Forwards and Futures

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WORKBOOK RAMON RABINOVITCH
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DERIVATIVES ARE CONTRACTS Two
parties Agreement Underlying security
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DERIVATIVES FORWARDS FUTURES OPTIONS SWAPS
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• FORWARD MARKET
• THE MARKET FOR DEFERRED DELIVERY
• SELLER SHORT BUYER LONG
• THE TWO PARTIES MAKE A CONTRACT THAT DETERMINES
  THE TRANSACTION TO BE MADE ON A FUTURE DATE
• DELIVERY AND PAYMENT WILL TAKE PLACE IN THE
  FUTURE AS SPECIFIED BY THE CONTRACT BETWEEN THE
  SHORT AND LONG.

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• A FORWARD CONTRACT
• THE SHORT WILL SELL TO THE LONG 25,764 AUNCES OF
  GOLD OF A CERTAIN QUALITY. THE PRICE WILL BE
  300/AUNCE. DELIVERY AND PAYMENT WILL TAKE PLACE
  IN A SPECIFIED PLACE EXACTLY 95 DAYS FROM TODAY.
• RISKS

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• THE FUTURES MARKET
• FUTURES CONTRACTS ARE STANDARDIZED FORWARDS
  TRADED ON ORGANIZED EXCHANGES
• STANDARDIZED ARE THE COMMODITIES, THE DELIVERY
  PROCEEDURES, THE DELIVERY TIMES AND PAYMENTS.
• CLEARINGHOUSE

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• THE MARKET FOR SWAPS
• A SWAPS IS AN AGREEMENT BETWEEN TWO PARTIES
  ARRANGING CASH FLOWS DISTRIBUTION BETWEEN THE TWO
  PARTIES. THE CASH FLOWS ARE BASED ON THE VALUE OF
  AN UNDERLYING COMMODITY

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• THE MARKET FOR SWAPS
• FOR THE NEXT 5 YEARS, PARTY B WILL PAY PARTY C
  THE FIXED RATE OF 7 ON 100,000,000, WHILE PARTY
  C WILL PAY PARTY B THE 6-MONTHS LIBOR. PAYMENTS
  WILL TAKE PLACE TWICE A YEAR.
• ONLY THE NET CASH FLOW WILL EXCHANGE HANDS

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• THE MARKET FOR SWAPS
• 3,500,000 gt
• B C
• lt 6-month LIBOR
• THE UNDERLYING SECURITY IS 100,000,000

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• OPTIONS
• A contingent claim
• The options value is contingent upon the value
  of the underlying asset
• Two Types of Options
• Calls
• Puts

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Option Buyer, holder or long. In exchange for
making a payment of money the premium, the owner
of an option-the long-has a call-the right, but
not the obligation, to buy or a put-the right to
sell- a specified quantity of the underlying
commodity at the execise price before the
options expiration date.
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Option Seller, writer or short. In exchange for
receiving the premium, the options writer has
the obligation to sell the underlying asset (in
case of a call) or purchase the underlying asset
( in case of a put) at the predetermined exercise
price upon being served with an exercise notice
during the life of the option, I.e., before the
option expires.
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WHY TRADE DERIVATIVES? PRICE RISK
UNCERTAINTY VOLATILITY IS THE FUNDAMENTAL REASON
FOR TRADING DERIVATIVES
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PRICE RISK At time zero, the assets price at
time t is not known.
Probability distributio
St
S0
0
t
time
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OPTIONS NOTATIONS S the underlying assets
market price X- the exercise price
t the current date T the
expiration date T-t - time till expiration c,p -
European call, put premiums C,P American
call,put premiums
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Buyer of a call option has the right to buy the
underlying at the strike price, X, before the
call expires at T. Thus gt expects the price of
the underlying commodity to increase during the
period of the option contract.
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Seller of a call option must sell the underlying
asset for X, if exercised. Thus gt expects the
price of the underlying asset to remain below or
at the exercise price during the options
life.This way the writer keeps the premium
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Buyer of a put option has the right to sell the
underlying for X before the put expires at T.
Thus gt expects the market price of the
underlying commodity, S, to decrease during the
puts life.
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Seller of a put must buy the underlying for X if
exercised. Thus gt expects the S to remain at or
above X during the puts life. This way the put
writer keeps the premium.
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X X t T S
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• Types of Options
• American Options
• exercisable any time before expiration
• European Options
• exercisable only on expiration date
• Asian Options
• European style on the underlying average price
  during its life

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At-the-money S X In this case the intrinsic
value for both calls and puts is zero S-X X-S
0 and the premium consists of the extrinsic
value only.
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In-the-money Calls Puts S gt X S lt
X S-X gt0 X-Sgt0 The Intrinsic value is
positive
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Out-of-the-money Calls Puts SltX
SgtX S-Xlt0 X-Sgt0 The intrinsic value is
zero and the premium consists of the extrinsic
value only
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Option Price Premium Two Components
Intrinsic Value The amount by which an option is
in-the-money. Extrinsic (time) Value The
amount by which the price of an option exceeds
its intrinsic value.
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Option Market Price Premium Intrinsic value
extrinsic value Intrinsic value Calls Max0,
S-X) Puts Max0, X-S) Intrinsic value cannot be
negative
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Option Parameters Underlying Price S Strike
Price X Time to Expiration T-t Annual
Volatility s Annual Interest Rate
r Annual Dividend Rate q
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• Summary
• CALLS PUTS
• S X Feb Mar May Feb Mar Apr
• 50 35 18 19 21 .05 .15 .27
• 50 40 12 13.5 16 .25 .34 .50
• 50 50 6.5 8.25 12 .75 1 1.15
• 50 60 3 4 9 11 12 15
• All prices are in s
• Expirations At the market close on the 3rd
  Friday of the expiration month

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INTEL Thursday, September 21, 2000. S
61.48 CALLS - LAST PUTS - LAST X
oct nov jan apr oct nov jan
apr 40 22 --- 23 --- ---
--- 0.56 --- 50 12 --- ---
--- 0.63 --- --- --- 55 8.13
--- 11.5 --- 1.25 --- 3.63
--- 60 4.75 --- 8.75 --- 2.88
4 5.75 --- 65 2.50 3.88 5.75
8.63 6.00 6.63 8.38 10 70 0.94
--- 3.88 --- 9.25 --- 11.25
--- 75 0.31 --- --- 5.13 13.38
--- --- 16.79 80 --- --- 1.63
--- --- --- --- --- 90
--- --- 0.81 --- --- ---
--- --- 95 --- --- 0.44 ---
--- --- --- ---
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WHY TRADE OPTIONS ON ORGANIZED EXCHANGES? IN THE
OVER-THE-COUNTER MARKET (OTC) INVESTORS ARE
EXPOSED TO Credit risk Operational risk Liquidity
risk
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• CREDIT RISK
• Does the other party have enough resources to
  meet its obligation?

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2. Operational risk Will the other party deliver
the underlying if I exercise my call? Will the
other party take delivery of the underlying if I
exercise my put ?
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• Market liquidity.
• In case the long wishes to get out of the market,
  what are the obstacles?
• In case the short wishes to quit its obligation,
  what to do?
• In other words how can the two parties come out
  of their respective obligations?

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THE GUARANTEE The exchanges understood that there
will exist no efficient options markets until the
above problems are resolved. So they have created
the OPTIONS CLEARING CORPORATION
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THE OPTION CLEARING CORPORATION PLACE IN THE
MARKET
EXCHANGE CORPORATION
OPTIONS CLEARING CORPORATION
NONCLEARING MEMEBRS
CLEARING MEMBERS
OCC MEMBER
BROKERS
CLIENTES
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The Options Clearing Corporation (OCC) gives all
the LONGS the absolute guarantee of the
completion of its side of the contract You will
always be able to exercise your option!!! The
OCCs absolute guarantee provides traders with
a default-free market. Thus, any investor who
wishes to engage in options buying knows that
there will be no operational default.
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The OCC also clears all options trading and
maintains the list of all long and short
positions. Every long position must be MATCHED
with a short position. Hence, the total sum of
all options traders positions must be ZERO at all
times. The OCCs absolute guarantee together
with the trading list makes the market very
liquid. ONE traders are not afraid to enter
the market TWO traders can quit the market at
any point in time by OFFSETTING their original
position.
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OFFSETTING POSITIONS A trader with a LONG
position who wishes to get out of the market must
open a SHORT position with equal number of the
same options on the same underlying asset for the
same month of expiration and for the same
exercise price. EX LONG 5, SEP, 125, IBM puts
offsets this position by SHORT 5, SEP,
125, IBM puts. The above trades offset each
other and zero out the traders position with the
OCC. This trader is out of the market. The
premiums paid and received upon entering the
above positions determine the traders profit or
loss.
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OFFSETTING POSITIONS A trader with a SHORT
position who wishes to get out of the market must
open a LONG position with equal number of the
same options on the same underlying asset for the
same month of expiration and for the same
exercise price. EX SHORT 25, JAN, 75, BA
calls offsets this position by LONG
25, JAN, 75, BA calls. The above trades offset
each other and zero out the traders position
with the OCC. This trader is out of the market.
The premiums paid and received upon entering the
above positions determine the traders profit or
loss.
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Some Financial Economics Principles

• Arbitrage A market situation whereby
• an investor can make a profit with
• no equity and no risk.
• Efficiency A market is said to be
• efficient if prices are such that there exist
• no arbitrage opportunities.
• Alternatively, a market is said to be
• inefficient if prices present arbitrage
• opportunities for investors in this market.

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• Valuation The current market value (price) of
  any project or investment is the net present
  value of all the future expected cash flows from
  the project.
• One-Price Law Any two projects whose cash flows
  are equal in every possible state of the world
  have the same market value.
• Domination Let two projects have equal cash
  flows in all possible states of the world but
  one. The project with the higher cash flow in
  that particular state of the world has a higher
  current market value and thus, is said to
  dominate the other project.

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• A proof by contradiction is a method of proving
  that an assumption, or a set of assumptions, is
  incorrect by showing that the implication of the
  assumptions contradicts these very same
  assumptions.
• Risk-Free Asset is a security of investment
  whose return carries no risk. Thus, the return
  on this security is known and guaranteed in
  advance.
• Risk-Free Borrowing And Landing By purchasing
  the risk-free asset, investors lend their capital
  and by selling the risk-free asset, investors
  borrow capita at the risk-free rate.

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• The One-Price Law
• There exists only one risk-free rate in an
  efficient economy.

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Compounded Interest

• Any principal amount, P, invested at an
• annual interest rate, r, compounded
• annually, for n years would grow to
• An P(1 r)n.
• If compounded Quarterly
• An P(1 r/4)4n.
• In general, with m compounding periods
• every year, the periodic rate becomes
• r/m and nm is the total compounding
• periods. Thus, P grows to
• An P(1 r/m)nm.

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• Monthly compounding becomes
• An P(1 r/12)n12
• and daily compounding yields
• An P(1 r/12)n12.
• EXAMPLES
• n 10 years r 12 P 100
• 1. Simple compounding yields
• A10 100(1 .12)10  310.58
• 2. Monthly compounding yields
• A10 100(1 .12/12)120   330.03
• 3. Daily compounding yields
• A10 100(1 .12/365)3650 331.94.

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• In the early 1970s, banks came up with
• the following economic reasoning Since
• the bank has depositors money all the
• time, this money should be working for
• the depositor all the time!
• This idea, of course, leads to the concept of
• continuous compounding.
• We want to apply this idea in the formula

Observe that continuous time means that the
number of compounding periods, m, increases
without limit, while the periodic interest rate,
r/m, becomes smaller and smaller.
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• This reasoning implies that in order to impose
  the concept of continuous time on the above
  compounding expression, we need to solve

This expression may be rewritten as
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• EXAMPLE, continued
• First, we remind you that the number e
• is defined as

For example x e 1 2 10 2.59374246 100
2.70481382 1,000 2.71692393 10,000 2.718145
92 1,000,000 2.71828046 In the limit e
2.718281828..
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• Recall that in our example
• N 10 years and r 12 and P100.
• Thus, P100 invested at a 12 annual
• rate, continuously compounded for ten
• years will grow to

Continuous compounding yields the highest return
to the investor Compounding Factor Simple 3.
105848208 Quarterly 3.262037792 Monthly 3.30
0386895 Daily 3.319462164 continuously
3.320116923
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Continuous Discounting
This expression may be rewritten as
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• EXAMPLE, continued
• First, we remind you that the number e
• is defined as

Recall that in our example P 100 n 10
years and r 12 Thus, 100 invested at an
annual rate of 12 , continuously compounded
for ten years will grow to
Therefore, we can write the continuously
discounted value of 320.01 is
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PURE ARBITRAGE PROFIT A PROFIT MADE 1.
WITHOUT EQUITY and 2. WITHOUT ANY RISK.
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Risk-free lending and borrowing

• Arbitrage A market situation in
• which an investor can make a profit
• with no equity and no risk.
• Efficiency A market is said to be
• efficient if prices are such that there
• exist no arbitrage opportunities.
• Alternatively,
• a market is said to be inefficient if
• prices present arbitrage opportunities
• for investors in this market.

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• Risk-free lending and borrowing
• PURE ARBITRAGE PROFIT
• A PROFIT MADE
• 1.WITHOUT EQUITY INVESTMENT
• and
• WITHOUT ANY RISK
• We will assume that
• the options market is efficient.
• This assumption implies that one cannot make
  arbitrage profits in the options markets

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• Risk-free lending and borrowing
• Treasury bills are zero-coupon bonds, or pure
  discount bonds, issued by the Treasury.
• A T-bill is a promissory paper which promises its
  holder the payment of the bonds Face Value (Par-
  Value) on a specific future maturity date.
• The purchase of a T-bill is, therefore, an
  investment that pays no cash flow between the
  purchase date and the bills maturity. Hence, its
  current market price is the NPV of the bills
  Face Value
• Pt NPVthe T-bill Face-Value
• We will only use
• continuous discounting

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• Risk-free lending and borrowing
• Risk-Free Asset is a security whose return is a
  known constant and it carries no risk.
• T-bills are risk-free LENDING assets. Investors
  lend money to the Government by purchasing
  T-bills (and other Treasury notes and bonds)
• We will assume that investors also can borrow
  money at the risk-free rate. I.e., investors may
  write IOU notes, promising the risk-free rate to
  their buyers, thereby, raising capital at the
  risk-free rate.

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• Risk-free lending and borrowing
• The One-Price Law
• There exists only one
• risk-free rate in an efficient economy.
• Proof If two risk-free rates exist in
• the market concurrently, all investors
• will try to borrow at the lower rate
• and simultaneously try to invest at
• the higher rate for an immediate
• arbitrage profit. These activities will
• increase the lower rate and decrease
• the higher rate until they coincide to
• one unique risk-free rate.

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• Risk-free lending and borrowing
• By purchasing the risk-free asset,
• investors lend capital.
• By selling the risk-free asset, investors borrow
  capital.
• Both activities are at the
• risk-free rate.

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• Continuous Discounting
• Recall that continuous compounding and
  discounting use the number e, which in itself is
  used as the result of continualizing the simple
  compounding formula as follows

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• EXAMPLE
• First, we remind the reader that the
• number e is defined as

On your own calculator you may try x
e 1 2 10 2.59374246 100 2.70481382 1,
000 2.71692393 10,000 2.71814592 1,000,000 2
.71828046. In the limit e 2.718281828..
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This expression may be rewritten as
But first, QUESTION Given P and r, how long it
takes to double our money? - the 72
rule Ans. 2P Pert t ln2/r t
69.31/r. r 10 gt t 6.931yrs.
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Continuous Discounting
This expression may be rewritten as
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Again, P 100 n 10 years and r 12 Thus,
100 invested at an annual rate of 12 ,
continuously compounded for ten years will grow
to
The continuously discounted value of 332.01 is
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• We are now ready to calculate the current value
  of a T-Bill.
• Pt NPVthe T-bill Face-Value.
• Thus
• the current time, t, T-bill price, Pt , which
  pays FV upon its maturity on date T, is
• Pt FVe-r(T-t)
• Clearly, r is the risk-free rate in the economy.

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• EXAMPLE Consider a T-bill that promises its
  holder FV 1,000 when it matures in 276 days,
  with a yield-to-maturity of 5
• Inputs for the formula
• FV 1,000
• r .05
• T-t 276/365yrs
• Pt FVe-r(T-t)
• Pt 1,000e-(.05)276/365
• Pt 962.90.

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• EXAMPLE The yield-to -maturity of a bond which
  sells for 945 and matures in 100 days, promising
  the FV 1,000 is
• r ?
• Pt 945 FV 1,000 T-t 100 days.
• Inputs for the formula
• FV 1,000 Pt 945 T-t 100/365.
• Solving Pt FVe-r(T-t) for r
• r 365/100ln1,000/945
• r 10.324.

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• SHORT SELLING STOCKS
• An Investor may call a broker and ask to sell a
  particular stock short.
• This means that the investor does not own shares
  of the stock, but wishes to sell it anyway.
• The investor speculates that the stocks
• share price will fall and money will be
• made upon buying the shares back at a
• lower price. Alas, the investor does not
• own shares of the stock. The broker
• will lend the investor shares from the
• brokers or a clients account and sell it
• in the investors name. The investors
• obligation is to hand over the shares
• some time in the future, or upon the
• brokers request.

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• SHORT SELLING STOCKS
• Other conditions
• The proceeds from the short sale cannot be
• used by the short seller. Instead, they are
• deposited in an escrow account in the
• investors name until the investor makes
• good on the promise to bring the shares
• back.
• Moreover, the investor must deposit an
• additional amount of at least 50 of the
• short sales proceeds in the escrow account.
• This additional amount guarantees that there
• is enough capital to buy back the borrowed
• shares and hand them over back to the
• broker, in case the shares price increases.

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• SHORT SELLING STOCKS
• There are more details associated with short
  selling stocks. For example, if the stock pays
  dividend, the short seller must pay the dividend
  to the broker. Moreover, the short seller does
  not gain interest on the amount deposited in the
  escrow account, etc.
• We will use stock short sales in many of
• strategies associated with options
• trading.
• In all of these strategies, we will assume that
  no cash flow occurs from the time the strategy is
  opened with the stock short sale until the time
  the strategy terminates and the stock is
  repurchased.
• In terms of cash flows
• St is the cash flow from selling the stock
• short on date t, and
• -ST is the cash flow from purchasing the
• back on date T.

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• Options Risk-Return Tradeoffs
• PROFIT PROFILE OF A STRATEGY
• A graph of the profit/loss as a function of all
  possible market values of the underlying asset
• We will begin with profit profiles at the
  options expiration I.e., an instant before the
  option expires.

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• Options Risk-Return Tradeoffs At Expiration
• 1. Only at expiry T-t 0
• 2. No time value T-t 0
• 3. At maturity
• CALL is exercised If SgtX expires worthless If S
  ? X
• Cash Flow Max0, S X
• PUT is exercised If SltX
  expires worthless If S?X
• Cash Flow Max0, X S

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• 4. All legs of the strategy remain open till
  expiry.
• 5. A Table Format
• Every row is one leg of the strategy. Every row
  is analyzed separately.The total strategy is the
  vertical sum of the rows.The profit is the cash
  flow at expiry plus the initial cash flows at the
  of the strategy, disregarding the time value of
  money

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• 6.A Graph of the profit/loss profile
• The profit/loss from the strategy as a function
  of all possible prices of the underlying asset at
  expiration.

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• The algebraic expressions
• Of profit/loss at expiration
• Cash Flows
• Long stock ST S0
• Short stock S0 - ST
• Long call-c Max0, ST -X
• Short callc Min0,X- ST
• Long put-p Max0,X- ST
• Short putp Min0, ST -X