Max[ST - K, 0] = 0
Equivalent to close a short position

Closing a position
Standard FX options contract
Customized for investor in terms of option type, foreign currency, strike rate, maturity
and quantity
A bilateral agreement entered between an investor and a bank
Cannot be transferred
No secondary market
An investor can NOT sell his FX options contract
Market maker
A bank which is always ready to quote a premium to enter a FX options contract with
an investor
An investor can always enter a position or a reverse position with a market marker
By entering a reverse position, an investor can always reverse and close his existing
FX option position
A liquid FX options market
Selling a FX options contract in fact means closing a FX option position by entering a
reverse position
Market markers create a liquid FX options market

Close out netting agreement
Covering all FX options entered between the
two parties
Allowing FX option positions with same
underlying currency and maturity to offset
among one another

Over-the-counter
An investor and a bank enter a FX option through
private negotiation
High degree of customization
High confidentiality
Loosely regulated
Short position subject to default risk
Primarily protected by contract law through civil litigation
Resulting unknown systemic risk to regulators
Nowadays, banks in many developed countries must
report their FX options positions to the Central Trade
Repository

Bank business model – service based
A commercial bank
Enters a FX option with an investor
Receives the premium and a higher service fee
Immediately enters a reverse position with an investment
bank
Pays the premium and a lower service fee
Profit
Higher service fee received - Lower service fee paid

Bank business model – position based
An investment bank
Enters a FX option with a commercial bank
Receives the premium and a service fee
Hedge the position through dynamic hedging
Profit
Service fee - Dynamic hedging cost

Default risk
Default
At maturity, the short position obligated to pay
does not pay the bank
Regular settlement
Before maturity
For a FX call option, the short position deposits S0 - K
to the long position when S0 > K
For a FX put option, the short position deposits K - S0
to the long position when K > S0

Practical issues
Black-Scholes formulas fails in the FX market
during disasters where the log-normal FX rate model
does not hold
Volatility is not a constant
For a fixed maturity, volatility exhibits a smile shape
relative to the strike rate
Different volatility for different maturity
Volatility data
Downloaded from financial information services
ATM volatility as the rule of thumb choice

Volatility smile
Strike rate
Volatlity

Term structure of volatility
Time to maturity
Volatility

Volatility surface

Chapter 6 fx options

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