Topics in Foreign Exchange Markets

by “kakuhen”

2019 September 20

Contents

1 Motivation 1

1.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Target Audience and Objective . . . . . . . . . . . . . . . . . . . 1

2 Market Structure and Derivatives 2

2.1 Microstructure of Foreign Exchange . . . . . . . . . . . . . . . . 2

2.2 A Model for Exchange Rates . . . . . . . . . . . . . . . . . . . . 3

2.3 Currency Forwards and Futures . . . . . . . . . . . . . . . . . . . 6

2.4 Currency Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Currency Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

A Further Reading 13

B Review of Probability Theory 14

1 Motivation

1.1 Historical Overview

Modern foreign exchange began in 1880—the year when the gold standard be-

gan. The gold standard was a monetary system where paper money was deﬁned

and exchanged at ﬁxed quantities of pure gold. Exchange rates were generally

allowed to ﬂuctuate ±1%. Since the end of the gold standard in 1971, however,

exchange rates have been mostly determined by market forces. Some countries

decide to “peg” their currency, to ﬁx the exchange rate to a certain amount of

another currency. The ﬂuctuation of exchange rates (or the stability of a peg)

presents a signiﬁcant risk to foreign investment, overseas revenue, and trade.

1.2 Target Audience and Objective

Intended for mathematics undergraduates with little ﬁnance knowledge, this

lecture provides a detailed introduction to the largest market in the world.

1

2 Market Structure and Derivatives

2.1 Microstructure of Foreign Exchange

Exchange rates are quoted as a currency pair X/Y , where X is the base currency

and Y is the quote currency. For instance, in USD/JPY, the base currency is

the U.S. dollar (USD) whereas the quote currency is the Japanese yen (JPY).

The value of this currency pair denotes a quote for U.S. dollars in Japanese yen.

Below is a table of some popular currency pairs.

Currency Pair Description Value

EUR/USD Euro/U.S. Dollar 1.10 USD

USD/JPY U.S. Dollar/Japanese yen 106.98 JPY

GBP/USD British Pound/U.S. Dollar 1.23 USD

USD/CHF U.S. Dollar/Swiss Franc 0.99 CHF

Table 1: Exchange rates as of 2019 September 7 at 18:15 JST

To buy currency, one needs a seller, and vice versa. There is often someone

called a market maker, who takes the other side of the buyer or seller’s trans-

action. Market makers are incentivized to “make a market” by collecting the

spread between the price they buy currency—the bid price—and the price they

sell currency—the asking price. In short, market makers proﬁt from facilitating

trade. The majority of market makers are banks and trading ﬁrms. Foreign

exchange is mostly over-the-counter, so exchange rates may diﬀer across mar-

ket makers. In practice, they are almost identical due to arbitrage, but there

are exceptions. PayPal, for instance, quotes dollar-yen rates about 2% lower

than banks since arbitrage through PayPal is impossible. This makes foreign

exchange a highly proﬁtable business for PayPal.

U.S. Consumer PayPal Japanese Company

Market Maker

1 USD 104.84 JPY

106.98 JPY1 USD

The best exchange rate is not ensured when dealing with individual market

makers, as shown above. Instead of market makers, one can use an electronic

communications network (ECN): a network where quotes are constantly posted

by market participants and orders are matched. The main advantage of ECNs

is that they oﬀer the best bid/ask price possible, given the quotes from all

market participants connected to the network. ECNs proﬁt from commissions

on trades. There are two types of ECNs: retail and institutional. The former

is for individuals whereas the latter is for large corporations and investment

companies. In either case, ECNs provide a quick way to trade currency.

Prices are ultimately set by the chaotic actions of market participants. While

economic sentiment inﬂuences rates in the long-term, the short-term is random.

2

2.2 A Model for Exchange Rates

Suppose there are riskless American and Japanese assets with returns r

1

and

r

2

, respectively. Then we can model these assets as follows.

dA = r

1

A(t) (2.2.1)

dB = r

2

B(t) (2.2.2)

Notice this setup as is normalizes the initial values of A(t) and B(t) to 1. Let

the dollar-yen exchange rate follow geometric Brownian motion (GBM), that is,

satisfy the stochastic diﬀerential equation

dY

t

= µY

t

dt + σY

t

dW

t

(2.2.3)

where µ is the percentage drift and σ is the percentage volatility. We use GBM

to avoid negative values and model relative changes in Y

t

as Brownian motion.

To solve equation (2.2.3), we will use Itˆo calculus, but ﬁrst deﬁne some things.

Deﬁnition 2.2.1. Let W (t) be a Wiener process (Brownian motion) with ﬁl-

tration F

t

, and let 0 = t

0

< ··· < t

n

= t be a partition P . For a locally bound,

left-continuous and F

t

-adapted process X(t), we deﬁne the Itˆo integral

Z

t

0

X(s) dW

s

= lim

||P ||→0

n−1

X

k=0

X(t

k

)(W (t

k+1

) − W (t

k

))

where ||P || is the mesh of the partition P .

Deﬁnition 2.2.2. An adapted process X(t) is an Itˆo process if it can be written

as the sum of an integral with respect to time and an Itˆo integral.

Theorem 2.2.3. Let dX

t

= µ(t, X) dt + σ(t, X) dW

t

be an Itˆo process and

f(t, X(t)) be a twice-diﬀerentiable scalar function. Then

df = f

0

(t, X(t)) dX

t

+

1

2

f

00

(t, X(t)) (dX

t

)

2

=

∂f

∂t

+ µ(t, X)

∂f

∂X

+

1

2

σ(t, X)

2

∂

2

f

∂X

2

dt + σ(t, X)

∂f

∂X

dW

t

.

We are now ready to solve equation (2.2.3). Apply Itˆo’s lemma (Theorem

2.2.3) with f(Y (t)) = ln(Y (t)) to obtain

df =

∂f

∂t

+ µY

∂f

∂Y

+

1

2

σ

2

Y

2

∂

2

f

∂Y

2

dt + σY

∂f

∂Y

dW

t

(2.2.4)

=

µ −

1

2

σ

2

dt + σ dW

t

. (2.2.5)

Integrating both sides and simplifying, we obtain

Y (t) = Y (0) exp

µ −

1

2

σ

2

t + σW (t)

. (2.2.6)

The value of the riskless American asset in Japanese yen is then

A(t)Y (t) = Y (0) exp

µ + r

1

−

1

2

σ

2

t + σW (t)

. (2.2.7)

3

In order to price derivatives, it is often useful to look at asset prices under

an equivalent martingale measure such as the following.

Deﬁnition 2.2.4. Let X(t) be a stochastic process on the probability space

(Ω, F, P). A probability measure Q is a risk-neutral measure if Q is equivalent

to P and X(t) is a martingale under this measure.

The motivation behind the risk-neutral measure is to be able to discount

expected values of assets with only one value—the risk-free interest rate—as

opposed to several values. But what do we mean by “discount?”

Deﬁnition 2.2.5. Suppose r(t) is an interest rate process. Then the process

D(t) = exp

−

Z

t

0

r(s) ds

is called a discount process.

Note that r(t) is either random or deterministic, depending on the time

horizon. Over short periods of time, our return is certain. We are uncertain over

long periods of time due to not knowing the future interest rate. The motivation

behind this deﬁnition is to determine the present value of an investment, since

we need consistent times and dates in order to compare payoﬀs from diﬀerent

investments.

Let Q

1

and Q

2

represent the risk-neutral measure for the American and

Japanese investor, respectively. To compute the exchange rate (equation (2.2.3))

under Q

2

, we require the process

exp(−r

2

t)A(t)Y (t) = exp

µ + r

1

− r

2

−

1

2

σ

2

t + σW (t)

(2.2.8)

to be a martingale. Since exp(−

1

2

σ

2

t + σW (t)) is already a martingale, we only

need the condition µ = r

2

−r

1

. Now let’s consider the exchange rate under Q

1

.

This time we have

exp(−r

1

t)

B(t)

Y (t)

= exp

r

2

− r

1

− µ +

1

2

σ

2

t − σW (t)

(2.2.9)

which is a martingale when µ = r

2

− r

1

+ σ

2

. Unless the risk-free interest rate

in the U.S. and Japan are identical, both measures necessarily disagree on the

drift coeﬃcient! This paradoxical outcome reminds to rely on risk-free measures

not as a price process, but as a tool for arbitrage-free pricing.

We can generalize risk-neutral pricing with the equation

D(t)S(t) = E

Q

[D(T )S(T )|F(t)] (2.2.10)

where 0 ≤ t < T . This equation implies that the discounted price process is

a Q-martingale. When S(t) follows GBM, we can derive risk-neutral pricing

through the discounted price process as follows.

4

We will compute d[D(t)S(t)], but ﬁrst deﬁne a product rule for Itˆo processes.

Apply Itˆo’s lemma on f(X(t)) = X(t)

2

to obtain

df = f

0

(X(t)) dX

t

+

1

2

f

00

(X(t)) (dX

t

)

2

= 2X(t) dX

t

+ (dX

t

)

2

. (2.2.11)

In general, the product xy is equivalent to

1

2

((x + y)

2

−x

2

−y

2

). By linearity of

the diﬀerential operator, we have for two arbitrary Itˆo processes X(t) and Y (t)

d[X(t)Y (t)] =

1

2

d[X(t) + Y (t)]

2

− dX

2

t

− dY

2

t

. (2.2.12)

Simplifying the above and using equation (2.2.11), we obtain Itˆo’s product rule.

Theorem 2.2.6. For two arbitrary Itˆo processes X(t) and X(t), it follows that

d[X(t)Y (t)] = X(t) dY

t

+ Y (t) dX

t

+ d[X(t), Y (t)]

where d[X(t), Y (t)] = dX

t

dY

t

is the quadratic covariation of X(t) and Y (t).

We can now apply Itˆo’s product rule on D(t)S(t) to obtain

d[D(t)S(t)] = D(t) dS(t) + S(t) dD(t) + d[D(t), S(t)] (2.2.13)

= D(t)[µS(t) dt + σS(t) dW

t

] − rD(t)S(t) dt (2.2.14)

= (µ − r)D(t)S(t) + σD(t)S(t) dW

t

(2.2.15)

= σD(t)S(t)[λ dt + dW

t

] (2.2.16)

where λ =

µ−r

σ

is the market price of risk. By Girsanov’s theorem, there exists

an equivalent measure Q such that dW

t

= dW

Q

t

− λ dt. Then

d[D(t)S(t)] = σD(t)S(t)dW

Q

t

implies D(t)S(t) is a Q-martingale. This easy derivation of risk-neutral pricing

may be why GBM is a popular choice in price models, but it should be noted

a major pitfall is its assumption of constant volatility. However, stochastic

volatility models are beyond the scope of this lecture and will not be discussed.

0 50 100 150 200 250

1

1.5

2

2.5

Days since 2018 August 30

1-month USD/JPY Volatility (%)

5

2.3 Currency Forwards and Futures

A forward contract is an agreement to buy or sell an asset at a speciﬁc future

time and price. The price of a forward depends on several factors, such as the

risk-free interest rate and the cost of carrying the asset. Assuming no carrying

cost, we have the function

F (t) = S(t)e

r(T −t)

(2.3.1)

where S(t) is the price of the asset at time t, r is the risk-free interest rate, and

(T −t) is the time till delivery. We require r to equal the risk-free interest rate

since arbitrage would exist otherwise. When S(t) follows geometric Brownian

motion (GBM), there is an interesting relation between the spot and forward

price.

Let S(t) be the solution to the stochastic diﬀerential equation

dS

S

= µ dt + σ dW

t

. (2.3.2)

Applying Itˆo’s lemma on F (t, S(t)), we obtain

dF =

∂F

∂t

+ µS

∂F

∂S

+

1

2

σ

2

∂

2

F

∂S

2

dt + σS

∂F

∂S

dW

t

(2.3.3)

= [µSe

r(T −t)

− rSe

r(T −t)

] dt + σSe

r(T −t)

dW

t

. (2.3.4)

By equation (2.3.1) we have the diﬀerential equation

dF

F

= (µ − r) dt + σ dW

t

. (2.3.5)

The forward price process is also a GBM, but with a percentage drift of µ − r.

We can also derive forward prices under a risk-neutral measure. Consider

a zero-coupon bond paying 1 at time T . We will denote its value at time t as

B(t, T ). It follows that

B(t, T ) =

1

D(t)

E

Q

[D(T )|F(t)]. (2.3.6)

The discounted payoﬀ of a forward under Q equals

1

D(t)

E

Q

[D(T )(S(T ) − K)|F(t)] (2.3.7)

where K is the delivery price of the forward. Since the above is a martingale,

it follows that the above equals

1

D(t)

E

Q

[D(T )S(T )|F(t)] − E

Q

[D(T )K|F(t)]

= S(t) −KB(t, T ) (2.3.8)

Forwards have an initial value of 0, so we necessarily have

F (t) = K =

S(t)

B(t, T )

(2.3.9)

6

as the price of the forward. One can also prove equation (2.3.9) is the forward

price by contradiction—if it does not hold, then arbitrage exists. The proof is

simple and therefore left as an exercise.

We will now apply this discussion to currencies. If we interpret currencies

as assets with a known yield, then we can deﬁne a currency forward as

F (t) = S(t)e

(r−r

f

)(T −t)

(2.3.10)

where r and r

f

are the domestic and foreign risk-free interest rates, respectively.

Notice that if we assume S(t) is deﬁned as a GBM (equation (2.3.2)), then F (t)

has the drift rate of r − r

f

percent. This helps explain why exchange rates

depend on interest rates between countries.

Although currency forwards can hedge against foreign exchange risk, it might

be diﬃcult to ﬁnd a counterparty, and forward contracts are generally non-

cancellable. These shortcomings are ﬁxed in the futures market. A futures

contract is an exchange-traded derivative for buying or selling a standardized

asset at a certain future time and price. Market participants can enter and exit

futures positions. Overall, futures diﬀer from forwards due to being cancellable,

exchange-traded, and standardized.

Futures can be replicated by buying and selling forward contracts daily and

settling cash one day after entering a contract. However, this requires negligible

default risk and ample market liquidity. Even if these conditions are met, the

extent at which the cash ﬂows will hedge against price movements is uncertain.

We will denote the price at time t of a futures contract delivering the asset

S(t) at time T as Fut

S

(t, T ). Since the replication results in a contract worth

zero, the futures price satisﬁes the equation

1

D(t)

E

Q

[D(T )(S(T ) − Fut

S

(T, T ))|F(t)] = 0. (2.3.11)

If the interest rate process in D(t) is deterministic, then we immediately have

Fut

S

(t, T ) = E

Q

[S(T )|F(t)] = S(t)e

r(T −t)

(2.3.12)

as the price of the futures contract. This looks identical to a forward, but there

is a diﬀerence. Forwards use a zero-coupon bond as their num´eraire whereas

futures use a risk-free asset. For this reason, we distinguish forward prices with

a T-forward measure deﬁned by the Radon-Nikod´ym derivative

dQ

T

dQ

=

D(T )

E

Q

[D(T )]

.

In general, we have the following relation between forwards and futures.

F (t) = E

Q

T

[S(T )|F(t)] = E

Q

T

[Fut

S

(T, T )|F(t)]

= B(t, T )E

Q

[D(T )Fut

S

(T, T )|F(t)] = Fut

S

(t, T )e

σ

B

σ

Fut

ρ

The e

σ

B

σ

Fut

ρ

term results from B(t, T ) and Fut

S

(t, T ) being correlated log-

normal random variables. Deterministic interest rates imply σ

B

= 0 and thus

F (t) = Fut

S

(t, T ). Stochastic interest rates imply diﬀerent prices.

7

2.4 Currency Swaps

Yet another way to manage foreign exchange risk is through currency swaps.

Many types of currency swaps exist due to interest rate swaps and the ﬂexibility

of OTC derivatives. In this lecture, we will only discuss ﬁxed-for-ﬁxed currency

swaps, where two parties lend each other currencies at a ﬁxed interest rate.

The ﬁxed-for-ﬁxed currency swap begins with an agreed notional exchange.

Company A Company B

N

1

N

2

Then both parties pay a ﬁxed interest rate for an agreed period of time.

Company A Company B

r

1

%

r

2

%

At the end of the swap, notional amounts are returned.

Company A Company B

N

1

N

2

Table 2 is an example of such currency swap. Overall, Company A pays Japanese

yen to receive U.S. dollars whereas Company B pays U.S. dollars to receive

Japanese yen.

Year A pays B B pays A

0 $1,000,000 U110,000,000

1 U275,000 $22,500

2 U275,000 $22,500

3 U110,275,000 $1,022,500

Table 2: Hypothetical 3-year currency swap transactions

In order to value the above swap, compute the present value of each trans-

action. We will focus on Company B, who receives 0.25% annually in yen and

pays 2.25% in dollars annually. Let the risk-free interest rate in the United

States and Japan be 2% and −0.10%, respectively. Further suppose that the

dollar-yen exchange rate is initially U110. Then the present value of the initial

exchange is trivially zero. The forward dollar-yen exchange rate each year is, by

equation (2.3.10), U107.71, U105.48, and U103.28. The next step is to convert

the yen cash ﬂows to dollars with the appropriate forward exchange rate. After

doing so, discount all dollar cash ﬂows by the risk-free interest rate in the United

States. The sum of these discounted cash ﬂows is the value of the swap. The

value of this swap turns out to be $3, 893.05.

The purpose of these derivatives is usually to transform a form of debt into

another form of debt. For instance, companies can borrow foreign currency

with an interest rate is not achievable otherwise. Another usage is to reduce

foreign exchange risk when handling certain cash ﬂows in a foreign currency.

Overall, currency swaps are highly ﬂexible instruments used to transform various

parameters in debt and the value of foreign cash ﬂows.

8

2.5 Currency Options

While currency forwards, futures, and swaps are useful for hedging risk in certain

future cash ﬂows, they can not hedge risk in uncertain future cash ﬂows. Options

are able to do this kind of hedging. An option gives the right, but not the

obligation, to buy or sell an asset at a certain price (called the strike price)

before a certain expiration date. An option giving the right to buy is a call

option, whereas an option giving the right to sell is a put option. Options are

versatile derivatives, allowing many ways to transfer, mitigate, or assume risk.

One of the most fundamental questions in mathematical ﬁnance is options

pricing. How do we “correctly” calculate the price of an option? We will begin

with the famous Black-Scholes model, a continuous-time model published in

1973 by economists Myron Scholes and Robert C. Merton. The model makes

the following assumptions.

1. There is a riskless asset earning the risk-free interest rate and a risky asset

following geometric Brownian motion (GBM) that pays no dividends.

2. Assets can be bought and sold in arbitrary amounts.

3. There are no arbitrage opportunities or transaction costs.

Let S(t) denote the value of the risky asset. We seek a formula for the value of

a portfolio holding one option V (t, S(t)) at time t. Itˆo’s lemma yields

dV =

∂V

∂t

+ µS

∂V

∂S

+

1

2

σ

2

S

2

∂

2

V

∂S

2

dt + σS

∂V

∂S

dW

t

. (2.5.1)

Suppose the portfolio also holds a certain quantity ∆ of the risky asset S(t).

Then the value of our portfolio is

d(V +∆S) =

∂V

∂t

+ µS

∂V

∂S

+

1

2

σ

2

S

2

∂

2

V

∂S

2

+ µ∆S

dt+

σS

∂V

∂S

+ σ∆S

dW

t

(2.5.2)

which loses randomness when ∆ = −

∂V

∂S

. In other words, we have

d(V + ∆S) =

∂V

∂t

+

1

2

σ

2

S

2

∂

2

V

∂S

2

dt (2.5.3)

and therefore a riskless portfolio! Since riskless portfolios earn the risk-free

interest rate, it follows that

∂V

∂t

+

1

2

σ

2

S

2

∂

2

V

∂S

2

dt = r

V − S

∂V

∂S

dt (2.5.4)

resulting in the famous Black-Scholes equation

∂V

∂t

+

1

2

σ

2

S

2

∂

2

V

∂S

2

+ rS

∂V

∂S

− rV = 0. (2.5.5)

9

For call options, use the boundary condition V (T, S(T )) = (S(T ) −K)

+

, where

K is the strike price and T is the time at expiration. As for put options, use

the boundary condition V (T, S(T )) = (K − S(T ))

+

. While it is possible to

solve this equation using risk-neutral measures and the Feynman-Kac formula,

we will skip to the solutions for sake of brevity.

For a call and put option, we have

C(t, S(t)) = S(t)Φ(d

1

) − Ke

−r(T −t)

Φ(d

2

) (2.5.6)

P (t, S(t)) = Φ(−d

2

)Ke

−r(T −t)

− S(t)Φ(−d

1

) (2.5.7)

where Φ(·) is the cumulative distribution function for the standard normal dis-

tribution, and

d

1

=

1

σ

√

T − t

ln

S(t)

K

+

r +

1

2

σ

2

(T − t)

d

2

= d

1

− σ

√

T − t

where r is the risk-free interest rate, σ is the underlying asset’s volatility, and

the quantity T − t is the time to expiration. Notice d

1

and d

2

are standard

normal variables (this is easily veriﬁed with equation (2.2.5)). We assumed no

dividends, but we will now weaken that assumption.

Assuming the underlying asset continuously pays dividends, we can deﬁne

S(T ) under the risk-neutral measure Q as

S(T ) = S(t) exp

r − q −

1

2

σ

2

τ + σ

√

τZ

(2.5.8)

where τ = T −t is the time to expiration, Z =

W

Q

(T )−W

Q

(t)

√

T −t

is a standard normal

variable, and q is a constant dividend rate. Notice S(t) is F(t)-measurable

whereas exp

r − q −

1

2

σ

2

τ + σ

√

τZ

is independent of F(t). By equation

(2.2.10), the price of a call option on S(t) equals

C(t, S(t)) = E

Q

[exp(−rτ)(S(T ) − K)

+

|F(t)] (2.5.9)

and after some expansion

E

Q

"

exp(−rτ)

S(t) exp

r − q −

1

2

σ

2

τ − σ

√

τZ

− K

+

#

. (2.5.10)

Redeﬁne d

1

as the quantity

1

σ

√

τ

ln

S(t)

K

+

r − q +

1

2

σ

2

τ

.

10

Then equation (2.5.10) is non-zero if and only if Z < d

2

. It follows that

C(t, S(t)) =

1

√

2π

Z

d

2

−∞

S(t)e

−

(

q+

1

2

σ

2

)

τ−σ

√

τ z

− Ke

−rτ

e

−

1

2

z

2

dz (2.5.11)

=

1

√

2π

Z

d

2

−∞

S(t)e

−

(

q+

1

2

σ

2

)

τ−σ

√

τ z−

1

2

z

2

dz − Ke

−rτ

Φ(d

2

) (2.5.12)

=

S(t)

√

2π

Z

d

2

−∞

e

−qτ−

1

2

(z+σ

√

τ)

2

dz − Ke

−rτ

Φ(d

2

) (2.5.13)

=

S(t)e

−qτ

√

2π

Z

d

2

+σ

√

τ

−∞

e

−

1

2

z

2

dz − Ke

−rτ

Φ(d

2

) (2.5.14)

= S(t)e

−q(T −t)

Φ(d

1

) − Ke

−r(T −t)

Φ(d

2

) (2.5.15)

is the price of the call option. Calculations similar to the above result in

P (t, S(t)) = Ke

−r(T −t)

Φ(−d

2

) − S(t)e

−q(T −t)

Φ(−d

1

) (2.5.16)

as the price of the put option.

If we treat a country’s risk-free interest rate as their currency’s “dividend

rate,” we can use equations (2.5.15) and (2.5.16) to price currency options. Let

r = r

d

be the domestic risk-free interest rate and q = r

f

be the foreign risk-free

interest rate. One interesting aspect of this pricing model is when the put and

call prices are identical. In other words, when

C(t, S(t)) − P (t, S(t)) = 0.

Subtracting equation (2.5.18) from (2.5.17), we have

C(t, S(t)) − P (t, S(t)) = S(t)e

−r

f

(T −t)

− Ke

−r

d

(T −t)

(2.5.17)

which is referred to as put-call parity. This equation implies the prices of put

and call options are equal when

K = S(t)e

(r

d

−r

f

)(T −t)

which is the forward exchange rate (equation (2.3.10))! Buying a call and writing

a put with such K is identical to buying a forward. For instance, according to

Table 3, the 90-day dollar-yen forward exchange rate was U107.50.

Put Price Expiration Strike Price Call Price

U1.21 2019 December 17 U107.00 U1.72

U1.41 2019 December 17 U107.50 U1.41

U1.65 2019 December 17 U108.00 U1.14

Table 3: Dollar-yen options as of 2019 September 17 at 16:40 JST

Put-call parity is not limited to currency options. In general, the relationship

can be expressed as

C(t, S(t)) − P (t, S(t)) = D(t)[F (t) − K]

11

assuming a forward exists (or can be replicated) and the market is liquid (i.e.

quick and easy to buy and sell the same asset). Note that these assumptions

are the only ones we need. This relationship holds under assumptions that are

even thinner than the Black-Scholes model. The following table is yet another

example of put-call parity observed in real life.

Call Price Expiration Strike Price Put Price

$725 2020 December 18 $7,900 $625

$696 2020 December 18 $7,950 $696

$669 2020 December 18 $8,000 $719

Table 4: Nasdaq-100 option prices as of 2019 September 16

So far we have covered European-style options, which do not allow exercise

until expiration. However, there are also options allowing “early exercise.” These

are called American-style options. Since early exercise may be optimal and the

exercise strategy of the option buyer is unknown, American-style options lack

general analytical solutions. We can derive solutions by either strengthening our

assumptions or approximating the solution through numerical methods. Such

topics are beyond the scope of this lecture and will not be discussed. Instead

we will turn our attention to binary options, which oﬀer only two outcomes.

Binary options are broadly categorized into cash-or-nothing options and

asset-or-nothing options. In both types of options, the buyer of the option

is paid if and only if the underlying asset’s price is above the option’s strike

price. Otherwise they earn nothing. Consider a cash-or-nothing call option

paying one unit of cash. Under the risk-neutral measure, we have

C(t, S(t)) =

1

D(t)

E

Q

[D(T )1

{S(t)>K}

(T )|F(t)] (2.5.18)

= e

−r(T −t)

Φ(d

2

) (2.5.19)

as the price of the call option. Similarly, the price of an asset-or-nothing call

equals

C(t, S(t)) = S(t)e

−q(T −t)

Φ(d

1

). (2.5.20)

Equations (2.5.20) and (2.5.21) imply European-style call options are equivalent

to a long asset-or-nothing option with a short cash-or-nothing option paying the

strike price. These computations readily apply to currencies when deﬁning r

and q as before. Binary options are easily accessible but are considered as

a form of gambling due to malicious brokerages advertising binary options as

low-risk investments and proﬁting by taking the opposite side of their clients’

trades. Consequently, they are outlawed in many countries. Even if they are

not outlawed, the legal framework in countries such as Singapore makes binary

options trading virtually impossible without untrustworthy oﬀshore brokerages.

Needless to say, options are high-risk instruments not suitable for all in-

vestors. We will conclude this lecture with a discussion on call option risk

management. Recall the formula for a European call option. Several variables

12

inﬂuence the option’s price, such as interest rates, time, and volatility. Each

variable in equation (2.5.15) will change over time, so it is important to under-

stand the sensitivity of an option’s price to changes in these variables.

Computing partial derivatives of C(t, S(t)) yields the so-called “Greeks.” For

instance, the delta of an option is deﬁned as

∂C

∂S

= e

−q(T −t)

Φ(d

1

)

which is interpreted as the extent an option price will change in response to

movements in the underlying asset’s price. Traders often reduce their delta to

zero by the end of the day in order to hedge against gap risk—the risk that an

asset will dramatically fall or rise in price overnight. A portfolio with a delta of

zero is called a delta-neutral portfolio.

Traders also monitor the second-order partial derivative gamma, which is

deﬁned as

∂

2

C

∂S

2

=

e

−q(T −t)

ϕ(d

1

)

S(t)σ

√

T − t

.

Gamma is important to monitor while delta-neutral because large values of

gamma mean that the delta-neutral position holds for a narrow price range.

Since options are inﬂuenced by not only the underlying asset price, but also

its volatility, traders monitor vega, deﬁned as

∂C

∂σ

= S(t)e

−q(T −t)

ϕ(d

1

)

√

T − t.

Vega is often used to ﬁne tune volatility exposure. For instance, a short-vega

portfolio proﬁts from the underlying asset’s price trading within a relatively

narrow range, whereas a long-vega portfolio proﬁts from the exact opposite.

Many long and short-vega portfolios are delta-neutral, since they risk a price’s

range rather than its direction.

There are also other Greeks such as theta (the quantity

∂C

∂t

) and rho (the

quantity

∂C

∂r

). Rho is seldom used since most options expire under three months

and short-term interest rates do not signiﬁcantly change over such period. As

for theta, it reminds us that options contain an extrinsic value that vanishes as

the expiration date approaches. Theta is almost always negative when buying

options. Overall, the partial derivatives of option prices are central to options

risk management, as they help people understand the nuances of options.

A Further Reading

1. John C. Hull, Options, Futures, and Other Derivatives (10

th

ed.), Pearson,

2017.

2. Steven E. Shreve, Stochastic Calculus for Finance II: Continuous-time

Models, Springer-Verlag New York, 2004.

13

B Review of Probability Theory

Deﬁnition B.0.1. Let Ω be a non-empty set and F be a σ-algebra of Ω. Then

a probability measure P is a mapping P : F → [0, 1] such that P (Ω) = 1 and

P

[

i∈I

A

i

!

=

X

i∈I

P (A

i

).

Deﬁnition B.0.2. The triple (Ω, F, P ) is called a probability space.

Deﬁnition B.0.3. Let (Ω, F, P ) be a probability space. A function X : Ω → R

is an F-measurable random variable if X

−1

(B) ∈ F for every Borel set B of R.

Deﬁnition B.0.4. The expectation of a random variable X is a probability-

weighed average deﬁned as

E[X] =

Z

Ω

X dP.

Deﬁnition B.0.5. A random variable is integrable if

Z

Ω

|X|dP < ∞.

Deﬁnition B.0.6. Let (Ω, F, P ) be as before. Then two sets A, B ∈ F are

independent if P (A ∩ B) = P (A)P (B).

Deﬁnition B.0.7. Two random variables X, Y are independent if for any two

Borel sets A, B of R it follows that X

−1

(A) and Y

−1

(B) are independent.

Deﬁnition B.0.8. Let X be an integrable random variable on the probability

space (Ω, F, P ) and G ⊆ F be a sub-σ-algebra. Then E[X|G] is a G-measurable

random variable such that

Z

A

E[X|G] dP =

Z

A

X dP

for all A ∈ G.

Deﬁnition B.0.9. A sequence of σ-algebras F

1

⊆ ··· ⊆ F

n

is called a ﬁltration.

Deﬁnition B.0.10. A stochastic process X : Ω ×T → S is a family of random

variables indexed by T and deﬁned on a common probability space. Stochastic

processes are often denoted as X(t), X(t, ω), or X

t

, where t ∈ T and ω ∈ Ω.

Deﬁnition B.0.11. A stochastic process X is a martingale if E[X

t

|X

s

] = E[X

s

]

for all s ≤ t.

Deﬁnition B.0.12. An F

i

-adapted process is a stochastic process that is mea-

surable by some ﬁltration (F

i

)

i∈I

.

14