investment banking - f-d 97

7/31/2019 Investment Banking - F-D 97

1/139


2/139

Contents1) Derivatives

2) Derivatives Pricing

1) Derivatives Market in Pakistan


3/139

Derivatives

Some say the world will end in fire, Some say

This is what the Derivative world is?


4/139

Financial Derivatives

Financial instrument whose price is dependent upon or derivedfrom the value of underlying assets

The underlying not necessarily has to be an asset. It could beany other random/uncertain event like temperature/weather etc.

The most common underlying assets includes:

Stock BondsCommoditiesCurrencies

Interest rates


5/139

Financial Derivatives

Derivatives are mainly used to mitigate future risks, however they donot add value since hedging is a zero sum game and secondly investorsmostly use them on do it yourself alternatives basis.

Derivatives markets can be traced back to middle ages. They weredeveloped to meet the needs of farmers and merchants. First futureexchange was established in Japan in 16 th century. The Chicago Boardof Trade was established in 1848.The international Monetary market

was established in 1972 for future trading in foreign currencies

Insurance is a kind of mitigating risk but it has its limitations. To avoidzero NPV it tries to cover administrative costs, adverse selection, andmoral hazard risks in its premium. Apart from this simple format, thederivatives on the other hand have lot of varieties, i.e. from simple tohighly exotics, creating a world of their own. Size of its activities aremanifold as compared to total world GDP


6/139

International Markets and their risks

In global world, Foreign exchange markets of each country play an importantrole. The market embodies a spot market with a forward market. The forwardmarket represents the selling and buying committed or due on some futuredates. To understand the dynamics of foreign exchange market in Pakistan orin any country we have to understand the difference and relationship betweenexchange rates and interest rates. This gives birth to four questions i.e. (1) why the dollar rate of interest is different from say PKR (2) Why the forward rate of exchange (F-PKR/$) is different from the spot rate (S-PKR $) (3) Whatdetermines next years expected spot rate of exchange between $ and PKR (E (S-PKR/$)) (4) What is the relationship between inflation rate in the US and theinflation rate in Pakistan. Suppose that individuals are not worried about riskor cost to international trade than (a) difference in interest rates =1 + r PKR/ 1+r$ must be equal to expected difference in inflation rates = E (1 + Inf Pak)/ E ( 1+ inf US). Further difference between forward and spot rates = F-PKR per $/S-PKR per $ must be equal to Expected change in spot rate = E (S-PKR per S)/ (S-PKR per $)

Prepared by: Farrukh Aleem Mirza


7/139

International Markets and their Risks

Interest rate parity theory says that the difference in interest rate must be equalto the difference between the forward and spot exchange rates i.e. 1 +r PKR/ 1 +r$ = difference between the forward and spot exchange rates. In case of Pakistan it would give the following results = 1.12/1.01 X 85 = forward rate of PKR/$ after one year = 94.25Expectation theory of exchange rates tells us that the difference in % betweenthe forward rate and today's spot rate is equal to the expected change in thespot rate = difference between forward and spot rates = F-PKR per $ / S-PKR per $ = expected change in spot rate E (S PKR per $) / (SPKR per $).Purchasing power parity implies that difference in the rates of inflation will beoffset by a change in the exchange rate =Expected difference in inflation rate =

E (1 + Inf-PKR)/ E (1 + Inf-US) = Expected change in spot rate = E (S-PKR/$) / S-PKR/$. This would give a result of Rs 90.77/US$ considering inflation inPakistan as 10. in US as 3 and spot rate of PKR/US$ as Rs 85.However life is not very simple and apart from these exercises inflows andoutflows mostly dictate the spot and forward rates that can alter.

Prepared by: Farrukh Aleem Mirza


8/139

Derivatives Markets

Exchange Traded Contracts

Over The Counter Market


9/139

Market ParticipantsHedger

Hedgers face risk associated with the price of an asset. Theyuse futures or options markets to reduce or eliminate this risk

SpeculatorsSpeculators wish to bet on future movements in the price of anasset. Derivatives can give them an extra leverage to enhancetheir returns

ArbitrageursArbitragers work at making profits by taking advantage of discrepancy between prices of the same product acrossdifferent markets


10/139

Types of Financial DerivativeCan be plain vanilla or exotic

Forward

Futures

Options

Forward Rate Agreement (FRAs)

Swaps


11/139

Forwards

Forward contract is a binding contract which fixes now thebuying/selling rate of the underlying asset to bebought/sold at some time in future.

Long ForwardBinding to buy the asset in future at the predeterminedrate.

Short ForwardBinding to sell the asset in future at the predeterminedrate.


12/139

Basic Features

Long/ShortPay Off Binding ContractOTC Transaction/ContractRisk/Uncertainty Elimination

Zero Cost ProductCredit Risk


13/139

Pay Off

Pay Off = S T K (for the long forward)Pay Off = K ST (for the short forward)

T = Time to expiry of the contractST = Spot Price of the underlying asset at time T

K = Strike Price or the price at which the asset will bebought/sold


14/139

Forward Contract Payoff

K


15/139

Futures

A futures contract is a standardized contract, traded on afutures exchange, to buy or sell a certain underlying instrumentat a certain date in the future, at a specified price

Specification of a standard contract: GoldCommodity NameExchange NameSize of Contract : 100 troy ounceDelivery month: Feb/April/June/Aug/Oct/Dec


16/139

DISTINCTIONS BETWEENFUTURES & FORWARDS

ForwardsTraded in dispersedinterbank market 24 hr a day.Lacks price transparency

Transactions are customizedand flexible to meetcustomers preferences.

FuturesTraded in centralizedexchanges during specifiedtrading hours. Exhibits pricetransparency.

Transactions are highlystandardized to promotetrading and liquidity.


17/139

DISTINCTIONS BETWEEN

FUTURES & FORWARDSForwards

Counter party risk is variable

No cash flows take placeuntil the final maturity of thecontract.

FuturesBeing one of the two parties,the clearing housestandardizes the counterpartyrisk of all contracts.

On a daily basis, cash mayflow in or out of the marginaccount, which is marked tomarket.


18/139

Margin Requirements/Daily Settlemen

Initial MarginAn Initial margin is the deposit required to maintain either

a short or long position in a futures contract.

Maintenance MarginMaintenance margin is the amount of initial margin thatmust be maintained for that position before a margin callis generated.


19/139

Margin CallIf the amount actually falls below the maintenancemargin, a margin call is given to the investor to replenishthe account to the initial margin level, otherwise theaccount is closed.

Variation MarginThe additional funds deposited to make up to the initialmargin.


20/139

Futures Price Close to Expiry In the delivery month the Futures Price is almost equal to theprevailing Spot Price.

Or we can say that with the passage of time the Futures Pricegradually approaches the prevailing Spot Price.

Because of No Arbitrage Principle


21/139

No Arbitrage Principle

Assume NotationsT = Delivery/Expiry time of the futures contract

F0 = Futures Price now to be delivered at time TST = Spot Price of the underlying at delivery time TIf F0 > ST there is an arbitrage opportunity

Short the futures contract, buy the asset and make thedeliveryPay Off = F 0 - ST > 0


22/139

If F0 < ST

Parties who want to buy the asset, would immediately go longthe futures contract and wait for the delivery.

Because of increased demand for the futures, the futures pricewould go up to the actual market spot price of the asset toremove the anomaly.


23/139

Hedging StrategiesShort Hedge

When the hedger owns an asset or expects to own anasset in future and wants to sell it.By shorting an appropriate futures contract, the hedgercan lock in a price now to sell the asset at some time infuture.


24/139

HEDGING WITH FUTURESSuppose

Commitment to sell 1000 barrels after 3 months at the thenprevailing spot price say S T Futures Price for delivery after 3 months = 18.75

StrategyGo short a 6 months future contract to lock in a price nowAt maturity go long a futures to close the position

Scenario Cash flow from sale atspot rate

Gain/Loss on futures

If ST = 19.5 19.5 18.75-19.5If ST = 17.5 17.5 18.75-17.5If ST = 18.75 18.75 0


25/139

HEDGING WITH FUTURES Long Hedge

When a company knows it will have to purchase a certainasset in the future and wants to lock in a price now.By going long an appropriate futures contract the hedgercan lock in a price he will be paying after time T to buythe asset.


26/139

HEDGING WITH FUTURES Suppose

Commitment to buy 1000 barrels after some time T at thethen prevailing spot price say S T Futures Price = 18.75

StrategyGo long a 6 months future contract to lock in a price nowAt maturity go short a futures contract to close the position

Scenario Price paid after 6 monthsat spot rate Gain/Loss onfutures

If ST = 19.5 - 19.5 19.5-18.75

If ST = 17.5 - 17.5 17.5-18.75

If ST = 18.75 -18.75 0


27/139

BARINGS BANKA bank with a history of 233 years collapsed because of imprudent use of derivatives.One of the traders Nick Leeson, who was basically responsible tomake arbitrage profits on Stock Index Futures on Nikkei 225 onthe Singapore and Osaka exchanges.Instead of looking for arbitrage opportunities, the trader startedmaking bets on the index and went long the futures.Unfortunately the market fell by more than 15% in the 1995

leading to margin calls on his positions.Because of his influence on the back office, he was able to hidethe actual position and sold options to make for the margin calls.But his view on the market proved to be wrong and losses

mounted to an unmanageable amount.


28/139

METALLGESELLSCHAFTA US Subsidiary of a German Company used hedging strategies,which went against them, resulting in heavy cash outflows.Being an Oil Refinery and Marketing company, they sold forwardcontracts on oil maturing up to 10 years.

To hedge their position, they went long the available futurescontracts with maturities up to 1 year.They planned to use Stack & Roll strategy to cover their shortforwards contracts.But due to decreasing oil prices they had margin calls on their futurescontracts.They were unable to meet those heavy cash outflows.Ultimately the US team of the company were kicked out and a newteam from Germany came to USA and liquidated the contracts

making heavy losses.


29/139

SOCIT GNRALE On January 24, 2008, the bank announced that a single futurestrader at the bank had fraudulently caused a loss of 4.9 billionEuros to the bank, the largest such loss in history.

Jerome Kerviel, went far beyond his role -- taking "massivefraudulent directional positions" in various futures contracts.He got caught when markets dropped, exposing him incontracts where he had bet on a rise.

Due to the loss, credit rating agencies reduced the bank's longterm debt ratings: from AA to AA-by Fitch; and from Aa1/B to Aa2/B-by Moody's.The alleged fraud was much larger than the transactions by

Nick Leeson that brought down Barings Bank.


30/139

Options

An option is a contract that gives the buyer the right, but notthe obligation, to buy or sell an underlying asset at a specificprice on or before a certain date

Unlike a forward/future, this contract gives the right but notthe obligation. So its not a binding contract.

The holder will exercise the option only if it is profitable.


31/139

Types Of Options

On the basis of payoff structures

Call optionA call gives the holder the right to buy an asset at a certainprice within a specific period of time.

Put option A put gives the holder the right to sell an asset at a certain pricewithin a specific period of time.


32/139

Strike Price TerminologyThe type of option and the relationship between the spotprice of the underlying asset and the strike price of the optiondetermine whether an option is in-the-money, at-the-moneyor out-of-the-money.Exercising an in-the-money call or in-the-money put willresult in a payoff. Neither a call nor put that is at-the-moneywill produce a payoff.

Call Option Put OptionIn-the-Money Spot > Strike Spot < Strike

At-the-Money Spot = Strike Spot = StrikeOut-of-the-Money Spot < Strike Spot > Strike


33/139

Pay-Offs


34/139


35/139

Types Of OptionsOn the basis of versatility

Vanilla Option

A normal option with no special or unusual features

Exotic OptionA type of option that differs from common American orEuropean options in terms of the underlying asset or thecalculation of how or when the investor receives a certainpayoff.


36/139

Types of Exotic options Bermuda Option

A type of option that can only be exercised on predetermineddates, usually every month

Compound Option An option for which the underlying is another option. Therefore,there are two strike prices and two exercise dates. These arethe four types of compound options:

- Call on a call- Put on a put- Call on a put- Put on a call


37/139

Asian OptionAn option whose payoff depends on the average price of theunderlying asset over a certain period of time as opposed to atmaturity. Also known as an average option.

Digital OptionAn option whose payout is fixed after the underlying stock exceeds the predetermined threshold or strike price.

Also referred to as "binary" or "all-or-nothing option."


38/139

Shout OptionAn exotic option that allows the holder to lock in a definedprofit while maintaining the right to continue participating ingains without a loss of locked-in monies.

Barrier OptionA type of option whose payoff depends on whether or not theunderlying asset has reached or exceeded a predetermined

price.


39/139

Chooser OptionAn option where the investor has the opportunity to choosewhether the option is a put or call at a certain point in timeduring the life of the option

Quantity-Adjusting Option (Quanto Option)A cash-settled, cross-currency derivative in which theunderlying asset is denominated in a currency other than the

currency in which the option is settled. Quantos are settled at afixed rate of exchange, providing investors with shelter fromexchange-rate risk.


40/139

Trading Strategies Involving Options


41/139

Bull Spreads

Buy a Call Option with strike X 1, and sell a call optionwith strike X2 (whereas X2>X1). Date of maturity for

both the options is same

Pay Off

If ST X1 : ZeroIf X1 < S T X2 : ST X1If ST > X2 : S T X1 (ST X2) = X2 X1


42/139

Bull Spread


43/139

Bear Spreads

Buy a Put Option at strike price X2 and sell a put optionat a lower strike price of X1. Both options have same

expiry

Pay Off

If ST X2 : ZeroIf X1 S T < X2 : X2 - S T If ST < X1 : X2 - S T ( X1 - S T)


44/139

Bear Spread


45/139

Butterfly Spreads

Buy a call option at strike price X1 , and buy another calloption at a higher strike price of X2 . Sell 2 call optionsat strike price X3 , exactly halfway between X1 and X2

Pay Off

If ST X1 : ZeroIf X1 < S T X3: ST X1If X3 < S T X2: ST X1 2(ST X3) = X2 - S T If ST > X2: Zero


46/139

Butterfly Spread


47/139

Straddle

Buy a call option and a put option both at strike price X

Pay Off

If ST > X : S T X

If ST < X : X- S T If ST = X : Zero


48/139

Straddle


49/139

Strangle

Buy a Put Option with strike price K1= and buy a CallOption at strike price K2 > K1

Pay Off

If ST > K2 : S T K2If ST < K1 : K1 - S T If K1 S T K2 : Zero


50/139

Strangle


51/139

Forward Rate Agreement (FRA) An over-the-counter contract between parties that determinesthe rate of interest, to be paid or received on an obligationbeginning at a future start date. On this type of agreement, it is

only the differential that is paid on the notional amount of thecontract.

Also known as a "future rate agreement".


52/139

Forward Rate Agreement

Example

Client has a six month borrowing requirement in three months

and wants to hedge its floating interest rate exposureClient can enter into a 3*9 FRA Agreement with bank whereby:Bank will pay 6m KIBOR rate to the customer.Customer will pay a fixed rate to bank.Since the exchange will take place at the start of the borrowingperiod, only the net discounted amount will be exchanged.


53/139


FRA Rate Calculation:

Current KIBOR rates:3 month 9.30% = i39 month 9.80% = i9

FRA(3*9) rate={[1+(i9*9/12)]/[1+(i3*3/12)]}*[12/6]=9.82%


54/139


A client needs to borrowRs.100m for 6 months in 6months time. The clientsborrowing rate isKIBOR+1.Suppose,today 6

month KIBOR is [email protected]%.

AfterMonth

DoNothing

Buy 6*[email protected]%

KIBORrises to

11%

Borrow@12%

Borrow@12%,get 0.5% from

seller. your cost11.5%

Kiborfall to

9%

Borrow@ 10%

Borrow@9%,pay 1.5% to

seller. your cost11.5%


55/139

SWAP

A swap is a derivative in which two counterparties agree toexchange one stream of cash flows against another stream.These streams are called the legs of the swap

There are two basic types of swaps:

Interest Rate Swap Currency Swaps


56/139

Interest Rate Swap

An agreement between two parties where one stream of futureinterest payments is exchanged for another based on a specifiedprincipal amount. Interest rate swaps often exchange a fixedpayment for a floating payment that is linked to an interest rate.


57/139

Interest Rate SwapEXAMPLE

Counter parties:: A and BMaturity:: 5 yearsA pays to B : 6% fixed p.a.

B pays to A : 6-month KIBORPayment terms : semi-annualNotional Principal amount: PKR 10 million.

Party A Party B6% p.a. Fixed

6 Month KIBOR


58/139

Cash flows in the above swap are represented as follows: Payments Fixed rate Floating rate Net Cash

at the end payments Payments From

Half year 6m Kibor A to B

period1 300000 337500 -37500

2 300000 337500 -37500

3 300000 337500 -37500

4 300000 325000 -25000

5 300000 325000 -25000

6 300000 325000 -250007 300000 312500 -12500

8 300000 312500 -12500

9 300000 312500 -12500

10 300000 325000 -25000

-250000


59/139

Interest Rate Swap

Typical Characteristics of the Interest Rate Swaps:

The principal amount is only notional.

Opposing payments through the swap are normally netted.

The frequency of payment reflects the tenor of the floating rateindex.


60/139

Currency Swap

A swap that involves the exchange of principal and interest inone currency for the same in another currency. It is consideredto be a foreign exchange transaction and is required by law tobe shown on the balance sheet.


61/139

To further elaborate the modus operandi and underlyingfundamentals of a swap transaction we shall discuss a simpleready against six month forward swap. As the first leg of thetransaction is in ready therefore it would be executed today atthe today prevailing exchange rate. For sake of simplicity weshall make the following assumptions:

READY USD/PKR Rate 60.00PKR 6-month interest rate 11.00%USD 6-month interest rate 6.00%


62/139

In order to ensure that no opportunities of arbitrage arise theTherefore, interest rate differential being (11-6) 5% thetheoretical depreciation in rupee would be

60.00 * 5% * (as interest rates are quoted on annual basis thesix months impact would be roughly )

= 1.5

This will result in a six month forward rate of 60 + 1.5=61.5


63/139

BANKA

BANKB

TODAY

USD 1 MILLION

PKR 60 MILLION

6 MONTHS FROM NOW

PKR 61.5 MILLION

USD 1 MILLION


64/139

Cross Currency Swap

Cross-currency swaps offer companies opportunities to reduceborrowing costs in both domestic and foreign markets

A currency swap involves the exchange of paymentsdenominated in one currency for payments denominated inanother. Payments are based on a notional principal amount the

value of which is fixed in exchange rate terms at the swap'sinception


65/139

Cross Currency Swap

Consider a Pakistani Exporter having exports proceeds inUSD. He/She has a PKR(KIBOR) loan liability on itsbalance sheet and he/she wants to convert this PKR liabilityto USD(LIBOR) liability to exploit the low interest rates ascompared to that of PKR.

In doing so he/she is taking on exchange risk but with hisexports proceeds in FX he/she has a natural hedge.


66/139

Mechanics of a Cross Currency Swap

@ - Initial Liability before swap is KIBOR @ - After swap net Liability is LIBOR

Exporter Bank

PKR Loan Liability

KIBOR

LIBOR

KIBOR


67/139

Derivative Pricing


68/139

Pricing ObjectivesMarket price, i.e. the price at which traders are willing to buyor sell the contract.

For exchange-traded derivatives, market price is usuallytransparent.

Complications can arise with OTC or floor-traded contractsthough, as trading is handled manually, making it difficult toautomatically broadcast prices.

Arbitrage-free price, meaning that no risk-free profits can bemade by trading in these contracts


69/139

Forward Contract Pricing


70/139

Future Contract Pricing


71/139

Option Pricing Theory The value, or premium, of an option is determined by the future

price of its underlying asset.

Of course, no one can really know for certain what an assets future price will be.

To help estimate this price, techniques have been developedusing probabilities and statistics.


72/139

Determinants of Option Prices

Current Price of Asset

Strike Price

Time to Expiration

Volatility Risk-Free rate


73/139

Option Pricing Models The Black-Scholes model and the Cox, Ross and Rubinsteinbinomial model are the primary models used for pricing options.

The binomial option pricing model uses an iterative procedure,allowing for the specification of nodes, or points in time, during thetime span between the valuation date and the option's expiration date.

The Black Scholes Model is regarded as one of the best ways of determining fair prices of options The model assumes that the priceof heavily traded assets follow a geometric Brownian motion withconstant drift and volatility. When applied to a stock option, themodel incorporates the constant price variation of the stock, the timevalue of money, the option's strike price and the time to the option'sexpiry.


74/139

Simpler models i.e. constructing option equivalents from commonstocks and borrowings

DCF model does not work in case of option since in that our standardprocedures for valuing an asset is to (1) figure out expected cash flowand (2) discount them at the opportunity cost. This is not practical foroption , however by combining stock investment and borrowing, theoption equivalent can be calculated. In this case, suppose we have astock with an exercise price of Rs 60, than assuming the day when theoption is at the money and by taking the short term risk free rate as1.5% for 8 months and 1.0 % for a year, we assume that the price of thestock may fall by a quarter to 45 or rise by one third to 80. In the case if price falls to 45 than the option is worthless but in case of moving to 80the pay off is 80-60 = 20. Now suppose you buy 4/7 shares and borrow25.46 from the bank than it would turn in to given scenario. 4/7 sharesof valuing 45 would value 25.71 and valuing 80 would value 45.71. Therepayment of loan with interest would come out as 25.71.


75/139

Simpler models i.e. constructing option equivalents from commonstocks and borrowings

In this method we borrowed money and exactly replicate it with the pay off from a call option. The number of shares needed to replicate one call is calledhedge ratio or option delta which is = spread of possible option prices/spread of possible share prices = 20-0/80-45= 20/35= 4/7

Now to value call option we would use given formula =value of 4/7 shares -25.46 bank loan = 60 x (4/7) 25.46 = 8.83 (value of call).To calculate value of put we would go reverse, that in the case stock at 80 wouldgive a payoff of 0 and at 45 would give a pay off of 15. This would give optiondelta of 0-15/80-45 = -3/7. The delta of put option would always be negative.Now with this case, the option payoffs can be replicated by selling 3/7 sharesand lending 33.95 making it 34.29 after one year. Now at sale of 3/7 shares of 45

you get -19.29 and against shares of 80 you get -34.29 (that you would lend).This give 0 payoffs for stock valuing 80 and 15 against stock valuing 45. Fromthis, value of put would be -3/7 shares + 33.95 as lending = -3/7 x 60 +33.95 =8.23


76/139

Simpler models i.e. constructing option equivalents from commonstocks and borrowings-Risk neutral valuation

In the above case the call option should sell at 8.83. If the option price is higherthan 8.83 you can have payoff by buying 4/7 shares, selling a call option andborrowing 25.46. Similarly if the prices are less than 8.33 you can make equally payoff by selling 4/7 shares, buying a call and lending the balance. In eithercase there would be an arbitrage opportunity. This provides us another way to

value the option. In this we pretend, that investors are indifferent about risk, work out the expected future value of the option and discount it back at therisk free rate to get the current value. If investors are indifferent about risk thanExpected return on stock = 1.0% per 8 months. In this case we have presumedthat stock can either rise by 33.33 % to 80 or fall by 25 % to 45. The probability of price rise is = Expected return = (probability of rise x 33.3) + ( (1-probability of rise) x (-25) ) =1.00%. Probability of rise can be ascertained as =interest rate downside change/ upside change downside change =.01 (- .25)/.333 (-.25)= . 446. The expected value of the call option would be (probability of rise x 20+ ( (1-probability of rise) x 0) = (.446 x 20) + (.554 x 0) =8.92. PV of call wouldbe = 8.92/1.01 = 8.83


77/139

Simpler models i.e. constructing option equivalents from commonstocks and borrowings-Risk neutral valuation

valuing the put option through risk neutral valuation method we would proceed as

(probability of rise X 0 ) + ( ( 1 probability of rise ) X 15 ) = (.446 X 0)+ ( .554 X 15) = 8.31 PV of put option would be 8.31/1.01 = 8.23

Relationship of call and putIn European option there is a simple relationship between call and theput i.e. value of put = value of call share price + PV of exercise price =8.83 -60 + 60/1.01 = 8.23


78/139

Binomial Pricing TheoryOne of the most widely used options pricing techniques is theBinomial Pricing Theory. It involves the construction of abinomial tree, and this tree is used to represent all of the

possible paths that the price of an underlying asset may takeduring the life of the option.

An example of a two-step option valuation of a European Call

will be used here to demonstrate the general functioning of thispricing method. For this model a few assumptions have to bemade:


79/139

1. The direction and degree of the underlying assets fluctuationis given. Assume for each stage of the tree that the underlyingassets price may either go up or down by 10%.

2. The risk free interest rate is known with certainty. For thisexample assume 12%.

3. The current price of the stock is used to project forwardpossible movements through the binomial tree. For the purposeof this example a price of $10.00 is used, with the optionhaving a strike price of $10.50. Thus, the option is out of themoney

4. No arbitrage .


80/139

The following notation will be used in this example:

So = Current price of the stock, Let So = $10.00

X =Strike or Exercise price of option

r = Risk-free rate interest, Let r = 12%

T = Time between periods in years. This example has two 3-month periods, making= 3months/12months or .25. T = .25

U = Degree or percentage change in upward movement, Let U = 1.1

D = Degree or percentage change in downward movement, Let D = 0.9

Pu = Probability of an upward movement

Pd = Probability of a downward movement


81/139

A two-step binomial tree model would, in theory, look like this:

figure 1

U = 1 upward movement in the price of the asset.UU = 2 consecutive upward movements in the price of the asset.

D = 1 downward movement in the price of the asset DD = 2 consecutive downward movements in the price of the

asset.UD = An upward or downward movement in the price of the

asset followed by an upward or downward movement in theprice of the underlying asset.


82/139

The tree with all of the possible asset prices would look like

this:


83/139

It is assumed that the strike price of the option is $10.50, andthe following possible option prices are then calculated asfollows:

Cuu = 12.10 10.50 = 1.60Cud = 9.90 10.50 = -.60, nil

Cdd = 8.10 10.50 = -2.40, nil

These prices are two periods forward, and the valuation occursat the present period. Thus, the prices of the options may becalculated through working a path back through the tree,

eventually ending up at the first node. For example, the optionprice at node U is calculated as the expected value of the twonodes that follow it discounted by the risk-free rate of interest.


84/139

However, to calculate the expected value one needs to calculate

the probability of an upward and downward movement. Thisprobability of an upward movement may be calculated throughusing the following formula:


85/139

Subbing in the values assumed at the beginning of thisexample, Pu is found to be 0.6523 and Pd is 0.3477.

= $1.04 This expected value has to be discounted back one period tonode U. This is the present value equation:

= $1.01


86/139

The tree would now look like this:

Where the value of the call at time zero is found through this equation:

Therefore, the price of this call is $0.97


87/139

Advantages & Limitations

The big advantage of binomial model over the Black-Scholes model is that it can be used to accurately price

American options. This is because with the binomialmodel it's possible to check at every point in an option'slife

The main limitation of the binomial model is its relativelyslow speed


88/139

Black Scholes ModelThe Black-Scholes Model was first discovered in 1973 by FischerBlack and Myron Scholes, and then further developed by RobertMerton. It was for the development of the Black-Scholes Model

that Scholes and Merton received the Nobel Prize of Economicsin 1997 (Black had passed away two years earlier). The idea of the Black-Scholes Model was first published in "The Pricing of Options and Corporate Liabilities" of the Journal of PoliticalEconomy by Fischer Black and Myron Scholes and thenelaborated in "Theory of Rational Option Pricing" by RobertMerton in 1973


89/139

Black Scholes ModelThe Black and Scholes Call Models

C = Theoretical Call Premium

r = Risk-Free Interest Rate

sigma = Standard Deviation of Stock Returns

T = Time until expiration (years)

X = Option Strike Price

N = Cumulative Standard Normal Distribution

S = Current Stock Price

e = Exponential Function


90/139

Assumptions of the Black Scholes Mod 1) The stock pays no dividends during the option's life2) European exercise terms are used3) Markets are efficient4) No commissions are charged5) Interest rates remain constant and known

6) Returns are log normally distributed


91/139

Lognormal Distribution Normal Distribution


92/139

Other Models for pricing Options

For rapid calculation of a large number of prices, analyticmodels, like Black-Scholes, are the only practical

option. However, the pricing of American options (other thancalls on non-dividend paying assets) using analytic models ismore difficult than for European options

To handle American option pricing in an efficient manner othermodels have been developed. Three of the most widely usedmodels are:


93/139

Roll, Geske and Whaley analytic solution : The RGWformula can be used for pricing an American call on a stock paying discrete dividends.

Black's approximation for American calls : Black'sapproximation basically involves using the Black-Scholesmodel after making adjustments to the stock price andexpiration date to take account of early exercise.

Barone-Adesi and Whaley quadratic approximation: Ananalytic solution for American puts and calls paying acontinuous dividend.


94/139

Advantages & Limitations

Advantage:The main advantage of the Black-Scholes modelis speed -- it lets you calculate a very large number of option

prices in a very short time

Limitation: The Black-Scholes model has one majorlimitation: it cannot be used to accurately price options with an

American-style exercise as it only calculates the option price atone point in time -- at expiration


95/139

What Is Volatility?

Volatility is a measure of the rate and magnitude of the changeof prices (up or down) of the underlying

If volatility is high, the premium on the option will berelatively high, and vice versa

Once you have a measure of statistical volatility (SV) for anyunderlying, you can plug the value into a standard optionspricing model and calculate the fair market value of an option


96/139

Implied Volatility The estimated volatility of a security's price

In short, the more a stock is expected to make a big move due

to some important news release or earnings release in the nearfuture, the higher will be the implied volatility of that stock

In fact, implied volatility rises as the date of that importantrelease approaches

Under such circumstances, market makers also hike impliedvolatility in order to charge a higher price for the higherdemand


97/139

Volatility Measures

Standard Deviation

Beta

R-Squared

Alpha


98/139

Standard Deviation

A measure of the dispersion of a set of data from its mean. Themore spread apart the data, the higher the deviation. Standard

deviation is calculated as the square root of variance

Standard deviation is a statistical measurement that sheds lighton historical volatility. For example, a volatile stock will have a

high standard deviation while the deviation of a stable bluechip stock will be lower. A large dispersion tells us how muchthe return on the fund is deviating from the expected normal

returns


99/139

Beta

A measure of the volatility, or systematic risk, of a security or aportfolio in comparison to the market as a whole. Also knownas "beta coefficient"

Beta is calculated using regression analysis, and it indicates thetendency of a security's returns to respond to swings in themarket. A beta of 1 indicates that the security's price will movewith the market. A beta of less than 1 means that the security

will be less volatile than the market. A beta of greater than 1indicates that the security's price will be more volatile than themarket. For example, if a stock's beta is 1.2, it's theoretically20% more volatile than the market


100/139

R-Squared

A statistical measure that represents the percentage of a fund orsecurity's movements that can be explained by movements in abenchmark index

R-squared values range from 0 to 100.A higher R-squaredvalue will indicate a more useful beta figure. For example, if afund has an R-squared value of close to 100 but has a betabelow 1, it is most likely offering higher risk-adjusted returns.A low R-squared means you should ignore the beta


101/139

Alpha

A measure of performance on a risk-adjusted basis.Alpha takes the volatility (price risk) of a security

and compares its risk-adjusted performance to a benchmark index. The excess return of the fund relativeto the return of the benchmark index is a fund's alpha

A positive alpha of 1.0 means the fund has outperformed itsbenchmark index by 1%. Correspondingly, a similar negativealpha would indicate an underperformance of 1%


102/139

Hedging Greeks

Delta

Gamma

Vega

Theta

Rho


103/139

Hedging Greeks Delta

Options Delta measures the sensitivity of an option's price to achange in the price of the underlying stock

In layman terms, delta is that options Greek which tells youhow much money a stock options will rise or drop in valuewith a $1 rise or drop in the underlying stock

This means that the higher the delta value a stock option has,the more it will rise with every $1 rise in the underlying stock


104/139

Hedging Greeks

Gamma

Options Gamma is the rate of change of options delta with asmall rise in the price of the underlying stock

Just as options delta measures how much the value of an optionchanges with a change in the price of the underlying stock,Options Gamma describes how much the options delta changesas the price of the underlying stock changes


105/139

Hedging Greeks

Vega

Options Vega measures the sensitivity of a stock option's priceto a change in implied volatility

When implied volatility rises, the price of stock options risesalong with it

Options Vega measures how much rise in option value withevery 1 percentage rise in implied volatility


106/139

Model's fair market value, however, is often out of line with theactual market value for that same option

This is known as option mispricing

The answer can be found in the amount of expected volatility(implied volatility) the market is pricing into the option

H dgi g G k


107/139

Hedging GreeksTheta

Options Theta measures the daily rate of depreciation of a stock option's price with the underlying stock remaining stagnant

In layman terms, Theta is that options Greek which tells youhow much an option's price will diminish over time, which isthe rate of time decay of stock options.

Time decay is a well known phenomena in options tradingwhere the value of options reduces over time even though theunderlying stock remains stagnant

H d i G k


108/139

Hedging Greeks Rho

Options Rho measures the sensitivity of a stock option's priceto a change in interest rates

Options Rho is definitely the least important of the OptionsGreeks and have the least impact on stock options pricing

In fact, this is the options Greek that is most often ignored byoptions traders because interest rates rarely changedramatically and the impact of such changes affect optionsprice quite insignificantly


109/139

Price Value of a Basis Point (PVBP)

A measure used to describe how a basis point change in yieldaffects the price of a security

Also knows as the "value of a basis point" (VBP)


110/139

Value at Risk

The Question Being Asked in VAR

What loss level is such that we are X % confident it will not beexceeded in N business days?


111/139

VAR vs. C-VAR VAR is the loss level that will not be exceeded with a specifiedprobability

C-VAR (or expected shortfall) is the expected loss given thatthe loss is greater than the VAR level

Although C-VAR is theoretically more appealing, it is not

widely used


112/139

Advantages of VAR

It captures an important aspect of risk in a single number

It is easy to understand

It asks the simple question: How bad can things get?


113/139

Methods of Calculating VAR There are three methods of calculating VAR:

The Historical Method

The Variance-Covariance Method

The Monte Carlo Simulation


114/139

Historical Method

The historical method simply re-organizes actual historicalreturns, putting them in order from worst to best

It then assumes that history will repeat itself, from a risk perspective


115/139


116/139

With 95% confidence, we expect that our worst daily loss willnot exceed 4%

If we invest $100, we are 95% confident that our worst dailyloss will not exceed $4 ($100 x -4%)


117/139

The Variance-Covariance Method

This method assumes that stock returns are normally distributed

It requires that we estimate only two factors:

Expected (or average) return Standard deviation

It allow us to plot a normal distribution curve.


118/139


119/139

The advantage of the normal curve is that we automaticallyknow where the worst 5% and 1% lie on the curve. They are afunction of our desired confidence and the standard deviation


120/139

Monte Carlo Simulation

A Monte Carlo simulation refers to any method that randomlygenerates trials, but by itself does not tell us anything about the

underlying methodology

A Monte Carlo simulation amounts to a "black box" generatorof random outcomes


121/139


122/139

Converting One Time Period to Anothe

Because of the time variable, users of VAR need to know howto convert one time period to another

They can do so by relying on a classic idea in finance:

The standard deviation of stock returns tends to increase withthe square root of time


123/139

Value at Risk


124/139

Derivatives Market

in Pakistan

Derivatives Market in Pakistan


125/139

Derivatives Market in Pakistan

Commodity futures contracts have recently been introducedfrom the platform of National Commodity Exchange LimitedKarachi. Currently they only trade in Gold futures and plan to

expand the contracts on agricultural commodities and interestrates.

SBP took initiative in 2004 by granting Authorized Derivative

Dealers (ADD) license to five commercial banks.


126/139

Need & Scope of Derivatives in Pakista

Volatile financial markets due to:

1. Political Uncertainty2. Monetary Policy3. Fiscal Policy4. Foreign Investment/Disinvestment5. War On Terror

Wh N d Th M ?


127/139

Who Need Them Most?

Equities/Interest Rate/Currency PortfoliosMutual Funds

Pension FundsBanksMajority Stockholders/OwnersForeign InvestorsCorporateFarmers

SBP APPROVED DERIVATIVES


128/139

SBP APPROVED DERIVATIVESFX Options Dealing in FX option is permitted in G-7 currencies only with 1

year tenor and the restriction to cover them on back to back basis by the ADDs/NMIs

Interest Rate Swaps Dealing in IRS is permitted in PKR Rupees only up to 5 years

tenor

Forward Rate Agreement Dealing in FRAs is permitted in PKR only up to 24 months

tenor

Recent Trend in Derivative Transaction


129/139

Recent Trend in Derivative TransactionAmounts in PKR Millions

31-Dec-06 31-Mar-07 30-Jun-07 30-Sep-07 31-Dec-07 Interest Rate Swaps(PKR) 13,748 19,350 19,023 18,339 22,726

Interest Rate Swaps(FX) 64,759 62,131 61,038 60,996 61,345

Cross Currency Swaps 30,114 71,411 89,689 122,180 156,391Foreign CurrencyOptions 10,983 37,151 42,561 41,321 47,312Forward RateAgreements 250 - 300 300 -

Total 129,165 190,043 213,263 268,716 316,401


130/139

(20,000)

-

20,00040,000

60,000

80,000

100,000

120,000

140,000

160,000180,000

31-Dec-06 31-Mar-07 30-Jun-07 30-Sep-07 31-Dec-07

FRA PKR IRS FX IRS FX Options CCS


131/139

C KSE P d /M k


132/139

Current KSE Products/Markets

Product Settlement Basis Since

Ready Market T+1 2001

Deliverable Futures 30 Days 2003

Cash Settled Futures 30, 60 & 90 Days 2007

COT (Stopped in 2006) T+22 1994

CFS (Stopped in 2009) T+22 2005

KSE plans for Derivatives Trading


133/139

KSE plans for Derivatives TradingProduct Settlement Basis Month

2008

SIFC Trading 90 Days March

Sector Index Trading 90 Days April

Options 90 Days August

Within 5 years, the volume of trades in Derivatives is expected toreach 50%of the total trading volume at the KSE

KSE : Futures


134/139

KSE : Futures Some analysts suggest that the March 2005 crisis was due todelivery pressures on account of Deliverable Future Contracts,concluding in the fourth week of March 2005

To prevent another crisis, the Securities & ExchangeCommission had advised the KSE to introduced Cash SettledFutures

Non-Deliverable Future Contracts have been introduced in 2007 Non-Deliverable Futures have not gained momentum due to:1. Absence of price discovery & convergence mechanics2. Presence of scrip level circuit breakers, as opposed to Market

halts

N i l C di E h Li i


135/139

National Commodity Exchange Limite

NCEL is the first technology driven, de-mutualized,on-linecommodity futures exchange in Pakistan

NCELs shareholders are Karachi Stock Exchange, LahoreStock Exchange, Islamabad Stock Exchange, Pak KuwaitInvestment Company (Pvt.) Limited, and Zarai Taraqiati Bank Ltd

NCEL is regulated by Securities and Exchange Commission of Pakistan.

NCEL PRODUCTS


136/139

NCEL PRODUCTS

Pakistan's National Commodity Exchange Limited (NCEL) hasformally started rice trading based on three-month futures

contracts becoming the country's first electronic commoditytrading platform.

NECL had first started operations on May 11, 2007 with three-

month futures trading in gold and the first delivery wassuccessfully executed in mid-August

NCEL Operations v Rest of World


137/139

NCEL Operations v Rest of World NCEL

Online, Electronic Tradingup to client level withUnique Identification (UID)for allClient has direct access tothe market with pre-tradechecksMargining for each andevery account at ExchangelevelMonitoring up to client levelMargins in cash only

International PracticePhysical and ElectronicBrokers provide onlineaccess

All orders routed throughbrokersPost-trade marginingMembers responsible formargining clientsMarket-wide & broker levelsurveillance with client levelreportingSecurities are acceptable

F t Pl


138/139

Future PlansNCEL is planning to launched futures related to followingcommodities in the near future:

Cotton Seed

Oil Cake

Palm Oil

Rice Irri-6


139/139

THANK YOU

investment banking - f-d 97

Documents