More Derivative notes


FinanceProfessor.com Bringing the real world to the classroom and vice versa!

Intrinsic value = value if exercised today
finding the intrinsic value is easy. For example, for a call it is the Max (S-X,0)

However, prior to expiration, the option value is also made up of volatility and time value.

Restrictions on Option value

Upper bound (price no one would ever pay more than this for the option)
    stock price    (assuming limited liability)

Lower bound    = “adjusted intrinsic value”
        = Stock price - PV(x) - PV(div)
        = So - PV(x) - PV(div)

Logic:

Consider 2 portfolios:
    1. a call with strike price X
    2. a share of stock and borrowing the PV(x) as well as the PV(div)

                    At expiration:

State of Nature      if S<x          if s>x
value of call      0   s-x

Value of share   S   S
repayment of loan    -X-D                   -X-D
       total      S-X-d S-X-d

but since the option payoff is greater than leveraged stock position (why? because on the left S<X)
    therefore the price of the call must be greater the value is S - X - D or more correctly:

           S-PV(x)-PV(div)

Prior to expiration a call option holder can close out his position by either selling the call or by exercising the option, but it is almost always worthwhile to sell it rather than exercise. (For thoise who know me personally and are familair with my workout practices, it is indeed a rare occasion when I I tell soemone not to exercise ;) )

Why not exercise? You get more for selling it. (That is, the option value is above intrinsic value.)

Assume the call is in the money
You exercise and get S-X, but we just showed that you can sell it (lower bound) for S-PV(x)-PV(d)
Since PV(x)<X, you would rather sell it than exercise it.

Thus for a NONDIVIDEND PAYING STOCK A European option and an American option are virtually identical. (at least no dividends over the life of the option)

Early Exercise of American puts
Early exercise of American Put options is possible in the case of bankruptcy. Stock falls to zero. Now you want the money now as it can not fall past zero and waiting lowers PV.
What about for a few cents? Is it worth waiting? Depends on interest rates.

Early Exercise of American Calls
Why? Voting rights, maybe to capture dividends, spin-off etc

Pricing Models:

A. Binomial Option Pricing Model
    assume the stock can only be at one of two prices at expiration

    Current stock price =$100..stock can be either $200 or $50 at the end of period.
Call with a strike of $125, T=1 year

so the payoff from this call is either 0 or $75.

Alternatively you could buy the stock and Borrow $46.30 (PV($50) if i=8%)

    At year end     value of stock        50        200
        repay loan                  -50                -50
            total             0        150

this is exactly double the call....we know the outlay needed to initiate this position: $100-$46.30=$53.70.
    so since this is twice a call, the call price must be $26.85 If not there is an arbitrage position available.

The keys to this are the idea of a replicating portfolio and that you can perfectly hedge your position
write 2 calls....
    stock value                50        200
    -obligation from shorting calls        0        -2(75)
    total                    50        50

thus this is RF...do it should cost pv(50) = $46.30
        thus 100-2c=$46.30

B. Black Scholes Pricing model

The BSOPM for a non-dividend paying stock paying is as follows.
(1)     C = S0 N(d1) - X e-rT N(d2)
where:

d1 = [ln(S0/X) + (r +Sigma²/2)T]/( sqrt(T))
d2 = d1 - (Sigma*sqrt(T))

S₀ = stock price at time zero

X = strike price

T = time to expiration

r = risk-free rate

Sigma = volatility of the underlying asset

N(x) = the cumulative probability distribution function for a variable

that is normally distributed with a mean of zero and a standard

deviation of 1.0.

Very widely used model!
Succinct.
Need to know partial derivatives!
most widely used model
assuming no dividends until option date
i-rates and volatility are known and any changes is predictable
stock price movement is continuous

Implied Volatility-backing out projected volatility

Merton (1973), Latane and Rendlemand (1976), and Hull and White (1987) show that the implied volatility of a particular option represents the mean anticipated daily volatility, that is the anticipated risk, over the life of the option.

The key is to understanding B-S is the N(d) term.....
    risk adjusted probability of expiring in the money

much time is spent on trying to calculate variance(forecasted) grach-egarch, agrach models.

Test tip
Be able to follow a WSJ index listing and show that for a calloption the following reslationships hold
                    + - + + +    -
call value =f(S,X,σ,T,Rf,div)

Dealing with Dividends
    two main ways
        1. Assume early exercise at dividend date
        2. Use ex-dividend stock price

    take the higher of the two for a pseudo-American call option

Hedge ratios”
Delta

p 671

Portfolio insurance
        hedge ratio constantly changes...
        selling a portion of your stock and investing in T-bills. Get the same payoff as a protective put.

N(d1)= probability of ending up in the money.

also the delta hedge ratio
    usually between 0 and 1.

Option positions

spreads
    money spread
    Time spread
Strangles

Using (and seeing) options that are not “so labeled” is critical.
Example: levered equity is a call.
    Jr. Debt is similar to a bull spread.

Using B-S allows us to predict how participants will fight over things etc.

Futures:

similar to forward, but more standardized, liquid etc
Developed largely as agricultural product, now (since 1975, financial as well)

Futures exchange clearinghouse
•    Clearinghouse acts as counterparty to all trades
•    acts as a guarantor, oversees delivery, bookkeeper, and settlement treasurer

Profits:
    Long position
        Spot price at maturity-original futures price
    Short position
        Original futures price-spot price at maturity

Trading takes place in the “pits” (see Trading Places)

To get out of a contract: reversing trade

Maintain margin accounts. Maintenance margin is usually 75% of the initial margin

Futures are marked to market daily.

Basis = spot (t)-Future price (t,T)
where (t,T) is the future price of a contract at time t that matures at T
A buy contract specifies that the individual will accept delivery and hence “buy” the commodity. A sell contract specifies that the individual will make delivery and hence “sell” the commodity.

The units of trading vary with each commodity. For example, if the investor buys a contract for corn, the unit of trading is 5,000 bushels. If the investor buys a contract for eggs, then the unit of trading is 22,500 dozen.

Selected Commodities, Their Markets, and Their Units of Trading
Commodity        Market                        Unit of Contract
Corn            Chicago Board of Trade            5,000 bushels
Soybeans        Chicago Board of Trade            5,000 bushels
Barley            Winnipeg Commodity Exchange        20 metric tons
Cattle             Chicago Mercantile Exchange            40,000 pounds
Coffee            New York Coffee and Sugar Exchange    37,500 pounds
Copper            Commodity Exchange, Inc., of New York    25,000 pounds
Platinum        New York Mercantile Exchange        50 troy ounces
Silver            Commodity Exchange, Inc., of New York    5,000 troy ounces
Lumber        Chicago Mercantile Exchange            100,000 board feet
Cotton            New York Cotton Exchange            50,000 pounds

The commodity exchanges are subject to regulation. Federal laws pertaining to commodity exchanges and commodity transaction laws are enforced by the Commodity Exchange Authority, which is a division of the Department of Agriculture.

Commodities are paid for on delivery. Thus, a contract for future delivery means that the goods do not have to be paid for when the individual enters the contract. Instead, the investor (either a buyer or a seller) provides an amount of money, which is called a margin (good faith deposit). The margin should not be confused with the margin that is used in the purchase of stocks and bonds.

In the commodity markets the amount of margin does not vary with the dollar value of the transaction. Each contract has a fixed minimum margin requirement.   At least according to Mayo (investments)

Note that the actual numbers will not be on the test, but if you are looking for more recent numbers for your own personal use they can be found at http://WWW.FADC.COM/pdf/99specs.pdf combined with http://WWW.FADC.COM/trf_mar.htm.

Margin Requirements for Selected Commodity Contracts at the time of publication (1994)
            Margin            Financial            Margin
Commodity        Requirement        Futures                Requirement
Broilers        $ 500            S&P 500            $20,000
Cocoa            1,000            NYSE Composite Index        7,000
Cotton            1,000            Value Line Index        20,000
Hogs                800            Treasury Bonds            5,000
Lumber        1,200            Treasury Bills                1,500
Potatoes            500            Municipal Bonds            4,000
Soybeans        1,500
Wheat            1,000

* Small amount of margin is one reason why a commodity contract offers so much potential leverage.
* margins can change with market conditions. As volatility increase, the margins are increased
* Example
    * a contract to buy wheat at $3.50 per bushel
    * controls 5,000 bushels of wheat worth a total of $17,500 (5,000 X $3.50)
    * must remit $1,000
    * an increase of only $0.20 per bushel produces an increase of $1,000

* The percentage return on the investment is 100%.

* An increase of less than 6 percent in the price of wheat produced a return of 100 percent.

* Leverage, of course, works both ways. If the price of the wheat declines by $0.10, the contract will be worth $17,000. A decline of only 2.9 percent in the price reduces the investor’s margin from $1,000 to $500. To maintain the position, the investor must deposit additional margin. Failure to meet the margin call will result in the broker’s closing the position.

There are two margin requirements. The first is the minimum initial deposit, and the second is the maintenance margin. For example, the margin requirement for wheat is $1,000 and the maintenance margin is $750. The maintenance margin is usually 75% of the initial margin.

The margin adjustments occur daily. After the market closes, the value of each account is totaled. In the jargon of futures trading, each account is marked to the market.

Limits are imposed by the markets on the amount of price change permitted each day.

Taxes and Futures
All positions in futures are considered to have been closed at the end of the tax year. Open positions then must be marked to the market on the last day of the tax year and any paper profits taxed as if they were realized capital gains.

The profits are arbitrarily apportioned as 60 percent long-term capital gains and 40 percent short-term capital gains.

Programmed trading refers to the coordinated purchase or sales of an entire portfolio of securities.

Arbitrage refers to the simultaneous establishment of long and short positions to take advantage of price differentials.

Index arbitrage is no different, except the arbitrageur is buying or selling index futures and securities instead of pounds. If prices deviate in different markets, an opportunity for arbitrage is created.
Programmed trading index arbitrage - if stock index futures prices rise, the arbitrageurs will short the futures and buy the stocks in the index; if futures prices decline, the arbitrageurs do the opposite.

Three potential problems:
    * transactions costs
    * there is an obvious problem with buying or shorting all the securities in a broad-based
        index. To get around this program, the arbitrageurs have developed smaller
        portfolios called baskets that mirror the larger index.
    * for arbitrage to be riskless, both positions must be made simultaneously

More on futures to follow....

FinanceProfessor.com