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Derivatives
Options
Derivatives 2
Options
Hans Buysse
Derivatives
Options
Foreign exchange derivatives
– Options
– Combined Options
– Exotic Options
– Case 3 Futures Cheapest to Deliver
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Derivatives
Options
Plain vanilla FX Options
Using options as a FX hedge (CRO)
– Within 6 months BF Trading will receive USD 100.000. Our
treasurer still fears a dollar decrease (spot EUR/USD
1.1274-1.1284). He wishes to hedge this risk but he would
also like to benefit from a possible upside. Therefore he
buys a put with a premium of 0.0248 EUR per USD.
– Definition : you receive the right, during a defined period
(USA) or during a certain moment, to buy currencies (call) or
to sell currencies (put) at a determined price.
For this, you have to pay a premium.
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Derivatives
Options
The option family…
Profit from option strategy
versus doing nothing
Realised profit
Through option ecercise
0.84
0.86 0.8862
0.90
0.92
spot rate
At expiry
No excercise,
cost = premium
-1
OPTION
CALL
PUT
Buying
Sale
(write)
RIGHT to buy
OBLIGATION
to supply
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Buying
RIGHT to sell
Sale
(write)
OBLIGATION
to purchase
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Derivatives
Options
FX options as a hedge
Scenarios 6 months later:
– USD/EUR: 0.84 (ie. EUR/USD 1.1905), Excercise
option (sell dollars at 0.8862 or 1,1284). Realised
profit = intrinsic value. Intrinsic value > 0 ==> option
in the money: (88.620 – 2.480) – 84.000 = 2.140
EUR
– USD/EUR: 0.8862. Optie at the money. No
excercise.
– USD/EUR: 0.92 (ie. EUR/USD 1, 0870). Option
excercise implies negative intrinsic value: far out of
the money.
premium = intrinsic value + time value
call: formula B&S ==> Garman Kohlagen
put: calculate call + apply put-call parity (cf slide)
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Derivatives
4.2 Currency options
Options
FX options as a hedge
•
Call option: gives the holder the right to buy the underlying
asset by a certain date for a certain price
•
Put option: gives the holder the right to sell the underlying
asset by a certain date for a certain price
•
Exercise / strike price: the price in the option contract
•
Expiration date / maturity:
– The end date in the contract at which or before which
the option has to be exercised
– American options can be exercised any time up to the
expiration date
– European options can be exercised only on the
expiration date
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Derivatives
Options
A currency call option grants the holder the right to buy a specific currency
at a specific price (exercise or strike price) within a specific period of time
r = domestic rate
rf = foreign rate
T-t = option life (years)
S0 = spot rate
X = Strike rate
σ = volatility
d1=
d2 =
At expiration date a call option is
– in the money
if spot rate > strike price
– at the money
if spot rate = strike price
– out of the money
if spot rate < strike price
ln( S 0 / K ) + ( σ
σ
Black-Scholes (Garman Kohlhagen) model
Call option:
C 0 = S0e
Put option:
P0 = Xe
− qT
−rT
* N(d 1 ) − Xe
* N( − d2 ) − S 0 e
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d1 − σ
−rT
− qT
2
/ 2 )( T − t )
(T − t )
(T
− t)
* N(d 2 )
* N( − d1 )
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Derivatives
Options
Currency options
Input parameters:
– Fixed rate of exchange (Strike or exercise price)
– Interest rates (domestic and foreign)
– Volatility of exchange rate (σ)
– Call currency / put currency
– Settlement and expiry date
– Style (American or European)
Intrinsic time value
and expectation
value
Intrinsic value of a call option
Exercise rate
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Derivatives
Options
Currency options
Delta
Strike
Forward
Lognormal
distribution
with voly σ
Spot
rate
Trade date
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Exercise date
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Derivatives
Options
Currency options
Determinants of option premium, and their impact on the premium
Call premium
Put premium
Appreciation spot rate
higher
Lower
Depreciation spot rate
Lower
Higher
Increased volatility
Higher
Higher
Decreased volatility
Lower
Lower
Interest currency 1 higher /
currency 2 lower
Lower
Higher
Interest currency 1 higher /
currency 2 lower
higher
Lower
Decrease time to maturity
lower
Lower
* Currency 1 is the quotation currency of the option and currency 2 is the currency for the
quotation of the premium
Derivatives
Options
Selling a call option
Our treasurer of BF Trading likes options, but does not want to pay
a premium.
•Therefore, he is going to sell a call option to the bank (obligation to
deliver USD through exercise). This option gives the right to buy
dollars at a rate of EUR/USD 1.12840 over 6 months (call). BF
Trading receives a premium for selling this option: 0.0297 EUR per
USD.
•Example: BF Trading sells call USD 6 mth, excercise price 1.12840
EUR/USD or 0.8862 USD/EUR, for USD 100.000.
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Derivatives
Options
Selling a call (cont’d)
1. Scenarios 6 mths later
ƒ USD/EUR 0.84. No exercise. BF Trading sells her dollars in the market
and receives 84.000 + premium (EUR 2.970) = EUR 86.970 ==> this is
not a hedge!!!
ƒ USD/EUR 0.8862 Receipt of EUR 88.620 + EUR 2.970
ƒ USD/EUR 0.92. Excercise of the call by the bank. BF Trading receives
EUR 88.620 + EUR 2.970 instead of 92.000.
2. So selling options is trading and not hedging
Profit of this option strategy
Versus doing nothing
1
profit = premium because the
Option is not exercised
0.84
0.86 0.8862
0.90
0.92
Loss becausse
of exercise
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spot rate
At expiry
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Derivatives
Options
Foreign exchange derivatives
– Options
– Combined Options
– Exotic Options
– Case 3 Futures Cheapest to Deliver
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Derivatives
Options
Spreads & Combinations of options
•
Positions of different options can be used to create a tailor made pay off
scheme:
– Based on underlying exposure
– Based on own vision
•
Spreads are positions in two or more options of the same type (2 calls or
2 puts)
– Bull, bear, butterfly, calendar, etc
•
Combinations are positions in two or more options of both calls and
puts
– Strangle, straddle, strips and straps
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Derivatives
4.6 Bull spread
Options
Spreads
Spread of buying a call option with strike K and selling a call with strike L with
same expiration date
– K<L
This spread is used to limit the exposure of the hedger on the underlying asset
but leaves him with limited flexibility
– In return for giving up the potential of call option K the hedger receives
the premium of call option L
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Derivatives
Options
Risk Reversals or Cylinder Options
Both bull & bear combinations are also known as risk reversals or cylinders:
– The purchase of an out-of-the-money option of one type (put or call),
and the sale of an out-of-the-money option of the other type, with the
same currency pair, maturity and amount.
– The most popular version is a zero premium strategy, where the
premium of the option sold offsets the premium of the option
purchased.
– When sold to a hedger with an underlying exposure, this is called
variously a cylinder, collar or range forward, and it creates a risk
reward profile like a spread.
– As it offers some profit potential, limited risk and no out-of-pocket
cost, it is probably the most frequently used option hedge.
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Derivatives
Options
Combined option or Cylinder option
•
Combination: risk hedging + advantage of USD increase & not paying a
premium
•
Definition: forward contract with variable rate. Definition of a minimum and
maximum rate for a transaction.
•
i.e.: combined option of “buy call sell put” or “buy put sell call”
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Derivatives
Options
Example of a combined option
BF Trading contracts a combined option with the bank,
supplying USD within 6 months at a rate between 0.8820 –
0.8920 (provisie 0,2 of 177,24 EUR).
Scenarios 6 mths later:
– USD/EUR: 0.86 BF Trading sells USD at the minimum
rate of 0.8820
– USD/EUR > 0.8820, < 0.8920 ==> transaction at spot
rate
– USD/EUR: 0.91 BF Trading must sell USD at 0.8920
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Derivatives
Options
Second example of a Risk reversal (cylinder)
Risk reversal
– Notional:
– Maturity:
– Buy EUR put / USD call
Strike:
– Sell EUR call / USD put
Strike:
€10mln
2m
1,1700
1,2250
Buy a EUR put / USD call option struck below the forward rate
and fund the purchase of this option by selling a EUR call /
USD put with a strike above the forward.
– The current EUR / USD spot rate is 1,2000.
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Derivatives
Options
Risk reversal (cylinder)
Payoff diagram
Final
hedge
rate
EUR put/
USD call
1,2250
1,1980
1,2250
1,1700
EUR/USD
1,1700
Underlying
exposure
EUR call/
USD put
Final hedge rate
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Derivatives
Options
Risk reversal (cylinder)
Advantages
– Fully hedged below the strike of the purchased EUR put / USD call
option
– Participation in beneficial FX moves up to the strike of the sold EUR
call / USD put option
– Zero premium
Disadvantages
– Participation in beneficial moves above 1,2250, and is not hedged
until 1,1700
The purchaser of a cylinder expects a limited move of the rate in a favorable
direction, but also requires a hedge for unfavorable moves
– The cylinder is an effective instrument for the purchaser which is
willing to risk a limited loss in order to have the opportunity to make a
limited profit
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Derivatives
Options
Straddle & Strangle
Combination of a purchased put option and a call option at the
same exercise price
– This is called a straddle
– By purchasing both options, the speculator may gain if
the currency moves substantially in either direction, or if
it moves in one direction followed by the other
Combination of a purchased put option and a call option with
different exercise price
– Call strike is higher than put strike
– This is called a strangle or bottom vertical combination
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Derivatives
Options
Foreign exchange derivatives
– Options
– Combined Options
– Exotic Options
– Case 3 Futures Cheapest to Deliver
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Derivatives
Options
Exotic options
Exotic option: option offering a variation from the standard pay-offs of the
European or American call and put options
Variations from vanilla options:
– Path dependency explicitly in the pay-off
• The contract not only depends on what the underlying asset
price is on the expiration date but also on how it got there
(Asian options and barrier options)
– Pay-offs that depend on choices made by the holder at points in
time prior to the expiration date
• e.g. compound options
– Options that are a function of more than one asset price
• e.g. basket options
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Derivatives
Options
Exotic options
This section will cover the following types:
– Vanilla participating forward
– Barrier option
– Participation forward (American knock-in forward)
– Cancellation forward (American knock-out forward)
– Compound option
– Chooser option
– Digital option
– Asian option
• Average Rate Option
• Average Strike Option
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Derivatives
Options
Participating forward
Principle :
– Our treasurer wants a combined option but he wants to take advantage
of possible increases too.
– Thus, he wants to conclude a participating forward with his bank, i.e.
a forward contract for which the rate at the expiration date depends on :
• the guaranteed limit rate
• participating percentage on the spot rate volatility
– If the client defines the guaranteed limit rate, the bank defines the
participating percentage.
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Derivatives
Options
Example participating forward
•
BF Trading contracts a participating forward, 6 months, 100.000 USD. The
guaranteed selling limit rate : USD/EUR 0.8419; % of participation = 79%
•
Scenarios after 6 mths:
– USD/EUR 0.82. BF Trading sells USD at EUR 0.8419
– USD/EUR 92. BF Trading sells USD at 0.8419 + (0.79 x (0.92-0.8419) =
0.9036 USD/EUR or 1.1067 EUR/USD.
Korte Termijn Financieel Management
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Derivatives
Options
Participating forward
Participating forward
– A binding obligation to buy or sell a certain amount of
foreign currency, on a certain date. A Participating
Forward allows you the flexibility to specify a worst
rate at which you will deal for the total amount, while
enabling you to enjoy the full benefit of favorable
exchange rate movements on a pre-agreed proportion
of the total.
– You are therefore guaranteed the worst rate at which
you will deal on the total amount, with the ability to
enjoy unlimited upside on the pre-agreed proportion.
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Derivatives
Options
Vanilla participating forward
EUR put / USD call option
– Notional:
– Maturity:
– Strike:
– Premium:
zero
– Participation:
€10mln
2m
1.1800
50%
The current EUR/USD spot rate is again 1,2000 and the strike
price is 1.1800 for 2m forward. You buy a EUR/USD put
option for 100% and sell a EUR/USD call for 50% of the
notional.
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Derivatives
Options
Vanilla participating forward
Payoff diagram
Final
hedge
rate
EUR put/
USD call
1,1980
1,1980
EUR/USD
Underlying
exposure
Final hedge rate
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Derivatives
Options
Vanilla participating forward
Advantages
– Fully hedged below the strike of the option. The worst
case hedge rate is the strike of the option, here 1,1800
– The option, unlike the forward, allows to benefit for 50%
from upward movements in EUR/USD
– Zero premium strategy
Disadvantages
– Only benefit for 50% in favorable spot moves
Purchaser of a vanilla participating forward:
– The purchaser has an underlying position and expects
that the exchange rate will move in a favorable direction,
but requires at the same time a hedge.
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Derivatives
5.3 Barrier option
Options
Barrier options:
– Options where the payoff depends on whether the underlying
asset’s price reaches a certain level during a certain period of time.
– A number of different types of barrier options regularly trade in the
over-the-counter market.
– Can be classified as either knock-out options or knock-in options.
– A knock-out option ceases to exist when the underlying asset price
reaches a certain barrier; a knock-in option comes into existence
only when the underlying asset price reaches a barrier.
The price depends upon the probability of the barrier being reached, and
the value of the underlying option if it is reached.
– They are therefore sensitive to volatility
Barrier options are applicable for any investor/borrower where their
underlying rate view is dependent upon a trigger level in the same or
different market being reached.
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Derivatives
5.3 Barrier option
Options
Example
– Consider a knock-in call option with a strike price of
EUR 100 and a knock-in barrier at EUR 110.
– Suppose the option was purchased when the underlying
asset was at EUR 90. If the option expired with the
underlying asset at EUR 103, but the underlying asset
never reached the barrier level of EUR 110 during the
life of the option, the option would expire worthless.
– On the other hand, if the underlying asset first rose to
the EUR 110 barrier, this would cause the option to
knock-in. It would then be worth EUR 3 when it expired
with the underlying asset at EUR103.
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Derivatives
5.3 Barrier option
Options
Advantages:
–
–
–
–
Cheaper than standard options
Flexibility in setting the barrier level and thus the cost of the option
Can be linked to any underlying
Customized to clients exposures
Disadvantages:
– The rate protection is contingent upon an ‘independent’ event
– The option cease to exist if spot trades lower than knock-out level.
– In case knock-out, the purchaser will loose it’s protection and will
have to re-hedge it’s underlying exposure at a worse rate compared
to the initial rate.
– In case the purchaser re-hedges instantly, the purchaser loses the
difference between the knock-out level and the initial strike rate.
The currency view of the purchaser is that the currency will
appreciate (call), or depreciate (put), but to a limited extend (not
beyond barrier).
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Derivatives
Options
Foreign exchange derivatives
– Options
– Combined Options
– Exotic Options
– Case 3 Hedging with Options
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Derivatives
Options
Case 3 Hedging with options
We are in June, 2012 and you manage a portfolio of 40 million EUR representing the
assets of a pension fund of a major German company (reference currency: EUR).
20% of the portfolio is invested in U.S. equities. You think that the Euro Sovereign Debt
Crisis will recover and therefore by December you fear a further weakening of the US
dollar against the EUR. For this reason you decide to hedge your portfolio against this
eventuality.
The EUR/USD spot rate currently is 1.30 [1.30 USD for 1 EUR], the continuously
compounded interest rates for all maturities are REUR = 1.25% for the EUR, and RUSD =
0.15% for the dollar. You have at your disposal the following European options on the
EUR/USD exchange rate which expire in December. Answer the following questions
assuming – in order to determine the amount to be hedged – that the value in EUR of the
U.S. equities remains constant between June and December.
Call EUR/ Put USD
Strike K = 1.25
Strike K = 1.30
Price C
(in USD per 1 EUR
notional)
0.0612
0.0331
delta
0.69
0.48
Put EUR/ Call USD
Price P
(in USD per 1 EUR
notional)
0.0184
0.0402
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delta
-0.30
-0.51
36
Derivatives
Options
Question A
a) You decide to use options with strike K = 1.30. Which position do you take? On what
notional amount? Which premium do you receive/pay? [Specify the currency].
b) What is the break-even rate of the EUR/USD at maturity [i.e. the exchange rate beyond
which the hedge generates a profit]?
c) You are persuaded that in December the EUR/USD exchange rate will anyway lie above
1.25. Which further option position can you take in order to profit from your idea and to
reduce the initial cost of the hedging strategy? What is the total cost of this new hedging
strategy? And what is the new break-even rate?
d) Draw in a graph the final value of the hedging strategy as of c) (for 1 EUR of notional) as
a function of the EUR/USD exchange rate ST at maturity, inserting all the relevant levels
[strikes, break even points, initial cost, …].
e) Calculate the delta of the hedging strategy, and using this figure calculate the
approximated change in value of the option strategy when the spot rate increases from
1.30 to 1.31 (all other variables remaining the same).
f)
In order to hedge the USD denominated part of your portfolio against a drop of the USD
against the EUR, instead of using options you consider using a forward contract.
Calculate the forward rate (maturity 6 months) and detail the forward position you should
take. What are the pros/cons of such a hedge with respect to a hedge that uses
options?
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Derivatives
Options
Solution:
a) The current US stocks position amounts to 40 Mio EUR x 20% = 8 million EUR
[which corresponds to 8 million x 1.30 = 10.4 million USD]. Assuming that the
counter value in EUR of the US stock remains constant between June and
December, in order to hedge against a fall of the USD [which means the EUR
becomes more expensive] we have to BUY a put 10.4 million USD / call 8 million
EUR with strike K = 1.30, spending 0.0331 USD per EUR. Therefore the total cost is
8 million x 0.0331 = 264,800 USD.
b) 1.30 + 0.0331 = 1.3331.
c) Considering that 8 million x 1.25 = 10 million USD, we SELL a call 10 million USD /
put 8 million EUR with strike K = 1.25 collecting 0.0184 USD per 1 EUR. The total
cost of the hedging strategy becomes 8 million x [0.0331 - 0.0184] = 8 million x
0.0147 = 117,600 USD.
The break-even point is: 1.30 + 0.0147 = 1.3147.
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Derivatives
Options
Solution (cont’d):
Value in USD of the hedging strategy
EUR/USD ST
‐0.0147
1.25
1.30
1.3147
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Derivatives
Options
Solution (cont’d):
e) The delta of the strategy [long put USD K = 1.30 and short call USD K = 1.25] is:
[0.48 – (–0.30)] = 0.78.
If the spot exchange rate increases by 0.01 [i.e. the dollar weakens: more dollars are
needed for 1 EUR] the strategy increases in value by: 8 million · 0.01 x 0.78 = 62,400
USD.
f)
Therefore he has to buy forward 8 million EUR – sell 10.343200 million USD
[implying a forward rate of 1.2929] maturity 6 months
Pros: The hedge which uses forward contracts has no initial cost; at the contrary the
hedge which uses options has an initial cost. The break-even point of the ‘forward’
hedge (1.2929) is lower than the one with options (1.3147)
Cons: the ‘forward’ hedge does not allow to profit from a USD strengthening; at the
contrary the ‘options‘ hedge does, up to a certain limit (1.25).
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Derivatives
Clairfield International Options
Hans Buysse
Partner
ABAF-BVFA Deputy Chairman
EFFAS Board member and treasurer
ACIIA Council member
XBRL Europe Board member
ESMA CWG CRSC member
Tel: +32 475 444 632
email: [email protected]
www.clairfield.com