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1. (10 pt each) XYZ flour company is planning to buy wheat for the next year’s

production. 10,000 bushels of wheat will be purchased each quarter, on the 15th of

January, April, July, and October in 2022. It wants to lock the wheat price using

futures contract, and at the same time use a swap to prepare the expenditure. The

spot price of wheat today (April 15, 2021) is $3.1 per bushel, and the company

enters futures contracts today. The prevailing interest rate is 1.5%, compounded

continuously. Recall that each futures contract is for 5000 bushels and assume no

storage cost of wheat.

(a) If the company uses a prepaid swap, what is the prepayment?

(b) If the company uses a plain vanilla swap for the quarterly purchase next year,

what is the fixed amount?

2. (30 pt) Prove the Put-Call Parity C − P + d(0, T)K = S of European options.

Mathematics of Finance II-1

1

Forwards, Futures, and Swaps

Definition 1.1. A derivative is financial security whose payoff is explicitly tied to the value of

other variable(s), usually the price of another financial security or a group of securities.

Definition 1.2. An underlying security is the security that determines the price of a derivative.

1.1

Forward Contract

Throughout the chapter, we asume that there is no arbitrage profit opportunity.

Definition 1.3. A forward contract is an agreement to trade a specific amount of asset at a

predetermined price at a future date.

For example, an investor may enter a forward contract by promising to buy an asset at a future

time T . This invester is taking a long position or going long. Nothing changes hands at time 0,

and at t = T , this long position holder pays a price F (0, T ) predetermined at t = 0 and take the

delivery of the asset from his counterparty, the short position holder. The payoff the long position

holder is

S(T ) − F (0, T )

where S(T ) is the market price of the underlying asset at t = T and F (0, T ) the forward price of

the contract that was initiated at t = 0 and has a delivery time T . The payoff of the short position

holder is

− (S(T ) − F (0, T )) = F (0, T ) − S(t).

Theorem 1.1. (Forward price formula: basic version) If an asset can be stored at no cost and sold

short, then

S(0)

F (0, T ) =

or equivalently S(0) = F (0, T )d(0, T ),

d(0, T )

where S = S(0) is the current (t = 0) spot price, F (0, T ) is the current forward price for delivery

at time T , and d(0, T ) the discount factor for the time interval [0, T ].

Proof. Assume the contrary, then derive a contradiction using the strategy “buy low, sell high”.

Case 1. Suppose F >

S

d(0,T ) ,

then buy in the spot market and sell in the forward market.

t=0

t=T

S

d(0,T )

Spot Market:

borrow S to buy a unit

debt has increased to

Forward Market:

contract to short a unit

deliver the unit to receive F

©2021 by Youngna Choi. All rights reserved.

1

Mathematics of Finance II-1

S

S

Therefore, after paying off the debt plus interest d(0,T

) , there is an arbitrage profit of F − d(0,T ) ,

contradicting to no arbitrage profit opportunity assmption.

Case 2. Suppose F <
S
d(0,T ) ,
then buy in the forward markett and sell in the market.
t=0
t=T
Spot Market:
short a unit for S and invest it
Forward Market:
contract to long a share
investment has grown to
S
d(0,T )
take the delivery and pay F
After returning the share to close the short selling, there is an atbitrage profit of
contradicting to no arbitrage profit opportunity assmption.
S
d(0,T )
− F,
Borrowing money is equivalent to selling bonds. For notational purpose, we introduce a hypothetical zero coupon bond (ZCB) whose face value is one unit cucrrency, and call it a unit ZCB.
Denote by B(t, T ) the price at time t of a unit ZCB that matures at T . When we assume that there
is only one prevailing interest rate, we can consider B(t, T ) as the discount factor during the time
interval [t, T ]. For this reason, we will use the two notations B(t, T ) and d(t, T ) interchangeably.
Theorem 1.2. (Forward price formula: general version) If an asset can be stored at no cost and
sold short, then the price of a forward contract initiate at t ≤ T for delivery at T is
F (t, T ) =
S(t)
B(t, T )
where S = S(t) is the spot price at time t, F (t, T ) is the forward price for delivery at time T , and
d(t, T ) the discount factor for the time interval [t, T ].
Proof. Case 1. Suppose F (t, T ) >

S(t)

B(t,T ) ,

then

– at time t,

buy a share of underlying asset at S(t) which is financed by selling

S(t)

B(t,T )

unit ZCB

sell in the forward market, i.e. take the short position in a forward contract at F (t, T )

– and at time T ,

deliver the asset bought at t in the forward market at F (t, T )

pay

S(t)

B(t,T )

to the bond holder

2

Mathematics of Finance II-1

⇒ profit = F (t, T )−

S(t)

> 0,

B(t, T )

Case 2. Suppose F (t, T ) <
S(t)
B(t,T ) ,
contradicts to no-arbitrage profit opportunity assumption.
then
- at time t,
enter a long forward contract at F (t, T ) (i.e., buy in the forward market)
short a share at S(t) and buy
S(t)
B(t,T )
unit ZCB
- and at time T,
receive
S(t)
B(t,T )
in the bond market
pay F (t, T ) in the forward market to take a share
use the share to unwind the short position
⇒ profit =
S(t)
−F (t, T ) > 0,

B(t, T )

contradicts to no-arbitrage profit opportunity assumption.

Example 1.1. S(0) = $17, F (0.1) = $18, r = 8%.

a. Is there an arbitrage profit opportunity?

b. If short selling requires a 30% deposit at d = 4%, what is the artibrage profit (if there is any)?

When the underlying costs money to hold, we call it “cost of carry”. When the underlying

generates income by holding, we call it “dividend”. Cost of carry and dividend are negative of each

other.

Theorem 1.3. When the underlythg asset earns dividend D(t) at t < T ,
F (0, T ) =
S(0)
D(t)
−
.
B(0, T ) B(t, T )
Proof. Consider the following intuitively cash flow: buy a share of underlying asset at S(0) at time
0, received the dividend D(t) at t, and deliver the stock to the counterparty at time T to collect
F (0, T ). The cash flow equation is
−S(0) + D(t)B(0, t) + F (0, T )B(0, T ) = 0
or equivalently
3
Mathematics of Finance II-1
−
S(0)
D(t))
−
+ F (0, T ) = 0.
B(0, T ) B(t, T )
Therefore
F (0, T ) =
S(0)
D(t)
−
.
B(0, T ) B(t, T )
Precise proof goes as follows.
Case 1. Suppose F (0, T ) >

S(0)

B(0,T )

−

D(t)

B(t,T ) ,

then

– at t = 0,

borrow S(0) by issuing

S(0)

B(0,T )

bonds

buy a unit of underlying asset at S(0)

D(t)

take a long farward position on B(t,T

) unit bonds maturing at T with delivery date t

and forward price B(t, T ) per bond (this is to invest the dividend income D(t))

take a short forward position on the underlying asset for F (0, T )

– at time t,

receive the dividend div and use it to unwind the long forward on bonds:

D(t)

units · $B(t, T )/unit = $D(t)

B(t, T )

– at time T ,

sell the underlying asset for F (0, T )

collect $1 for each bind, totalling

=⇒

Case 2. Suppose F (0, T ) <
S(0)
B(0,T )
D(t)
B(t,T ) ,
F (0, T ) +
−
D(t)
B(t,T ) ,
pay off $1 each for
S(0)
B(0,T )
units
D(t)
S(0)
−
> 0.

B(t, T ) B(0, T )

then

– at time 0,

take a long forward on a unit of underlying asset with price F (0, T ), delivery at T

short a unit of asset at S(0), invest it in

4

S(0)

B(0,T )

unit ZCB maturing at T

Mathematics of Finance II-1

(to pay dividend at t), take a short forward positnon on

1

with a forward price B(t,T

) , delivery time E

D(t)

B(t,T )

unit ZCB maturing at T

– at time t,

sell

D(t)

B(t,T )

unlts of ZCB at $B(t, T ) per unit, to collect D(t)

Pay it to the asset holder who lent it for the short selling

– at time T ,

receive $1 each from

pay out $1 each for

S(0)

B(0,T )

D(t)

B(t,T )

units of ZCB (from time 0)

units of ZCB (from time t)

buy a unit of underlying asset stock for F (0, T ) and return it to clsoe the short position

=⇒

D(t)

S(0)

−

− F (0, T ) > 0.

B(0, T ) B(t, T )

Corollary 1.4. When there is a stream of dividends that are earned at the end of each periods,

T

X D(k)

S(0)

−

F (0, T ) =

d(0, T )

d(k, T )

k=1

where D(k) is the dividend per unit of underlying asset at time k, 1 ≤ k ≤ T , and d(k, T ) is the

discount factor between k and T .

Corollary 1.5. When there is cost of carry C(k) to be paid at the end of each periods,

T −1

X C(k)

S(0)

F (0, T ) =

+

d(0, T )

d(k, T )

k=0

where C(k) is the cost of carry per unit of underlying asset at time k, 0 ≤ k ≤ T − 1, d(k, T ) is a

discount factor between k and T .

Example 1.2. Consider a stock whose price on 2000 January 1 is $120 and which will pay a

dividend of $1 on 2000 July 1 and $2 on 2000 October 1. The interest rate is 12%, compounded

continuously. Is there an arbitrage opportunity if on 2000 January 1 the forward price for delivery

of the stock on 2000 November 1 is $131? If so, compute the arbitrage profit

5

Mathematics of Finance II-1

Example 1.3. The cunent price of sugar is 12 cents per pound. The carrying cost of sugar is 1 cent

per pound per month, to be paid at the beginning of each month, and the interest rate is constant

at 9% per annum, compounded monthly. What iis the forward price of sugar to be delivered in 5

months?

Example 1.4. (A bond forward) Consider a Treasury bond with a face value of $10,000, semianual

coupons at 8% per annum, and several years to maturity. Currently this bond is selling for $9,260,

and the previous coupon has just been paid. What is the forward price for delivery of this bond in

1 year? Assume that the yield for one year out is 9%(2).

1.2

Value of a Forward Contract

Given an asset, let S(t) be the spot price at t and F (t, T ) the forward price at t for delivery at T .

Theorem 1.6. For all t such that 0 ≤ t ≤ T , the time t-value of the “long” forward contract with

forward price F (0, T ) is

fl (t) = [F (t, T ) − F (0, T )]d(t, T ).

Proof. Without any faward contract, a buyer (long position holder) will pay S(T ) at trme T .

– at time 0, take the long forward position with forward price F (0, T )

– at time t, the same person takes the short position with forward price F (t, T )

– at time T , s/he buys the asset at F (0, T ), then sells it at F (t, T )

Then the total gain is

= {S(T ) − F (0, T )}

(from the long position)

+ {F (t, T ) − S(T )}

(from the short position)

= F (t, T ) − F (0, T ),

which is the risk-free arbitrage profit.

Corollary 1.7. For all t such that 0 ≤ t ≤ T , the time t-value of the “short” forward contract with

forward price F (0, T ) is

fs (t) = [F (0, T ) − F (t, T )]d(t, T ).

Example 1.5. Suppose that the price of a stock is $45 at the beginning of the year, the risk-free

rate is 6% compounded continuously, and a $2 dividend is to be paid after half a year. For a long

forward position with delivery in one year, find its value after 9 months if the stock price at that

time turns out to be a) $49, b) $51.

References

[1] M. Capiński, T. Zastawniak, Mathematics for Finance, 2nd ed., Springer, (2011), Ch.4

[2] D. G. Luenberger, Investment Science, 2nd ed., Oxford University Press, (2014), Ch.12

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