Class #6, 7: FX Options, page XXX
Class Currency Options Ch 8
Introduction 2
Definitions 2
Call options 2
Example of Call option 2
Payoff diagram for a call option 2
Put options 2
Example of a put 3
Example of fx put option 3
Payoff diagram for a put option 3
Advantages of options over forwards and futures 3
Payoff charts 3
Elementary positions 4
Ratio hedges 5
Combinations 6
Hedged positions 6
Applications of options and futures 6
Example 1: Using options to set a ceiling on a fx payment 6
Example 2: Using put options to set a floor on a fx receivable 7
Example 3: Writing options to hedge against fx risk. 8
Example 4: Using options to hedge a contingent CF 8
Notation: 9
Basic principles of fx option pricing 9
Simple relationships 9
7. Put-call parity relationship for fx options 10
Introduction
I am not sure how many of you have studied options before and in what detail. They are very interesting securities, different from all the other financial securities we have seen so far for they allow for a non-linear (kinked) pattern of returns. Also they are are very simple to understand - especially since you have all seen options in real life. Let us try and remind ourselves about these examples.
Definitions
A call option is a right to buy the underlying security (in our case the fx) for a fixed price (strike or exercise price) on or before a certain date (maturity date). A simple example of this is a rain-check.
The normal logic will work if the price of the option is quoted as HC/FC. Otherwise, either convert the price, or think of the call on the HC as put on the FC, and vice-versa.
Call options
Example of Call option
Suppose that you went shopping during a X'mas sale for a Sony camcorder, selling for $700 - a must have item in todays yuppie world and the store had run out of this item. Then the store might issue you with a rain-check which would permit you to got back to the store within a month and buy the camcorder for $700. Suppose the day you went back to the store camcorders were selling for $680 then would you use your rain-check? No the rain-check would be worthless and you would just throw it away.
Going back to call options, suppose you have an option on the £, with a strike price of $1.75, and a life of 3 months. This means that during the next 3 months you may buy a £ by paying $1.75, and if the £ is selling for less than that then your option is worthless. Thus,
c = max {S - K, 0 }
Payoff diagram for a call option
Payoff diagram (at maturity) looks like - see next page
Put options
The story with put options is the similar. Put options give you the right to sell the underlying security for a fixed price (strike price) on or before a certain date (the expiration date).
Example of a put
A simple example of a put option is car insurance. Suppose you buy a new BMW for $30,000 and you have it insured for $25,000. This means that in the vent of an accident you have the right to sell the car to the insurance company for a price of $25,000. However, if the damage done to the car is slight and the car is worth $28,000 after the accident then you would obviously not exercise the option to sell your car.
Example of fx put option
Going back to options on fx, suppose you have a put option on the DM with a strike price of $2.00, and a life of 3 months. This means that during the next 3 months you have the right to sell your DM for $2. You will obviously do this only if the DM price in the market is less than $2. Otherwise, it would be to your advantage to throw away the put option and sell the £ in the open market. Thus,
p = max { K - S, 0 )
Payoff diagram for a put option
And the payoff diagram (at maturity) for a put option looks like this:See next page
European option: can be exercised only at maturity
American option: can be exercised at any time
Advantages of options over forwards and futures
1. Use options when time of CF is not known - American options can be exercised at any time.
2. Use options when cash-flow is contingent, that is, not certain
3. When want an asymmetric cash flow pattern, that is a knik in the payoff pattern. (Compared to payoff from fwd/futures, which are symmetric.)
Disadvantage of using options: Have to constantly monitor the position, for changes in risk (for delta is not constant).
Payoff charts
Let us look at the ways we can combine options with existing positions in fx, and options with options to get different pattern of returns. This is all very simple, all it requires is a knowledge of 7th grade geometry. Also, you should go through the handout I have given you, and which uses the same kind of graphical analysis.
Elementary positions
Long fx
Short fx
Long (buy) call
Short (written) call
Long (buy) put
Short (written) put
Elementary Postions
Ratio hedges
Combinations
Hedged positions
Long fx and write a call
Short fx and buy a put
Applications of options and futures
Example 1: Using options to set a ceiling on a fx payment
Suppose a Can$ has to pay £5 m sometime during the next 3 months. To hedge this the importer buys a call options on the £, and the option premium is $0.0220/£, for options with K = $1.50/£
Q1. What option should the importer buy?
Since the importer is has to make a £ payment, he should buy options that give him the right to buy £: CALL options
Q2. What is the cost incurred today?
£ 5 m . 0.0220 = $0.11 m
Q3. What is the ceiling that the importer has set on the price of the £?
The max that he will have to pay for each £ is $.022/£ + $1.50/£ = $1.552/£
Q4. What is the actual amount that the importer will pay if the spot rate at the end of 3 months is $1.46/£?
Since ST < K, the options are worthless and the importer can do better by buying at the market rate of $1.46/£. Thus, his total cost, ignoring time value of the payments, is $1.46 +$.022 = $1..482/£
Q5. What is the actual amount that the importer will pay if the spot rate at the end of 3 months is $1.55/£?
Now, ST > K and therefore it is worth exercising the options. The importer will pay his ceiling price, $1.522/£.
Payoff Chart for this strategy:
Example 2: Using put options to set a floor on a fx receivable
Suppose a Japanese company, Matsushita, has to sell Can$ 50 m sometime during the next 6 months, ans would like to lock in a minimum ¥ value for this. The price of a put option with a strike price of K = ¥ 230/$ is ¥ 4/$
Q1. What option should the importer buy?
Since Matsushita wishes to sell $, it should buy a put option on the $. This is, of course, the same as wanting to buy ¥, and therefore, an call option on the ¥.
Q2. What is the cost incurred today?
$ 50 m . ¥ 4 /$= ¥ 200
Q3. What is the floor that the Matsushita has set on the price of the £?
The min that they will have to receive for each $ is
= K - premium
= ¥ 230 - ¥4 = ¥ 226/$
Q4. What is the actual amount that they receive if the spot rate at the end of 3 months is ¥ 245/$?Since ST >K, the options are worthless and Matsushita can do better by selling at the market rate of ¥ 245/$, rather than the exercise price of ¥ 230/$. Thus, their total receipts will be
= ¥ 245/$ - ¥ 4/$
= ¥ 241/$
Q5. What is the actual amount that they receive if the spot rate at the end of 3 months is ¥ 215/$?
Now, ST < K and therefore it is worth exercising the options. Matsushita will receive their floor price, ¥ 230 - ¥4 = ¥ 226/$
Payoff Chart for this strategy:
So far our examples have shown how buying options can help in hedging fx risk. However, we can also hedge fx risk by writing (same as selling) fx options.
Example 3: Writing options to hedge against fx risk.
Texaco, USA has a large fx exposure in the form of a Can$ cash inflow from its Canadian operations. The risk to Texaco is that the Can$ may depreciate, thereby decreasing the US$ value of Texaco's Can$.
Texaco can reduce its long position in the Can$ by writing options on the Can$. This strategy is called "fully covered call writing."
The advantage of this strategy is that when Texaco writes options it receives a positive cash flow today (from the premium on the options). If the value of the Can$ falls (S($/£) decreases) then this positive cash flow helps offset the loss from depreciation. The price of this strategy is that if the Can$ appreciates, then the option buyer reaps the gains from this - rather than Texaco.
As a financial officer, your job would be to pick the best strike price. There is the following trade-off between the risk and return:
As you increase K, the premium decreases, so your revenue falls, but the chance of the options being exercised against you decreases.
As you decrease K, ...
Payoff chart for this strategy:
Example 4: Using options to hedge a contingent CF
Suppose that you submit a tender to build the new Eiffel Tower. You are not sure that you will win this bid. If you win the bid, then you will be receiving FF cash flows, and therefore you would like to buy a put option to hedge against exchange risk; but if you do not win the bid you will not have any exchange risk to hedge. Thus, you can see that you will not like to be holding a forward contract in case you lose the bid.
Let us examine what the optimal exercise policy will be when you buy a put option. There are 4 possible outcomes:
Bid accepted / Bid rejectedS > K / do not exer, get ST / do not exercise, get 0
S < K / exer, get K / still exercise, get K-S
Notation:
C, c = HC value of an American,euoropean call on one unit of fx
P, p = HC value of an American,euoropean put on one unit of fx
K = strike price
t = date you buy the option
T = expiration date
t = life of option, T - t
B(t, T) = current HC price of a $1 domestic discount bond =
B*(t, T) = current FC price of a FC1 foreign discount bond =
You have probably anticipated my next comment that options can be used to hedge fx risk. They are particularly useful in hedging contingent cash flows, or cash flows whose date is not known with certainty.
They are different from fwd contracts in that you have the choice to exercise them, unlike the case for fwd contracts which you must honor. Of course you pay a price for the right to make this choice, and this is reflected in the price (premium) that you pay for a option. Moreover, their payoff pattern is kinked, and they can be exercised before maturity.
Examples: See below
Basic principles of fx option pricing
Simple relationships
1. c,C,p,P - all are > 0
2. At expirattion: c = C = max(0, ST - K) and p = P = max(0, K - ST)
3. Always, C > c; P > p
4. As t increases the value of the option (call or put) increases
5. As K increases, value of a call decreases, of a put increases
6. A call option on the FC can be considered a put option on the HC
7. Put-call parity relationship for fx options
Portfolio CF today(t) CF at Maturity (T)
if ST < K if ST > K
1. Sell put +p -(K-S) 0
2. Borrow FC, B*St -S -S
convert to HC
3. Buy call -c 0 +S-K
4. Lend PV(K) -KB +K +K
Total ? 0 0
Therefore,
p + B*S - c - KB = 0 , or
p + B*S = c + KB
Thus knowing three terms, you can get the fourth.