Future and Options

This assignment provides a comprehensive analysis of futures and options, covering topics such as arbitrage opportunities, forward and future prices, types of transactions in the futures market, interest rate derivatives, delta hedging, gamma hedging, theta, implied volatility, and the Ho & Lee model. It explores the concepts and applications of these financial instruments, offering a detailed understanding of their role in risk management and financial markets.

FUTURE AND OPTIONS

TABLE OF CONTENTS
INTRODUCTION...........................................................................................................................1
QUESTION 1...................................................................................................................................1
a. Presence of Arbitrage opportunity with its exploitation.....................................................1
b. Defining forward and future prices with its difference......................................................2
c. Types of transactions in the future market.........................................................................2
QUESTION 2...................................................................................................................................3
a. Assumption to buy or sell £1000000..................................................................................3
b. Actions for speculating in forward market with its expected dollar profit ........................3
c. Speculative profit in dollar when spot exchange rate turns $1.86/£...................................4
QUESTION 3...................................................................................................................................4
a. Explaining interest rate derivatives market........................................................................4
b. Alcoa case...........................................................................................................................4
b.1 Conversion of interest rate swap with obligation of interest into one synthetic fixed rate
loan.........................................................................................................................................5
b.2 Interest rate firm pays on synthetic fixed rate loan..........................................................5
QUESTION 4...................................................................................................................................5
a. Delta hedging......................................................................................................................5
b. Theta...................................................................................................................................5
c. Gamma Hedging.................................................................................................................6
d. Relationship among delta hedging, theta and gamma hedging..........................................6
QUESTION 5...................................................................................................................................6
a. Explaining Option's implied volatility and method for calculating....................................6
b. Explaining strength and weakness of Implied Volatility Function ...................................7
c. Explaining Ho & Lee model with graphical presentation and note on building interest rate
tree..........................................................................................................................................7
CONCLUSION................................................................................................................................8
REFERENCES................................................................................................................................9

INTRODUCTION
The future contracts are one which allows selling or buy underlying stock or index at
price which is set on prior basis for purpose of delivery on future date. Futures and option reflect
two and very common form of derivatives. Derivaties are considered as financial instruments
which derive value through underlying. It could be stock issued through currency, gold, company
etc. The present report will discuss about exploitation of arbitrage opportunity along with
elaboration of future and forward prices with there difference and types of transaction. In the
similar aspect, it will specify interest rate derivative market along with delta and gamma hedging
and theta.
QUESTION 1
a. Presence of Arbitrage opportunity with its exploitation
Current interest rate c(June) 12.00%
Future price of gold (December) 448
Future price 564
Spot price (December future price) 564/(1+12%)
503.57
spot price of gold in December future price
448/(1+12%)^.6
418.54
Spot price extracted from December future price is higher than spot price of gold in
December future price. Thus, actual future price of December is too high so there is presence of
arbitrage by selling its December future contract. The arbitrage could be exploited as stated
below:
Arbitrage position
Strategy Today December
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Take long position of gold today and short in
December -S(0) S(Dec)
Short position of December future contract today 0 564 – S(Dec)
Short position of December T-bills with its face
value of 564 today and hold till maturity 564/(1+12%)
503.57 -564
Short gold today and long in December S(0) -S(Dec)
Buy Dec T-bill with face value of 448 today and
hold till maturity 448/(1+12%)^.6
-418.54
Net cash flow 85.03 0
b. Defining forward and future prices with its difference
Forward price is referred as predetermined price of delivery with context of underlying
commodity, financial asset or currency as decided through seller and buyer of forward contract
which is to be paid at pre identified date in the future. At point of inception of forward contract,
the value of contract as 0 created through forward price but its alteration in price of particular
underlying will impact forward to undertake negative of positive value (Cao and et.al., 2018).
Future price is reflected as price of commodity on basis of current spot price, time till delivery,
storage costs and interest rate at future date. There is presence of various differences among
forward and future price which are stated below:
 The margin accounts on basis of future contracts are directly invested in securities of
short term interest as this variation through forward contracts aggregated as element to
return through future contracts impacting relationship of pricing.
 The regular loss and gain is settled on daily aspect on outstanding future contracts
through margin accounts which transfers credit risk is substituted. On the contrary,
forward contracts could accumulate credit risk through changes to price of any
underlying asset (Future Prices vs Forward Prices, 2018).
 The pricing of future contracts is directly impacted with correlation among future prices
and interest rates.
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 If there is presence of positive correlation then buyer of future contract is benefited as
gaining higher through margin account interest and increment in contract price.
c. Types of transactions in the future market
The trader who long future contracts has absences of other position of exchange is long.
If this purchase in not offset through any equivalent sale of futures then its buyer will undertake
delivery of its actual commodity (Tamvakis, 2018). On the contrary, trader who sell any future
contract with absence of offsetting purchase of future is referred to be short.
With this perspective, traders who undertake position in market with two methods of
liquidating it as first engages with actual delivery or good's receipt. Generally, second option is
selected with huge probability as canceling obligation to sell or buy through carrying any reverse
operation as offsetting transaction. Through purchasing any matching contract a future trader in
position of short would be released through obligation for delivering. Simultaneously, trader who
is long could offset the outstanding purchases through selling. The basic types of transaction in
future contract are:
 Against actual: It is highly possible for liquidating futures position in privately at spot
market with context of per-arranged trade. This transaction is replicated as against actual
trade and ignores complexities of creating physical delivery under future contract.
Conversely, these transactions should take place with various riles related to exchange
which supervises these future contracts (Chang, Ho and Hsiao, 2018).
 Open interest: The sum of clearing house's short and long position outstanding at
specified moment is replicated as open interest. In the end of every trading day, there is
assumption of clearing house of one side of every open contracts as if any trader has long
positioned where clearing house undertakes short position and vice versa situation
(Futures markets-Offsetting transactions, 2018).
 Cash and carry transaction: It is considered as kind of trade in future market where
price of any commodity is less than future contracts price. These transactions are
replicated as arbitrage and undertakes with cash or on spot market.
QUESTION 2
a. Assumption to buy or sell £1000000
Current spot exchange rate $1.95/£
3 month forward rate $1.90/£
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Expected spot exchange rate (in three months) $1.92/£
Assumption: Buy (Long position) 1000000
b. Actions for speculating in forward market with its expected dollar profit
Particulars Details
Speculation in forward market (buy)
Expected profit
Amount* (Expected spot rate – 3 months
forward rate)
1000000* (1.92 – 1.90)
Expected profit 20000
c. Speculative profit in dollar when spot exchange rate turns $1.86/£
Particulars Details
Spot exchange rate (in three months) 1.86
Loss from long position
Amount* (Turned spot rate – 3 months forward
rate)
1000000* (1.86 – 1.90)
Loss from long position -40000
QUESTION 3
a. Explaining interest rate derivatives market
Interest rate derivative is replicated as derivative where its underlying asset is right for
receiving or paying notional amount of money at particular interest rate. The interest rate
derivatives market is referred the largest derivative market throughout world. In simple words, it
is a derivative instrument where its underlying asset has right for receiving or paying money at
particular rate of interest (Interest rate derivatives, 2019). It could range through simply to very
complex as it could be used for decreasing or increasing exposure of interest rate. Interest rate
could be treasury, interbank and repo rates. The types of interest rate derivative are Vanilla,
Quasi vanilla and Exotic derivatives. The vanilla is very basic and standard kind of interest rate
derivative with reference to huge liquidity followed through Quasi vanilla is fairly liquid.
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Simultaneously, exotic derivatives are highly illiquid and complex comparatively to commonly
traded vanilla derivatives.
The example of Vanilla are interest rate swap, interest rate future, interest rate cap,
forward rate option and interest rate swapation. Quasi Vanilla as Constant treasury and maturity
swap, IRS on basis of two floating interest rate and in arrears swap (Rebentrost, Gupt and
Bromley, 2018). Furthermore, Exotic derivatives as strips of CMO, Cross currency swapations,
target redemption note and Bermudan swaptions.
b. Alcoa case
Floating rate bonds $10 million
Interest rate 1.00%
Bonds selling Par value
Swaps of LIBOR 7.00%
b.1 Conversion of interest rate swap with obligation of interest into one synthetic fixed rate loan
Let LIBOR be X
Interest rate payments on bond (X + 1%) * 10 million par value
Net cash flow through Swap (7% - X) * 10 million notional principal
Sum 8%*10 million
0.08*1,00,00,000
800000
b.2 Interest rate firm pays on synthetic fixed rate loan
The organization will pay 8% interest rate on synthetic fixed rate loan.
QUESTION 4
a. Delta hedging
Delta hedging is replicated as an option strategy whose objective is to decrease or hedge
risk which is directly associated to movement of price on basis of underlying asset, offsetting
short and long position. The options with huge hedge ratios is very profitable to purchase as
compared to write where greater the percentage and movement related to price of underlying and
corresponding erosion of little time value with greater leverage. In position of option, it could be
hedged with delta as it is opposed to current option position which maintains position of delta
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neutral. In simple words, delta neutral position is one where overall delta is zero and reduces
price movements related to underlying asset (Chen, 2018).
b. Theta
Theta helps in measuring exposure of price of option to particular time passage. In simple
words, rate has been measured at option price especially with context of time value, alterations
or decrement with time to expire has been approached. The option premium comprises time
value which decline to near time to expiration and declining with incur in expiration. In simple
words, it could be elaborated as measure of this time decay and reflected loss of time value per
day. It is minimal for purpose of long term option due to slowly time value decay but raises
expiration (Ghafarian, Hanafizadeh and Qahi, 2018). The theta of -.1 reflect option of losing $.10
of time value per day. Theta is at greatest when option is at money due to price where time value
is at greatest and has huge potential for decay.
In the same series, it measures alterations in option value of portfolio because of passage
of time and holdings of option has theta at negative position due to option value which declined
continuously along with time. Due to time decay which will favour option writer where short
position in option have positive position theta.
c. Gamma Hedging
Gamma hedging is referred as option of hedging strategy framed to eliminate, decrease
risk created with alteration in option delta. In simple words, it is change in delta for change in
each unit and underlying price. Furthermore, the absolute magnitude of delta raises time to
expiration of option decreases along with increment in intrinsic value.
d. Relationship among delta hedging, theta and gamma hedging
Delta gamma hedging is option strategy which mix both gamma and delta hedges for
purpose of mitigating risk of changes with context of underlying asset and delta as well. Delta
and gamma alters when stock price moves down or option moves in and out of money. The price
of at the money options will alter on significant aspect as compared to price of in and out of
money options with similar expiration. In this aspect, theta is very crucial concept as it gives
explanation about effect of time on option of premium sold or purchased.
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QUESTION 5
a. Explaining Option's implied volatility and method for calculating
Implied volatility is replicated parameter part with context of pricing model of option like
Black Scholes model which specifies market price of option. In simple words, it helps in
reflecting views of marketplace that where marketplace considers about volatility in the future.
The option's implied volatility is forward looking as it helps in gauging sentiment related to
stock's volatility or market as well (Markellos and Psychoyios, 2018). On the contrary, implied
volatility does not predict direction where option is headed. It is a dynamic figure which alters on
basis of activity in options market . Generally, when there is increment in implied volatility then
option price will also raise along with assumption that else all other things are constant
(Volatility, 2019). Thus, if there is raise in implied volatility after trade which is placed and good
for owner of option and worse for its seller.
Implied volatility could be extracted by undertaking market price of option, and
considering it in formula of black scholes model and solving for purpose of extracting volatility's
value. Simultaneously, there are multiple approaches for calculating implied volatility such as
application of iterative approach or trial and error method to calculate value of implied volatility
of option.
b. Explaining strength and weakness of Implied Volatility Function
Strength
It is very important input with context of valuation model of option as its input is referred
as user defined variable (not similar to dividend or interest rates). It is the only input which alters
because of demand and supply of its underlying options along with expectation of market
through direction of share price. In case expectations will raise then there will be increment in
demand of option along with rise in implied volatility (Fernandes and et.al., 2018). Generally,
options which have huge level of implied volatility whose outcome is related to premium of high
priced option. In simple words, it is very significant due to rise and fall in implied volatility will
identify about cheap and expensive time value to option and in turn it will give direct impact on
success options of trade.
Weakness
It had been evaluated that belief about prevailing sentiment about investor revealed with
price, open interest configuration and volume is wrong where predictive approach for activity of
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option where actual information reveals along with its challenges. It is used for determining
significant alteration in price direction such as signals of buy and sell is produced when reading
directly hit extreme level in reaction for identifiable event. Volatility is non directional which
signifies about magnitude and probability of price moving but not with direction and could not
be used for predictive indicator.
c. Explaining Ho & Lee model with graphical presentation and note on building interest rate tree
It is an interest rate option model which implies short rates related to pricing interest rate
derivatives like swapations and bond options or to value fixed income securities along with
embedded options like puttable and callable bonds etc. In simple words, it could be elaborated as
model with uncertain behaviour of structure of interest rate as whole but not with certain point on
curve. At the end, arbitrage free and term structure model assumes about interest rate with
context of yield curve correlated with irreversible to its mean (Kumar, 2018). The Ho-lee model
could be represented with this below stated tree:
Here u and u' is estimated for ensuring about short rate process which is consistent with
recent term structure of its interest rates. In this example, 0.5 is assumed with risk neutral
probability of its up movement.
Building interest rate tree
The derivation of interest rate follows process of four steps such as building perturbation
function as it is mathematical method which is used for gaining an approximate solution to a
problem because of inability to extract exact one. It will lead to build exact solution on basis of
related problem. In next step, calculation of risk neutral probabilities dependent on Cox-Ross-
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Rubinstein model and then deriving path dependence condition with application of binomial tree.
Atlast, combination of previous steps. In simple words, binomial tree is graphical representation
of all possible intrinsic values where option might undertake various nodes or time period.
CONCLUSION
On basis of above report it could be concluded that future and options are very important
for purpose of mitigating risk. In the same series, it has shown that arbitrage strategy is very
useful for reducing risk to gain benefit. Lastly, it could be said that Implied volatility function is
very important as it has both pros and cons as well but useful to eliminate issues.
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