Pricing forward & future contracts

THE FORWARD MARKET
I. INTRODUCTION
A. Definition of a Forward
Contract
an agreement between a bank and a
customer to deliver a specified amount
of currency against another currency at a
specified future date and at a fixed
exchange rate.

THE FORWARD MARKET
2. Purpose of a Forward:
Hedging
the act of reducing exchange rate risk.

THE FORWARD MARKET
B. Forward Rate Quotations
1. Two Methods:
a. Outright Rate: quoted to commercial
customers.
b. Swap Rate: quoted in the interbank
market as a discount or premium.

THE FORWARD MARKET
CALCULATING THE FORWARD PREMIUM
OR DISCOUNT
= F-S x 12 x 100
S n
where F = the forward rate of exchange
S = the spot rate of exchange
n = the number of months in the
forward contract

THE FORWARD MARKET
C. Forward Contract Maturities
1. Contract Terms
a. 30-day
b. 90-day
c. 180-day
d. 360-day
2. Longer-term Contracts

Meaning of Interest Rate Parity
 Uses nominal interest rates to analyze the relationship
between spot rate and a corresponding forward rate
 Relates interest rate differentials between home country
and foreign country to the forward premium/discount on
the foreign currency
 The size of the forward premium or discount on a currency
should be equal to the interest rate differential between
the countries of concern
 If nominal interest rates are higher in country A than
country B, the forward rate for country B’s currency should
be at a premium sufficient to prevent arbitrage

INTEREST RATE PARITY
THEORY
1. The Theory states:
the forward rate (F) differs
from the spot rate (S) at
equilibrium by an amount equal
to the interest differential (rh -
rf) between two
countries.

THEORY
2. The forward premium or
discount equals the interest
rate differential.
(F - S)/S = (rh - rf)
where rh = the home rate
rf = the foreign rate

THEORY
3. In equilibrium, returns on
currencies will be the same
i. e. No profit will be realized and
interest parity exists which can be
written
(1 + rh) = F
(1 + rf) S

THEORY
B.Covered Interest Arbitrage
1. Conditions required:
interest rate differential does not equal
the forward premium or
discount.
2. Funds will move to a country with a
more attractive rate.

THEORY
3. Market pressures develop:
a. As one currency is more demanded
spot and sold forward.
b. Inflow of fund depresses interest rates.
c. Parity eventually reached.

THEORY
C. Summary:
Interest Rate Parity states:
1. Higher interest rates on a
currency offset by forward
discounts.
2. Lower interest rates are
offset by forward premiums.

Covered Interest Arbitrage (CIA)

Uncovered Interest Arbitrage (UIA): The Yen
Carry Trade
In the yen carry trade, the investor borrows Japanese yen at relatively low interest rates, converts
the proceeds to another currency such as the U.S. dollar where the funds are invested at a higher
interest rate for a term. At the end of the period, the investor exchanges the dollars back to yen to
repay the loan, pocketing the difference as arbitrage profit. If the spot rate at the end of the period
is roughly the same as at the start, or the yen has fallen in value against the dollar, the investor
profits. If, however, the yen were to appreciate versus the dollar over the period, the investment
may result in significant loss.

Implications of IRP
 If domestic interest rates are less than foreign
interest rates, foreign currency must trade at a
forward discount to offset any benefit of higher
interest rates in foreign country to prevent
arbitrage
 If foreign currency does not trade at a forward
discount or if the forward discount is not large
enough to offset the interest rate advantage of
foreign country, arbitrage opportunity exists for
domestic investors. Domestic investors can
benefit by investing in the foreign market

Implications of IRP
 If domestic interest rates are more than foreign
interest rates, foreign currency must trade at a
forward premium to offset any benefit of higher
interest rates in domestic country to prevent
arbitrage
 If foreign currency does not trade at a forward
premium or if the forward premium is not large
enough to offset the interest rate advantage of
domestic country, arbitrage opportunity exists for
foreign investors. Foreign investors can benefit by
investing in the domestic market

Trading Underlying v/s Trading
Single Stock Futures
To trade securities ----
 a customer must open a security trading account with a
broker and a demat account with a depository.
 Buying security involves putting up all the money upfront.
 With the purchase of shares of a company, the holder
becomes a part owner of the company.
 The shareholder typically receives the rights and
privileges associated with the security --- dividends,
invitation to the annual shareholders meeting and the
power to vote.
 Selling securities involves buying the security before
selling it

To trade futures ----
 a customer must open a futures trading account with a
derivatives broker.
 Buying futures simply involves putting in the margin money.
 They enable the futures traders to take a position in the
underlying security without having to open an account with a
securities broker.
 With the purchase of futures on a security, the holder
essentially makes a legally binding promise or obligation to buy
the underlying security at some point in the future (the
expiration date of the contract).
 Security futures do not represent ownership in a corporation
and the holder is therefore not regarded as a shareholder.

 Selling security futures without previously
owning them simply obligates the trader to
selling a certain amount of the underlying
security at some point in the future.

Futures Payoffs
Futures contracts have linear or symmetrical
payoffs.
It means that the losses as well as profits for
the buyer and the seller of a futures contract
are unlimited.

Payoff for buyer of futures: Long
futures
 The payoff for a person who buys a futures
contract is similar to the payoff for a person
who holds an asset.
 He has a potentially unlimited upside as
well as a potentially unlimited downside.

Payoff for buyer of futures: Long
futures
 Take the case of a speculator who buys a two-
month Nifty index futures contract when the
Nifty stands at 2220.
 The underlying asset in this case is the Nifty
portfolio. When the index moves up, the long
futures position starts making profits, and when
the index moves down it starts making losses.

Payoff for a buyer of Nifty futures

Payoff for seller of futures: Short
futures
 The payoff for a person who sells a futures
contract is similar to the payoff for a person
who shorts an asset.
 He has a potentially unlimited upside as
well as a potentially unlimited downside.

Payoff for seller of futures: Short
futures
 Take the case of a speculator who sells a
two-month Nifty index futures contract
when the Nifty stands at 2220.
 The underlying asset in this case is the Nifty
portfolio. When the index moves down, the
short futures position starts making profits,
and when the index moves up, it starts
making losses.

Payoff for a seller of Nifty futures

Pricing Futures
 Pricing of futures contract is very simple. Using
the cost-of-carry logic, we calculate the fair
value of a futures contract.
 Every time the observed price deviates from
the fair value, arbitragers would enter into
trades to capture the arbitrage profit.
 This in turn would push the futures price back
to its fair value.

Pricing Futures
 The cost of carry model used for pricing futures is
given below:
where:
r Cost of financing (using continuously compounded
interest rate)
T Time till expiration in years
e 2.71828

Pricing Futures
 Security XYZ Ltd trades in the spot market
at Rs. 1150. Money can be invested at 11%
p.a. The fair value of a one-month futures
contract on XYZ is calculated as follows:

Pricing equity index futures
 A futures contract on the stock market index
gives its owner the right and obligation to
buy or sell the portfolio of stocks
characterized by the index.
 Stock index futures are cash settled; there is
no delivery of the underlying stocks.

The main differences between commodity and
equity index futures are that:
 There are no costs of storage involved in
holding equity.
 Equity comes with a dividend stream, which
is a negative cost if you are long the stock
and a positive cost if you are short the stock.

 Therefore, Cost of carry = Financing cost -
Dividends.
 Thus, a crucial aspect of dealing with equity
futures as opposed to commodity futures is an
accurate forecasting of dividends.
 The better the forecast of dividend offered by a
security, the better is the estimate of the futures
price.

Pricing index futures given
expected dividend amount
 The pricing of index futures is also based on
the cost-of-carry model, where the carrying
cost is the cost of financing the purchase of
the portfolio underlying the index,
 minus the present value of dividends
obtained from the stocks in the index
portfolio.

Nifty futures trade on NSE as one, two and three-
month contracts. Money can be borrowed at a
rate of 10% per annum. What will be the price
of a new two-month futures contract on Nifty?
1. Let us assume that ABC Ltd. will be declaring a
dividend of Rs.20 per share after 15 days of
purchasing the contract.
2. Current value of Nifty is 4000 and Nifty trades
with a multiplier of 100.

3. Since Nifty is traded in multiples of 100,
value of the contract is 100*4000 =
Rs.400,000.
4. If ABC Ltd. Has a weight of 7% in Nifty, its
value in Nifty is Rs.28,000 i.e.(400,000 *
0.07).
5. If the market price of ABC Ltd. Is Rs.140,
then a traded unit of Nifty involves 200
shares of ABC Ltd. i.e. (28,000/140).

6. To calculate the futures price, we need to reduce
the cost-of-carry to the extent of dividend received.
The amount of dividend received is Rs.4000 i.e.
(200*20). The dividend is received 15 days later
and hence compounded only for the remainder of
45 days. To calculate the futures price we need to
compute the amount of dividend received per unit
of Nifty. Hence we divide the compounded dividend
figure by 100.

7. Thus, futures price

expected dividend yield
If the dividend flow throughout the year is generally uniform, i.e. if there
are few historical cases of clustering of dividends in any particular
month, it is useful to calculate the annual dividend yield.
(r−q)T
F = Se
where:
F--- futures price
S--- spot index value
r--- cost of financing
q--- expected dividend yield
T--- holding period

A two-month futures contract trades on the NSE.
The cost of financing is 10% and the dividend
yield on Nifty is 2% annualized. The spot value
of Nifty 4000. What is the fair value of the
futures contract?
(0.1−0.02) × (60 / 365)
Fair value = 4000e
= Rs.4052.95

The cost-of-carry model explicitly defines the
relationship between the futures price and the
related spot price.
1. As the date of expiration comes near, the basis
reduces - there is a convergence of the futures price
towards the spot price. On the date of expiration,
the basis is zero. If it is not, then there is an
arbitrage opportunity.
2. There is nothing but cost-of-carry related arbitrage
that drives the behavior of the futures price.
3. Transactions costs are very important in the
business of arbitrage.

A futures contract on a stock gives its owner the right and obligation to
buy or sell the stocks. Stock futures are also cash settled; there is no
delivery of the underlying stocks. The main differences between
commodity and stock futures are that:
1. There are no costs of storage involved in holding stock.
2. Stocks come with a dividend stream, which is a negative cost if you
are long the stock and a positive cost if you are short the stock.
Therefore, Cost of carry = Financing cost - Dividends. Thus, a crucial
aspect of dealing with stock futures as opposed to commodity
futures is an accurate forecasting of dividends. The better the
forecast of dividend offered by a security, the better is the estimate
of the futures price.

Pricing stock futures when no
dividend expected
 The pricing of stock futures is also based on
the cost-of-carry model, where the carrying
cost is the cost of financing the purchase of
the stock, minus the present value of
dividends obtained from the stock.
 If no dividends are expected during the life
of the contract, pricing futures on that stock
is very simple.

dividend expected
 XYZ futures trade on NSE as one, two and
three-month contracts. Money can be
borrowed at 10% per annum. What will be
the price of a unit of new two-month futures
contract on SBI if no dividends are expected
during the two-month period?

dividend expected
Assume that the spot price of XYZ is Rs.228.
0.10× (60/365)
Thus, futures price F = 228e
= Rs.231.90

Pricing stock futures when
dividends are expected
 When dividends are expected during the life
of the futures contract, pricing involves
reducing the cost of carry to the extent of
the dividends.
 The net carrying cost is the cost of financing
the purchase of the stock, minus the present
value of dividends obtained from the stock.

XYZ futures trade on NSE as one, two and three-
month contracts. What will be the price of a
unit of new two-month futures contract on XYZ
if dividends are expected during the two-
month period?
1. Let us assume that XYZ will be declaring a
dividend of Rs. 10 per share after 15 days of
purchasing the contract.
2. Assume that the market price of XYZ is Rs.
140.

3. To calculate the futures price, we need to
reduce the cost-of-carry to the extent of
dividend received. The amount of dividend
received is Rs.10. The dividend is received
15 days later and hence compounded only
for the remainder of 45 days.

Thus, futures price =
0.1× (60/365) 0.1× (45/365)
F = 140e − 10e
= Rs.132.20

Application of Futures
Understanding beta
 Beta of a stock measures the sensitivity of the
stocks responsiveness to these market factors.
 Beta of a portfolio, measures the portfolios
responsiveness to these market movements.
 Beta is a measure of the systematic risk or
market risk of a portfolio.
 Using index futures contracts, it is possible to
hedge the systematic risk.

Hedging- Long security, sell
futures
 Futures can be used as an effective risk-
management tool.
 Case of an investor who holds the shares of
a company and gets uncomfortable with
market movements in the short run.
(Example)
 Index futures in particular can be very
effectively used to get rid of the market risk
of a portfolio.

futures
Hence a position LONG PORTFOLIO + SHORT NIFTY
can often become one-tenth as risky as the
LONG PORTFOLIO position!
Suppose we have a portfolio of Rs. 1 million
which has a beta of 1.25. Then a complete
hedge is obtained by selling Rs.1.25 million
of Nifty futures.

futures
 Hedging does not always make money.
 The best that can be achieved using hedging
is the removal of unwanted exposure, i.e.
unnecessary risk.
 The hedged position will make less profits
than the unhedged position, half the time.

Speculation- Bullish security,
buy futures
 A trader buys a 100 shares which cost him
one lakh rupees. His hunch proves correct
and two months later the security closes at
Rs.1010. He makes a profit of Rs.1000 on an
investment of Rs. 1,00,000 for a period of
two months. This works out to an annual
return of 6 percent.

Speculation- Bullish security,
buy futures
 The same trader buys 100 security futures
@1006 for which he pays a margin of
Rs.20,000. Two months later the security
closes at 1010.
 On the day of expiration, the futures price
converges to the spot price and he makes a
profit of Rs.400 on an investment of
Rs.20,000. This works out to an annual
return of 12 percent.

Speculation- Bearish security,
sell futures
 Stock futures can be used by a speculator who
believes that a particular security is over-
valued and is likely to see a fall in price.
 How can he trade based on his opinion? In the
absence of a deferral product, there wasn't
much he could do to profit from his opinion.
Today all he needs to do is sell stock futures.

Speculation- Bearish security,
sell futures
 A trader who expects to see a fall in the price of
ABC Ltd. He sells one two-month contract of
futures on ABC at Rs.240 (each contact for 100
underlying shares).
 He pays a small margin on the same. Two
months later, when the futures contract
expires, ABC closes at 220. On the day of
expiration, the spot and the futures price
converges. He has made a clean profit of Rs.20
per share. For the one contract that he bought,
this works out to be Rs.2000.

Arbitrage: Overpriced futures:
buy spot, sell futures
If you notice that futures on a security that
you have been observing seem overpriced,
how can you cash in on this opportunity to
earn riskless profits?
Say for instance, ABC Ltd. trades at Rs.1000.
One-month ABC futures trade at Rs.1025
and seem overpriced.

Arbitrage: Overpriced futures:
buy spot, sell futures
As an arbitrageur, you can make riskless profit by entering into the following set of
transactions.
1. On day one, borrow funds, buy the security on the cash/spot market at 1000.
2. Simultaneously, sell the futures on the security at 1025.
3. Take delivery of the security purchased and hold the security for a month.
4. On the futures expiration date, the spot and the futures price converge. Now
unwind the position.
5. Say the security closes at Rs.1015. Sell the security.
6. Futures position expires with profit of Rs.10.
7. The result is a riskless profit of Rs.15 on the spot position and Rs.10 on the
futures position.
8. Return the borrowed funds.
This is termed as cash-and-carry arbitrage. Remember however, that exploiting an
arbitrage opportunity involves trading on the spot and futures market.

Arbitrage: Underpriced futures:
buy futures, sell spot
 You notice the futures on a security you
hold seem underpriced.
 How can you cash in on this opportunity to
earn riskless profits? Say for instance, ABC
Ltd. trades at Rs.1000. One-month ABC
futures trade at Rs. 965 and seem
underpriced.

Arbitrage: Underpriced futures:
buy futures, sell spot
As an arbitrageur, you can make riskless profit by entering into the
following set of transactions.
1. On day one, sell the security in the cash/spot market at 1000.
2. Make delivery of the security.
3. Simultaneously, buy the futures on the security at 965.
4. On the futures expiration date, the spot and the futures price converge.
Now unwind the position.
5. Say the security closes at Rs.975. Buy back the security.
6. The futures position expires with a profit of Rs.10.
7. The result is a riskless profit of Rs.25 on the spot position and Rs.10
on the futures position.
If the returns you get by investing in riskless instruments is more than the
return from the arbitrage trades, it makes sense for you to arbitrage.
This is termed as reverse-cash-and-carry arbitrage.

Pricing forward & future contracts

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