Options and Futures

Most over the counter insurance policies (i.e., property, casualty, life, health, fire, etc.) underwritten by various insurance companies are put options, which provide protection against unforeseen losses.

Put options are the right to put (to sell) the underlying asset at strike price when the price falls below the strike price (exercise price) over a given period.

Call options are the right to buy (to call) the underlying asset at strike price when the price rise above the strike price (exercise price) over a given period.

The buyers and sellers in the option market are called:

Call or put buyers Call or put writers (seller) The buyers pay a premium for the right not an

obligation to buy (to call) or to sell (to put) the underlying asset to the writer.

For receiving premium the writers are obligated to sell or to buy when it pays off for the buyers to do so.

Determinants of option premium Call Put Strike price - + Volatility (risk) + + Time to expiration + + Time value of money (interest rate) + - Spot Price of the underlying asset + - Foreign interest rate (dividend) - +

Exhibit 4.4: Implied Volatility Rates for Foreign Currency Options August 31, 2009

  EUR 9.7 10.4 11.0 11.4 12.1 12.5 12.0 11.5 

  JPY 13.7 14.0 14.1 14.2 14.5 14.7 14.8 14.8 

  CHF 10.0 10.5 11.1 11.5 11.9 12.2 11.6 11.0 

  GBP 11.3 12.0 12.3 12.5 12.9 13.1 12.9 12.8 

  CAD 13.8 14.0 14.1 14.1 14.3 14.4 14.4 14.5 

  AUD 15.7 15.7 15.9 16.0 16.2 16.3 16.0 15.2 

  GBPEUR 8.9 9.3 9.7 10.0 10.6 11.1 11.5 11.8 

  EURJPY 15.0 15.2 15.7 15.8 16.4 16.9 17.6 17.9 

* This release provides survey ranges of implied volatility mid rates for the money options as of 11:00 a.m. The quotes are for contracts of at least $10 million with a prime counterparty.

Put and Call Quotations

In the exhibit, the buyer of a October call at strike price of 17, pays $1.20 per share, or $120 for one contract (100 shares). The seller on the same strike price receives $119 for one contract that is 1.19*100. Same principle holds for put options.

Pay off of Long Put and Short Put

Example: Suppose an investor wishes to buy simultaneously a long put and a short put at a strike price of $18 for October delivery in Exhibit 4.1 of the book. What is the pay-off of this strategy for one contract?

Verify that the pay-off is nearly zero, numerically and graphically

Pay off of October long and short put at strike price of $15 for various stock prices

Profit (loss)

Long Put (put buyer)

ITM ATM OTM

E= Strike price spot price

} Put Premium

Profit (loss) E=strike price } Put writer Premium Spot price Short Put (Put Writer)

Black and Scholes option pricing is estimated using option software, where you enter all 5 or 6 parameters in the determinants of option premium slide and the software calculates the price of call and put and other parameters of interest in options.

To access software go to S:\Ghomaifar\derivgem Click on Derivgem.exe Click on type of option and for stock or currency or

index Click on edit Enter the 5 parameters Compute You will see estimation like slide #11

Black-Scholes Option Pricing

--------------------------------------------------------------------------------------------- Call Put Price .045 .025 Delta .598 -.397 Gamma 4.41 4.41 Theta (per/day) 0 0 Vega .003 .003 Rho .002 -.002 Option type Regular Exercise European Asset Currency Interest Rate % 1.85 Foreign Rate % 2 Volatility (% ) 12 Time to Expiration (YRS) .25 Strike Price 1.43 Spot Price 1.45

As can be seen from the option calculation, the call is $.045, and put is priced at $.025. The call is in the money by 2 cents, where spot price is $1.45/pound and the buyer can buy it at strike price of $1.43, for $.045. The put on the other hand is out of money whose value is equal to time value of the option which is in this case $.025

Estimates of Historical Volatility for Major Currencies

Annualized Standard Deviations of Currency Return

Periods 1971-86 Periods 1987-2001

Aus Dollar 0.0810 0.075

Can Dollar 0.032 0.037 Swiss Franc 0.105 0.101 Jap Yen 0.092 0.10 Brit Pound 0.082 0.086 Euro 0.087

These historical volatilities are fairly comparable to those provided by the Federal Reserve bank of New York. For example, the historical volatility for yen is 10 percent which is very close to its implied volatility that is estimated by the Fed

Delta: Measures the price sensitivity of the call and put to changes in the spot price. Delta is always in the range of zero to 1, it is nearly zero for both OTM options (call and put).

the put option delta is in the range of zero to -1, the delta is in the median range for the ATM options and closer to unity when the options are ITM.

What is Delta?

Gamma: Exhibits the price sensitivity of the delta to changes in the spot price. That is, the rate at which delta is changing.

Theta: Measures the sensitivity of the options price (premium) to the unit change in time (rate of time deteriorations). As time approaches to the expiration of the contract, the time value not the intrinsic value approaches to zero.

Vega: Measures the sensitivity of the options to the changes in the volatility.

Rho: Measures the price sensitivity of the call and put options to changes in the interest rate differentials.

What is Gamma, Theta, Vega, and Rho?

Receivables: Maximize $ receivables Payables: Minimize $ Payables Reduce or eliminate foreign exchange risk Reduce or eliminate volatility of earnings Cost of hedging Premium paid to buy some form of protection Hedging Alternatives:

Forward Hedging Futures Hedging Options Hedging

Money market Hedging

Hedging objectives

Risk Management Process

1. Identification

2. Measurement/Assessment

3. Monitoring

4. Control/ Mitigation

In the identification phase, we need to figure out what is and how much we are at risk. For example, we expect to receive 3 million euro in three months whose dollar value is at risk. In this case we are concerned about possible devaluation of euro. This is followed by analyzing how much we are likely to lose, and the course of action to take to monitor, control and mitigate this risk.

Break down of Financial risks

Commercial Investment Treasury Retail Asset

Banking Banking Management Management Management

operational

Credit

Market

It is June 8 and Excom Corporation has submitted a bid on a contract in Paris for 12.5 million euros receivable in two months provided that the bid is accepted.

The company wishes to protect its contingent receivable by buying put options in CME. The put option premium on the August put at strike price of $.9450 is quoted at 1.84 cents in the CME as seen in Exhibit 4.6.

Hedging with Options

Option on Foreign Currency Futures

EURO FX (CME) 125,000 euros; cents per euro

Calls Puts Strike Price Jun Jly Aug Jun Jly Aug

9350 0.88 1.29 - 0.00 0.78 1.30 9400 0.38 1.02 1.57 0.00 1.01 1.56 9450 0.01 0.79 1.35 0.13 1.28 1.84 9500 0.00 0.61 - 0.62 - - 9550 0.00 0.47 0.98 - - - 9600 0.00 0.36 0.84 1.62 - - Estimated Volume Total: 6,191 Volume for Thursday Calls 2,598 Puts 2,419 The strike price in the above exhibit are in cents, for example 9350 is .9350 and so on. The buyer of .9450 strike price for call and put, will pay for August delivery, respectively, for call 1.35 cents *125000, and for put, 1.84 cents*125000. the size of one contract is 125000 units of euro at Chicago Mercantile Exchange CME.

June 8: buy 100 put options in euro futures at CME (each contract is for delivery of 125,000 units of euro) at strike price of $.9450/euro for a total premium of $230,000.

.0184*12,500,000=$230,000 Result Excom will be ensured that its receivables will be

$11,582,500.00 at its minimum (floor) when the put option is exercised on the August expiration date assuming at the expiration date spot rate is $.90/euro.

Hedging Strategy

Excom makes a profit of $3,325 per put option, the difference between the $.9266/€ (strike price minus the put premium) and $.90/€ times 125,000 (size of one contract to buy euros in the futures market).

125000*(.9266-.90)= $3,325 Assuming a spot rate of $1/€. In this scenario the

receivable will convert to $12,270,000, that is, equal to spot exchange of euro for $12.5 million less the cost of insurance ($230,000 premium paid for the put option that expires worthless).

Hedging with option Continued

Zero Collar

Sam Nissan, a major car dealership has sold 1,000 units of the Altima produced in the Smyrna, Tennessee plant to a British importer in Birmingham, England. The cars were shipped on October 30, 2001 and the importer has agreed to pay £10 million in January 2002.

Hedging Strategy: Buy collar Buy 90-day put option at a strike price of $1.50/£ at the offer price for 2-cents per unit of

British pound. Sell 90-day call At strike price of $1.52/£ at the offer price for 2-cents per unit of BP

The collar is created by combing a long put and short call or vise versa on the underlying instrument, such as stocks, bonds, interest rates, commodities, and so on.

The zero collar has to be structured so that by selling the underlying call short, that pays for the premium paid for buying the put. In this scenario, the upside potential is being sold for a fee to finance the protection sought in buying the long put.

The following exhibit shows the payoff from various hedging stragegy

Exhibit 4.11 Pay-offs from Various Hedging Instruments

14.214.414.614.8

1515.215.415.615.8

1616.2

1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62

Dollar per Pound

Do

llar

Receiv

ab

les

unhedged

forward

put option

collar

exchange unhedged forward put option collar1.44 14.4 14.785 14.8 151.45 14.5 14.785 14.8 151.48 14.8 14.785 14.8 15

1.5 15 14.785 14.8 151.53 15.3 14.785 15.1 15.21.55 15.5 14.785 15.3 15.2

1.6 16 14.785 15.8 15.2

Functions of Options and Futures

Risk management: Hedging Speculation Leverage Reduced transaction cost and increased

efficiency Price discovery Regulatory arbitrage

Short Selling

Short selling can be viewed from several different perspectives as a means of:

Speculating, when investor buy/sell call or put on the underlying asset

Financing, when a dealer sell a bond and agrees to repurchase it at slightly higher price.

Hedging, when an individual or a corporation takes an offsetting transaction to mitigate risk. For example, a corporation has receivable denominated in foreign currency, sell receivable forward at the prevailing forward price, that luck the company at that price. This is hedging.

Structure of a Repurchase Agreement

Repo Rate

Collateral

Loan

100(1-haircut)

Security Dealer

Municipality

In the above exhibit, a dealer sell security to a municipality over say one week and post the underlying security as collateral, and agrees to buy it back (repurchase). The collateral is subject to haircut depending on the quality of the collateral. For example if the underlying collateral is Treasury security, the collateral is risk free, and the haircut could be near zero. However, for risky bond the haircut could be 20 percent or more, meaning if the value of the collateral is $100, the dealer can post the collateral and secure a loan of $80 or less. The dealer is financing the purchase of a security by repo (repurchase) over time, the way Federal Government do to finance national debt.

Speculating is where the short seller will borrow the underlying asset to be shorted and sells with an explicit agreement to buy the asset back later at a lower or higher price in an speculative transaction.

For example, John Doe believes that the IBM stock priced at $95 per share is expected to drop in the next three months. He wants to short 100 shares of IBM and therefore he deposits $4,750. in his brokerage account and wishes to sell 100 shares of IBM short.

Speculating

Short selling as a means of financing

To finance the purchase of T/bill, a security dealer sells the underlying asset for example, T-bills to a municipality for one week and agrees to repurchase the security after one week at a slightly higher price.

This short selling can be viewed as a means of financing at a fully collateralized basis.

the interest rate that the dealer paid for financing the purchase of the T-bills and is known as a Repo Rate.

Delta Neutral Portfolio

Hedged Portfolio: Combining for example, 625,000 units of euro with 1 million short calls on the euro will create a hedged portfolio with the following pay-offs.

The number of units of euro of 625,000 is determined by the hedge ratio of .625, the delta of the call option in Exhibit 4.17. The current value of this portfolio is therefore:

625,000 (.93) – 1,000,000 (.03) = 551,250

If the spot price of euro goes up to $1.01/€, the call will be exercised and therefore the value of the hedged portfolio will be equal to 625,000 (1.01) – 1000,000 (.075) = $556,250.

However, if the spot price of euro in 90 days falls to $.89/€, the call will be equal to zero and expires worthless and the value of the portfolio is equal to exactly $556,250, that is, 625,000 units of euro at $.89/€ and zero call value. The return on the portfolio is therefore equal to:

($556,250/$551,250)-1 ~ .01 The return on the portfolio is equal to approximately 1

percent per 90 days which is equal to a 4 percent interest rate differential between domestic and foreign interest rates of 4 percent.

Constructing Synthetic Forward Contract

Barrick lease gold from a central bank for a period of 90 days and pay the lease rate

Sell gold at spot price and invest the proceed in a 90-day security that pays more than the Lease Rate

Sell the security in 90 days and return the leased gold to the central bank Extracted from the Barrick Gold Mines

Forward Gold = Spot Gold Compounded at an interest rates differential over period of 90 days

In the above exhibit, Central bank converts non dividend paying instrument such as gold into dividend paying by selling (leasing) gold to gold mining Co. Gold mining company will be able to mitigate gold price risk by selling gold it does not have (to be extracted from the mine in the next say three to six months).