• The market for FX options is centered mainly in five cities: Chicago, ________New York, Philadelphia and Tokyo.

*Expiration months:

*Expiration days:

*The role of the clearing house:

*Purpose/use of FX options:

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• The following principles will enable us to narrow down the possible range of FX (D/F) option values in relation to the value of other assets.

Principle #1: Limited liability for the long.

Principle #2: On the expiration date, the call option, C_t, will be worth:

C_t = max (0, S_t - E)

P_t = max (0, E - S_t)

Principle #4: Prior to expiration an valuable than a European option.

option will be more

Principle #8: options and risk:

• The development of the OPM: (calls only)

A. Assumptions Underlying the BSOP Model

1. FX values follow a _________ process.

2. Transactions costs and ___________ do not exist.

2. The risk-free rate and the variance (risk) of _______ returns are ________ while the option is alive.

4. Borrowing and lending take place at the risk-free rate.

B. The Model

C_o = FX_o x e^{-rf x t} x N(d₁) – E x N(d₂) x e^{–rd x t}

Equation #1 is a version the BSOP model and can be interpreted as follows: if an investor exercises a call on the______________date he will receive foreign exchange worth____________and will remit________ in exchange. The first term in the equation is the probability of _____given the _______ (i.e. t=0) exchange rate. The second term is the __________value of paying the exercise price if and only if ______________ (i.e. the call is ____ the-money). If FX is large relative to E (i.e. a way in-the money option) such that the probability of FX_t > E is a virtual

_________ then N(d_l) » 1 and costs C_o=FX_o^*e ^{–rf x t} - E*e ^{–rd x t}.

What are N(d₁) and N(d₂)?

N(d₁) and N(d₂) are __________ of observing a standard normal random variable within a given range. Recall that the standard normal random variable has a mean of _____ and a standard deviation of _____. Any ______ distributed _____variable (such as currency movements) can be converted to a standard normal as follows:

Z = X - m

s _x

e.g. if Z = 1.57 the corresponding standard normal table value is This means that the probability of observing a value of Z=1.57 is ______%

*d₁ and d₂ correspond to the Z-value above. The N(-) refers to the table value i.e. the ______ under the normal curve. More specifically ‘d’, can be found as follows:

d₁ = [ln(FX_o/E) + (r_d - r_f + 0.5 x s _s²)(t/365)]/[(t/365)^1/2 x s _s]

d₂ = d₁ – (t/365)^1/2 x s _s

e.g.

Put pricing is more complex than call pricing for several reasons. The complications are such that no simple closed-form solution exists for valuing a put option. Even so, in order for no arbitrage opportunities to exist there must be a well specified relationship between the put value and the call value on the same underlying currency with the same time to maturity and the same exercise price. This relationship is known as __________________ and is the only known useful methodology for valuing a put option.

P_o = [C_o + E x e^{-rd x t} – S_o]e^{-rf x t}

• What is hedging?

(1 ) Reduce ___________ in cash flows; and,

(2) Provide ________ protection during periods of ________.

• The problem:

• Throughout the following discussion let ____ represent the appropriate number of options to trade in order to __________.

1. Calls

If 'h' amount of FX is held (i.e. a long position in the cash market), would calls be bought or sold to hedge?

Under what conditions would a hedge be initiated?

The value of the hedged portfolio is:

V_o =hFX_o - C_o

What value of 'h' will leave the value of the portfolio unchanged for a given change in the stock price or call price? Taking the first _________ of both sides with respect to FX_o implies that:

h = dC_o/dFX_o = _________= the amount of FX held long (short) for each short (long) call.

NOTE N(d₁) is not the solution desired given the wording of the above question. How could the correct delta be defined?

2. Puts

If the hedger has a short position in 'h' amount of FX (i.e. the hedger anticipates the need for FX in the future), would puts be bought or sold to hedge?

Under what conditions would a hedge be initiated?

The value of the hedged portfolio is:

V_o =hFX_o + P_o

When using puts to hedge, the delta can be defined as:

h= 1 - N(d₁) = the amount of FX held long (short) for each _____( _____ ) put; or,

1 /(1 - N(d₁) = the number of puts long (short) for each unit of FX held _____ (_____).

Which alternative would provide the appropriate hedge for the above situation?

3. Puts and Calls

If the hedger has a long position in 'h' number of call contracts, would puts be bought or sold to hedge?

Under what conditions would a hedge be initiated?

The value of the portfolio is:

V_o =hC_o + P_o

When using only puts and calls (i.e. no FX is held or shorted) to hedge, the delta is defined as:

h=N(d₁)/[ 1 -N(d₁)] = the number of puts long (short) for each __________ (______) call held; or,

[1-N(d₁)]/N(d₁) = the number of calls long (short) for each _______ (______ ) put held.

Which alternative would provide the appropriate hedge for the above situation?

Question: Does it matter how large or small the existing position is when laying a hedge? If so, how should the delta be adjusted??

B. Summary of Delta Hedges

• Note: the symbol _____ means _____________.

1. N(d_l) = Amount of FX held long (short) x Q V ______(____) call contract.

(3) 1 /N(d₁) = # call contracts long (short) V _____ (_____) unit of FX held x Q.

(4) 1 /[1 - N(d₁) = #put contracts long (short) V _____ (____) unit of FX held x Q.

NOTE: To the right of the ‘V’ symbol defines the investor's current position. To the left of the ‘V’ symbol describes the hedging vehicle alternative given the current position. Q is the size of the option contract which differs depending on the currency being traded.

• The exchange rate between a domestic and foreign currency, should be equal to the ______ of the domestic and foreign call options for a given exercise price.

• Note that the equation below is just a combination of __________Parity and

___________________Parity.

S_d,f = [C_d + E_d x e^{–rd x t}]/[C_f - E_f x e^{–rf x t}]

Where:

S_d,f = the cash ______ rate (D/F);

C_d = the domestic call option _____;

C_f = the foreign call option___________;

E_d = the domestic call option's ________price;

E_f = the foreign call option's __________price;

e^{–rd x t} = the _________ value interest factor in continuous time from now until the option expires at time t (r_d is the risk-free rate of interest in the domestic country);