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
