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> 1) When you have an call option on futures at strike of X, does it mean > that the 'delivery price' of the futures contract you get is also X when > you exercise? That is effectively correct - but you must be careful of the distinction between a delivery price in a forward and in a futures. In a forward the delivery price never changes; in a futures, the contract is resettled, marked to market, and a new delivery price established every day. Suppose futures price is 65cents right now and you excercise a futures call with strike 60cents. Then you get an immediate mark-to-market gain of 5cents per unit of underlying. Gains or losses from then on are added to or subtracted from your margin account just like any other trader who is long futures. Your position is identical to someone who entered the futures contract at some prior date when the futures price really was 60cents. > Meaning X determines how much you can buy the futures contract for, and > how much you can buy the underlying of the futures for at maturity. Again, beware the difference between forwards and futures. With futures you do not buy at the forward price that existed when you entered the contract; with forwards you do. The gains/losses on the futures accumulate in your margin account and you end up with an accumulated position that means the final cost if you take delivery is as if you had actually purchased at the futures price that existed when you entered the contract, but you actually pay the futures price at the end of the life of the futures contract which is pretty much the same as the spot price. It is those mark-to-market gains/losses that when combined with the final delivery price produce a net cost as though you had simply bought at the futures price that existed when you entered the contract - see the numerical example using gold in the archive of F421 Q&A.
Making the step sizes smaller in the binomial model increases the number of possible paths that the stock price can follow through the tree and it increases the number of possible final stock prices that can be attained. These both have a very large effect on increasing the accuracy of the binomial option pricing. A second order smaller effect is that as we decrease the step size, the possible price paths look more like those that would be followed in a "continuous-time" process - the process assumed by the Black-Scholes model. This process is one where the prices are continuous variables, not discrete ones and the prices move smoothly through time not in little discrete jumps.
In the Monte-Carlo method, making the step sizes smaller serves the second purpose - it makes the price paths look more like those of a continuous-time process, but it does not have a strong effect on increasing accuracy of pricing. The SLLN is where the accuracy comes from, and this accuracy is a result of very many repetitions of the simulation procedure. That is, for the SLLN to kick in we need many price paths, not small time steps. In fact, if the option is not path dependent, then we only need one time step (equal to the entire life of the option) and with many repetitions of this single step we will get accuracy. We use intermediate time steps only when the option is path dependent (eg the SIIA oferred by Citi).
The binomial method is not driven by the SLLN, but the Monte-Carlo method is. In each case more paths give better pricing. Only in the binomial method does more paths necessarily mean smaller step sizes.
A bear spread can also be constructed using other options, and when using puts the payoff is zero or positive, so it would cost money up front to enter. However the profit diagrams are identical regardless of whether it costs or generates money up front.
Thus the underlining serves to reduce the time you spend trying to decide which question in section 2 and which in section 3 you will ignore.
Of course, once you have chosen which questions to do, you need to read all the words in those questions.
In fact, the covered write payoff was entirely at or above the horizontal (zero payoff) axis, and the short put payoff was entirely at or below the horizontal (zero payoff) axis. Thus the profit diagram for the covered write is lower than its payoff (because it must cost money up front), and the profit diagram for the short put is higher than its payoff diagram because it must generate money up front.
I don't know whether I told you that the profit diagrams are identical or not. They are! To show this you need to take account of the time value of money (TVM) correctly. Usually when we draw profit diagrams we just ignore TVM because the typical option life is so short.
Note that the contractual specifications of both the eurodollar futures contract and the t-bill futures contract mean that the futures contract value is given by
F=$10,000 times [100 - (90/360) d]where d is a number like 5.2 for example and is the settlement "price" for the contract. Alternatively, the futures contract value is given by
F=$10,000 times [100 - (90/360) (100-Q)]where Q is a number like 94.8 which is also a settlement "price" for the contract (and is equal to 100-d). That is, you put "90" into that fraction always! It is not a case of counting the number of days anywhere because the 90 represents the days to maturity of the underlying asset at delivery (which is 90 days for both the T-bill, and the EDO contract). Of course, we have to remember that delivery can not actually take place on either of these contracts because both are cash settled.
There is a similar looking formula for pricing T-bills, and this does require the counting of days, but that is because you are pricing a T-bill which has some number of days to maturity which is unlikely to be 90.
I just grabbed futures settlement prices for the near month 30-YR T-Bond and the near month Eurodollar deposit contracts from Feb 82 to Aug 99 on a daily basis - found them for free on the web. I plotted the closing prices on a daily basis for these 4406 days. what is neat about this is that you can see how extremely volatilie the tbond futures prices are. Yes, that's the long term interest rate contract. Conversely, look at how sedate the EDO futures prices are. Yes, that's the short term interest rate contract. Does that mean that long term interest rates are much more volatile than short term interest rates? No. What it means is that the Macaulay duration of the t-bond is so long and the Mac Duration of the EDO is so small, that even though short term interest rates are much mre volatile than long term rates, the futures prices behave quite differently. That is why we use duration matching.
The correlation between the settlement prices is +79%
First, corporate risk managers are exposed to interest rate risk on borrowing and lending. These interest rates upon which they face risk are corporate interest rates: they include default premia. Eurodollar rates also include default premia (the borrowing or kending is backed by a large bank). It follows that Eurodollar interest rates move more like corp borroing and lending rates than do the default-free t-bill rates. You want the best hedge possible, so eurodollar futures are going to match your exposure better than t-bill futures.
Second, futures speculators want rapid executions, low transactions costs, and liquidity. As corp hedgers migrated to eurodollar (EDO) futures in ever increasing numbers, that contract is where the liquidity went. Speculators would be attracted to the liquidity provided by the hedgers, and they would choose the EDO contract too. End result: 10,000 times as much volume in the CME's EDO futures contract as in the t-bill futures contract.
If you want to borrow USD overseas, other banks offer you the USD at interbank offer rates. If you borrow USD from a bank in London, you pay the London interbank offer rate for that bank: A USD LIBOR rate.
If it is not USD, but JPY, then the interest rates are JPY LIBID (earn on your deposit), and JPY LIBOR (pay on your borrowing).
Remember that no conversion of currency takes place if you deposit funds. That is, if you deposit USD in London and earn a LIBID rate, you earn it on a USD denominated deposit earning USD interest. There is no conversion to british pounds.
Now that 11 European countries have joined in currency union, there is a new currency: the EUR. If you borrow EUR in London, you pay EUR LIBOR.
If you borrow USD in Paris, you pay USD PIBOR.
The differences between bid and offer rates are typically approximately 1/16th to 1/8th of a percentage point per annum.
The new European Banking Federation now uses "euribor." This is the same concept as "EUR LIBOR" except the location is not London, but the euro zone.
Given the linking of euro zone currencies, I think that FRF LIBOR is now identical to DEM LIBOR. Indeed, checking on the Bloomberg terminal, the quotes are identical.
Different banks in London quote different LIBOR rates (just as different banks in Bloomington quote different mortgage borroing rates). When you see USD LIBOR reported in the WSJ, or FT, or on the news, it is an average of numbers from a sample of banks with the highest and lowest few observations cut out.
The convention for pricing T-bonds, T-bills, Eurodollar deposits and so on uses discretely compounded and quoted yields. We have to understand the market conventions for the pricing of these instruments outside of the derivatives world (using non-contintously compounded returns), but we have to know how to take those prices and yields and convert to our purposes in derivatives pricing so that we can use the formulae. It takes a while to see where you use one, the other, or both types of yields.
Appreciation/Depreciation of JPY IS due to the supply and demand for JPY. This in turn is related to macroeconmic factors like level of and changes in interest rates, unemployment, inflation, productivity, in Japan.
Forward premium/discount is a direct consequence of the difference in interest rates between two countries and is determined solely by no-arbitrage arguments (assuming no political risk problems).
This info is from their home page Spear, Leeds & Kellogg is a major presence in the trading arena, and is Specialist/Market Maker in the stocks and options of thousands of companies worldwide. While serving the professional trading community, we execute and clear approximately 60 million shares a day accurately and very cost-effectively through our specialist and market making divisions. SLK is the largest specialist firm on the NYSE where it serves 275 companies, including IBM, Gillette, Mobil, America Online, and Boeing, as well as the largest specialist firm on the AMEX, where it serves for more than 175 companies. The firm also serves as the specialist in the options of companies on the Philadelphia Options Exchange and the CBOE.
If you do not borrow money from them, and you have a cash balance in excess of $1000, then you earn 2.5% per annum on this balance. Each day they multiply the balance by 0.025 and then divide by 360. This is the interest earned on this day, and it accrues day-by-day until the first day of the next statement period when a single lump sum cash payment is made into your account representing the interest earned since the beginning of the previous statement period (which may be one month).
You can borrow money from Ameritrade, and combined with your own funds you may purchase securities and hold them in your brokerage account. This is "buying on margin." The amount you borrow is the "DEBIT BALANCE." Ameritrade has a helpful GLOSSARY of terms.
You are charged interest based on the NATIONAL PRIME RATE, and your DEBIT BALANCE (what you have borrowed). The NATIONAL PRIME RATE is quoted in the Wall Street Journal (WSJ) every day. It is a base lending rate quoted by commercial banks and used as a reference rate for loans and credit card debt too. My most recent WSJ gives the (BEY) yield on a one year T-bill as 5.23%. The same journal quotes the PRIME RATE as 8.25% (so it is 300BP over treasuries).
If your DEBIT BALANCE is under $25,000 you are charged PRIME+75BP. If $25,000-$49,000 you are charged PRIME+25BP. $50,000-$99,000 you are charged PRIME-75BP. $100,000-$249,000 you are charged PRIME-100BP. $250,000-$999,000 you are charged PRIME-125BP. Above $999,000 you are charged PRIME-175BP.
Thus the interest rate they use is linked to the daily PRIME RATE, but they charge you less the more you borrow. However, even the best rate is PRIME-175BP which is still T-BILL+125BP.
How do they calculate what you owe in interest payments? First, find the PRIME RATE, then use your balance and the above schedule to compute your rate. Let's say PRIME=8.25% today, and your DEBIT BALANCE (what you have borrowed) is $2000 (maybe you deposited $2000, and bought $4000 worth of stocks). Then they charge you interest at a rate of 9% per annum.
You charge for this day is the calculated rate times the DEBIT BALANCE divided by 360. That is, $2000 times 0.09 times (1/360). This interest charge for this day accrues (that is, it does not enter your account yet) on a day-by-day basis until the first business day of the following account statement period when it is posted to your account as a lump sum debit (increasing what you owe). It appears that they do not charge interest on your interest on a day-by-day basis, but only one a month-by-month basis.
Do note that the prime rate and thus the rate used to calculate your interest both move around all the time. So in the above example, if interest rates go up 25BP, and stocks fall, your debit balance of $2000 may be charged at 9.25%. Of course, if your stock falls a lot, they can make a margin call because they are worried that you moght not pay back all the money. They allow three days for you to meet the margin call, but they are allowed to sell your stock before then if they want to.
If the stock crashes, you still owe all the money you borrowed.
Note also that not all stocks/securities are eligible for purchase on margin, and not all securities can be posted as collateral against which you can borrow. For example, very low priced stocks (say $5 or under) or options on stocks are typically not eligible I think.
If the stock ticker is only three characters, eg IBM, then the option ticker symbol is usually the same.
See this page for full list: http://www.cboe.com/tools/symbols/
However, the liquidity risk is different. If I want to exit the position prior to maturity, then I can do so with a penalty (using the embedded put option) with the CD, but there is no such penalty with the T-bill. The T-bill is a highly liquid traded instrument, so I can just sell it quickly at a price not too different from that most recently quoted.
The t-bill is more versatile (no penalty!) , so I will pay extra (i.e. get a lower yield) if I want it. The CD is less versatile (penalty for exiting early), so I must be compensted (lower price or higher yield) for using it.
The difference in yields appears to be about 30 basis points - looking in a recent WSJ.
So yes, GLOBEX uses price limits, but they are not necessarily the same as during RTH.
I would like to know about the positives and negatives of convertible bonds/stocks.
I'm looking forward to hearing from you soon.
Thanks,
In class, I said that a convertible is a regular (or "straight") bond plus a warrant. Although true, these component parts are not separately identifiable, but rather are issued as one security.
There are also straight bonds that are issued with attached warrants where the separation is clear (they are two separate securities). However, for all intents and purposes, the two concepts are close enough to be considered the same, and convertibles may be analysed most easily as a package of straight bond plus warrant on stock.
Convertibles usually carry a lower coupon rate than "straight bonds" (bonds without any embedded options).
A low coupon is bad for the buyer. However, she is compensated by having the right to convert the bond into stock if the stock subsequently becomes more attractive to hold than the bond (for example, the company does well and the stock prices rises tremendously).
POSITIVE: participation in stock if stock rises POSITIVE: ownership of bond if stock price falls NEGATIVE: lower coupon on bond (than straight bond).
In a way you can think of this as stock market participation with a gauranteed downside protection. Why? Because if the stock falls, you still have the bond; and if the stock price rises, you get the stock.
Convertibles are often issued by riskier companies. They can help to reduce the conflict of interest between stock and bond holders by giving the bond holders "a piece of the action" in the stock (got that part from Brealy and Myers' "Principles of Corporate Finance" book).
The low coupon is also good for the issuing company itself, because they raise funds at a lower cost to them than with straight bonds.
Also (the plot thickens), when companies issue convertibles, there is usually another option involved. The company retains the right to call the bond back from the buyer at a preset price. The company would call the bond if the bond is worth more than the "call price" (the strike of the option the company has retained). That is, the company will call the bond if they can buy it back for less than it is worth.
If you buy a callable convertible you are
LONG a straight bond
LONG a call option on the stock (strike is in terms of bonds)
SHORT a call option on the bond (strike is a $ amount).
Which is a bit more complicated than we need to be at this stage of the semester!
I recommend Brealey and Myers' book "Principles of Corporate Finance" for further reading on convertibles.
Thank you TFC
knock, knock who's there, Impatient cow, Impati... (interrupting) MOOOOOOOO!
Once we know what a fair price is, we add our markup to it to make our profits. For forwards, there is no upfront commission. Instead, there is a spread around the fair price F: if you are buying forward, the bank will charge you F+a small amount; if you are selling forward, the bank will pay you F-a small amount. The bank makes the profit from the spread around the fair price.
So when you suggested the bank could make money, you are correct (otherwise why would they do it), however you are already assuming a spread that we have not yet included in the analysis. Before transactions costs, we cannot have that F-Sexp(rT)>0 or this is an arbitrage opportunity.
This is just like having a spread in the spot market where the ask price is above the fair value of the asset, and the bid price is below.
I pay the printer to copy, cover, and bind the book. I pay for postage and packaging and I must collect sales tax for the state on any sales I make. I figure the "construction cost" of the book works out at around $20. I add $10 markup and I charge around $30.
When we value forwards, futures, and options, all we are doing is figuring out the construction cost. This construction cost is a fair value in the absence of arbitrage opportunities. The term "construction cost" is very appropriate here because for both forwards and options, we construct a trading strategy that replicates the payoff to the derivative, and work out how much it costs to implement that strategy. It really is a construction cost! With no arbitrage opportunities available, the fair value of the replicating strategy must be the same as the fair value of the derivative.
Once we have this construction cost (or "fair value"), we can add a markup to cover other expenses like employee salaries, office space rental, computer costs, and a return on equity.
As the delivery period on a futures contract approaches, most people close their contracts out through "offset" (sell if you are long, buy if you are short). The few people who hang on have to start thinking about physical delivery.
The person in the short futures position has the option to choose when to make delivery during the delivery period. The delivery period can be as long as one full month. Should the short deliver at the beginning of the month, the middle, or the end? Or when?
Well, it depends upon the cost of carry. Let me denote the cost of
carry by "c" then c=r-q+u-y where:
r=interest rate
q=dividend yield on underlying
u=rate of storage costs, and
y=convenience yield.
If c>0 (positive cost of carry), then it must be that r>q+y-u (after some minor algebra). This means that your are better off in cash (returning yield r) than in the asset (returning yield q+y-u). In this case the short would wish to get out of the asset and into cash as soon as possible and would deliver at the beginning of the delivery period.
If the cost of carry is negative (c<0), the reverse is true. In this case the short is better off holding onto the asset as long as possible, and making delivery at the end of the delivery period.
Who cares when delivery is to be made? Well, you may need to stick "T" into a pricing formula to figure out a futures price. To do this, you need to know when delivery is likely to occur.
The speakers/instructors for the seminars/courses included Fischer Black, Myron Scholes, Robert Merton, John Cox, ChiFu huang and the *very top* people who had control for risk management or trading responsibilities at the corporations or investment banks mentioned above. I learned a great deal by working as a consultant and guide for seven semesters with teams of MIT students who were solving genuine financial engineering, or financial management, or risk management problems posed by the large investment banks and corporations mentioned above (8 different problems each semester for seven semesters). I have learned a lot by talking to people who were my students and now work in financial organisations or corporations (they are always calling me up or emailing me with questions comments, advice and stories - as I hope you will do). I ask them questions too.
I was also fortunate to have students who were working full time and studying also. For example, I was a TA in an investments course. One guy would come to see me to ask about this week's problem set or the upcoming midterm; meanwhile I would be asking him about the 4 billion dollar CMO pool he was managing. You can't help but learn!
I watch the business news almost every day for an hour, I spend 45 minutes with the WSJ everyday, I read any book that looks interesting, I talk to anyone who I know has real world experience. The one thing I do not do is trade derivatives for my own account. Why? Because it would be a damn fool thing for me to do (I have no true risk capital and I know from my trading friends who have lost tens of thousands of dollars just how risky it is).
Bottom line. As I told you clearly in the very first class, I am not here to turn you into investment bankers or traders because I am an academic and this is school, not the real world. What I can do is pass on what I have acquired, and give you the basic tools you need to form a foundation for whatever use you have in the real world for derivative securities.
If that does not answer your question use the anonymous email again and tell me what else you need to know.
4:30-5:00AM This Morning's Business Ch8 (?) OK 5:00-5:30AM Bloomberg Business News Ch30 (PBS) Excellent 5:30-6:00AM Bloomberg Business News Ch30 (PBS-Repeat) " " 5:30-6:00AM AG Day Ch59 (Fox) OK 6:00-6:30AM First Business Ch59 (Fox) GoodI'd recommend Bloomberg+First Business.
The formula implies that people, who convert dollars in Korunas, enter a short forward contract on Koruna, invest in the Check Republic, and then convert Korunas back to dollars, will earn a rate of return, which is equal to that of the US risk free securities over the same period. The forward rate is what implies a no arbitrage relationship. However, at the same time there are OTHER investors, who hold Check securities without combining them with short Koruna forwards. Those other investors, ON AVERAGE, must expect that future dollar/Koruna exchange rates will allow them to earn returns higher than the US 5.5%, simply because the REAL return on Check government securities has higher deviation, a result of hardships with predicting inflation. Therefore, ON AVERAGE, investors can expect that Koruna/dollar forward rates will decline as the exercise date approaches, because forward rates have to converge with the spot rates. It is clear that spot rates on average must be lower, so that US investors in Check securities (those who do not mess with forwards) are compensated for extra risk. However, if forward rates will have a straight tendency to decline as the exercise date approaches, this would be a clear money making opportunity, which also must not exist in efficient markets. Can you clarify what is going on?
You do NOT know that forward/futures rates will decline because you do not know what will happen to the spot rates. A fair price for delivery of a currency on Dec 1 (the Dec 1 forward price) must converge to the Dec 1 spot price as you move through time because the maturity of the delivery date apporaches spot delivery. However, without knowing where the spot rate is going, you cannot know whther that convergence occurs at a higher or lower rate than the current forward rate. If you knew where spot rates were going to be you would have not forex risk and you would not need forwards in the first place.
Let us suppose you sent a buy order to the floor via your broker. Your order is crossed with a sell order in the pit. The agreed upon fair price for future delivery (the futures price) is 68.00 cents/lb, say.
By the close of trade, the futures price for this contract falls to a settlement price of 67.90 cents/lb. The clearing house will now take $40 out of your margin account (0.10 cents/lb x 40,000 lb). You are down $40. This is not enough to breach the maintenance margin.
You can hold on to your position, and hope that the price will rise the next day, or some time in the not too distant future. If the settlement price is 68.10 cents/lb at the end of the next day (a change of +0.20 cents/lb), then $80 will be added to your margin account, and you will be $40 up.
This is why it is called marking-to-market. Every day the exchange looks at the settlement price and uses it to adjust the margin account balances of people who have futures positions.
So, you *can* hold on to the position, and your margin account *does* fall, but you have the potential to make it up again (or for you to lose even more) the next day.
The short is responsible for transportation costs for delivery of the goods to a CBOT-approved delivery point. The long is responsible for transportation costs involved with taking delivery of the goods at the CBOT-approved delivery point and transporting them from there. The short invoices the long for the futures price (or an adjusted price if something other than par grade is delivered); this invoice price does not include transportation costs. Take a look at section 6 of this source.
My answer is the only logical one, because it would not make sense for the long position to be invoiced differently for identical product depending upon its point of origin (it follows that the long does not pay to ship the goods to the delivery point). Similarly, it would not make sense for the short to invoice the long for different amounts for identical product depending upon how far from the delivery point the long is located (it follows that the short does not pay to deliver the goods from the delivery point to the long's warehouse).
Suppose that we are talking about a jewelery manufacturer who needs to buy 100 troy ounces of gold on June 15 for usage in production. The spot price of gold is $353.70 per troy ounce right now (Jan 28, 1997). The manufacturer does not know what the spot price of gold will be in June (neither do I). This is clearly an input price risk.
Buying (i.e. going long) a 100 oz gold futures contract for delivery in June will hedge this risk. The futures price for delivery of 100oz of gold in June is $358.00 right now (Jan 28, 97). Entering a long futures position now locks in a cost to you of $358 per ounce (for a total of $35,800) -- it is a hedge. Let us explore this.
Let me assume that the margin account does not earn interest. Let us suppose that by June 15 (in the middle of the delivery period) the spot price of gold has risen to $400 per oz.
What happens to the futures price of the June gold contract? Well, if we are in the delivery period, the futures price must equal (or be very close to) the spot price of $400 per oz. The arbitrage argument appears in the text (p22 text).
Let me assume that you do not close out the contract, and the clearing house notifies you that someone wishes to deliver gold to you. You will be invoiced for the futures price of $400/oz x 100oz = $40,000. Does this mean that you end up paying the spot price after all, and that this was not a hedge?
NO! Look at what you actually pay. Sure you are invoiced $40,000 and you have to pay this; but what about your margin account? You went into a long position at a futures price of $358. The future price rose to $400 by delivery. This is a change of +$42/oz in the futures price over the life of the contract. You were long, so your margin account gained by $42/oz x 100oz = $4200! This gain ocurred through day-by-day marking-to-market as the futures price rose. At delivery you pay the invoice price of $40,000, but you can use the $4,200 gain in your account to offset part of this cost. You need only come up with a total of $35,800 out of you own pocket to pay the $40,000 you are billed. You will notice that this is the price you locked in originally.
Other points: If you had used a forward instead of a futures, with a forward price of $358/oz, you would just have paid the $35,800 as agreed. The result is the same. Also, if the spot price fell to $320/oz at delivery, and the futures price converged to the spot, then you will be invoiced for $32,000, but your margin account would have posted a loss of $38/oz x 100oz = $3,800 (via margin calls). For a total cash outflow of $35,800. That is, a price rise or fall leads to the same cash outflow, and it equals that which you locked in initially.
Finally, stepping outside the box, futures are rarely taken to delivery. You could have closed out your position just prior to delivery and purchased gold in the spot market. The gain or loss on the contract, together with the spot market payment will be very close to $35,800.
So it is a hedge after all!
Hi! Just wondered if you could answer some questions for me.
First, on p.68 and p.90 the authors of our book talk about how interest rates and gold/silver cause a negative basis risk. I don't understand why this is and how the interest rates between two different countries affects the futures price.
I have a question that needs clarification from you. Would you please refer to Table 3.5 on page 56 (Hull's textbook).
The current spot price of the coupon bond is $900 and the 1-yr forward price is $930. Since the forward price is too high, there is an arbitrage opportunity and the net profit is $17.60 per forward contract. Of course, the market will not permit the existence of such opportunity, if at all, for a long time. On page 57 (last para), the market would then offer the forward price on the coupon bond for $912.40/contract (no arbitrage opportunity).
My question is:
Say I have $900 to invest. Can I, as a regular investor, short a forward
contract at $912.40? If so, assume I also purchases one bond at the
current
price of $900. For easy calculation sake, let's also assume that the
coupons
are not re-invested. Can I then say that I will earn $92.40 for a $900
investment? This is how I got it: I paid $900 for the bond (my intial
cash outflow) and received $992.40 ($80 from the 2 coupons and $912.40 from
the
contract). Now, wouldn't my return be about 10.27% per year?
The reason why I am asking you this question is that there are many investors who do not want to gamble their money in stock market or mutual funds, especially retirees and risk-averse people like my dad. Most of them would rather invest their money in CDs that merely pay 4% or 5% interest. Also, if it is fair to say that the above forward contract is a safe investment strategy and that the prices of the forward contract and the bond are realistic, why wouldn't portfolio managers who handle money market funds simply do this rather than invest in all kinds of money market instruments that pay only about 5% (like the Fidelity Cash Reserve which yields only 5.02%)? Wouldn't it be better to offer risk-averse customers a guaranteed 10.27% return on their investments? Am I thinking logically? Or are there any kind of costs or risk that I haven't taken into account? Please help me clarify this matter. Thank you so much for your time and have a good weekend.
Given this explanation, what you are really asking is: why should I invest safely at 5% when a much higher yield curve would give much higher safe returns. Answer: because the yield curve is much lower now than it used to be, and you have no choice.
In the 80's I had certificates of deposit that paid me (at their peak) 21.5% per annum compounded quarterly. When the market crash of 87 hit I was 100% in CD's that were paying about 15% per annum, and so I lost nothing. Inflation was higher, and so were my safe returns. Those days are gone - at least for now.
Also, the cap is automatically exercised if the cap price is hit, but the long-call/short-call position does not have this attribute. This is good if you do not know when to take your money and exit a position (mentioned in the WSJ article).
Also, In the lon-call/short-call, there is always the threat of exercise of the short call, that is, half your position can be pulled out from under you by someone who is long. You do not have this hanging over you with the cap.
I just added this link to the F421 glossary page, and the first definition is ADR's. The link has some interesting and unusual finance terms defined, and is worth a look.
Theoretically, if F=Se^(rT) (for a T-bill futures say), then variance of F will exceed variance of S (because e^(rT) is bigger than 1). To prove that, you need to know that if X is a random variable, and w is a constant, then VARIANCE(wX)=(w^2)*VARIANCE(X).
Although this result is theoretically correct, it is supply and demand that determines futures prices, so strange things can happen, and it need not always hold. Also, the pricing formula looks different for different types of underlyings.
Finally, when we come to Black-Scholes option pricing, we will use spot prices to calculate options prices. However, it certainly does not follow that options prices and spot prices have the same level of volatility.
Further to that, here are some numbers. A European put with 1 day left in its life, a strike of 100, when the stock price is 10 is worth about 90 (because you can exercise it tomorrow for this payoff). A European put with 12 months of life left, a strike of 100, when the stock price is 10 is worth about 85 (Black-Scholes - assuming r=0.05). If you pay an extra 10 to acquire the stock in addition to the 12 month put, you are locking in a payoff of 100 if the stock price is less than 100 in one year (the most likely scenario). The present value of the 100 is 95 (equivalent to the 10 plus the 85). This is worth 5 dollars more than the 1-day put because it locks in a payoff of 100 one year from now (of PV=95), instead of 90 now from the 1-day put.
So five dollars of the 10 dollar stock price represents the discount of the long-lived option compared to the short, and the other five represents the PV in the differences of the payoffs you are locking in.
I have a question for you. If I were a broker concerned about only getting nice monthly returns, would it be a wise decision to add out-of-the-money covered calls to stocks I own?
Yes - it would be wise. And if I correctly understand you, you need not be a broker for this to be sensible; you can be you too.
However, you don't get this income for free. There is no "free lunch." By writing out-of-the-money calls you are selling off some of the upside potential in your portfolio. If you have a stock worth $100, and you write a call with a strike of $120, then if your stock rises above $120 before the calls expire, you do not participate in this upside. The stock will be called away from you at $120. I think of it as selling of uncertain future upside for certain current income.
You use the phrase "getting nice monthly returns." This is a good choice of words because it connotes a reduction in potential returns that is acceptable because of the reduction in risk that accompanies it. When you write the (covered) call, you are reducing the range of possible payoffs that you can receive. This is both a reduction in potential returns, and a reduction in risk (risk in the sense of deviations from what you expect, not risk in the sense of "bad" events).
One of the most popular strategies - to be discussed in class next week - is writing a covered call, and using the proceeeds to buy a put, where both options are out of the money. This does not provide income, but it substantially reduces risk.
P(S,T;X) >= max(o,X-S) (R12)
Can you explain this in words and also, does an opportunity for arbitrage exist if the above holds true. I'm having a little trouble putting the restrictions into words and understanding the basic concept(s). Thank you.
First off, this is a "no-arbitrage" relationship (as were most of the ones we looked at that had similar notation). This means the relationship does *not* allow arbitrage opportunitites - because any that existed have already been found and exploited. Thus, if the relationship holds, arbitrage opportunites do *not* exist.
Second, what does it mean: The symbol P represents the value of an American put option. Inside the brackets "(S,T;X)" means we are talking about the value of the put option when the stock price has value S, the option has T years to maturity, and a strike price of X. It is important to identify these items because some of them appear on the other side of the relationship also. Let me just write "P" instead of "P(S,T;X)" because it is a bit easier to see.
Next, what about the max(0,X-S)? Well, an American put, like a stock, is a "limited liability" instrument. This means once you buy it, no one can ask you to pay any more money for it (your liability is limited to the initial price of the asset). This was not true for futures contracts for example, where a long futures position can bleed you dry if the futures price falls (because you get margin calls).
For the put option, the absolute worst that can happen is that the option expires worthless. So the absolute worst that can happen is the the value of the put (denoted "P") is zero. This means the value of the put is at least zero: denoted "P>=0."
Also, the American put can be exercised at any time. This means that at
any time you can choose - if it is attractive - to exercise your option
and sell the underlying stock for the strike price X. You can do this
even if you do not own the stock. If you exercie tha put, you either
give up the stock (of value S) if you already own it, or your give up
money (of value S) to acquire the stock, then you exercise your right to
sell at X. This means you lose S, and get X (the sort of thing you would
typically do only if S
If we know both that the put option is worth at least zero, and that it
is worth at least X-S, then we can write
"P >= 0, and P >= X-S."
Most of option pricing theory was derived by mathematicians.
Mathematicians love writing things as concisely as possible - using the
minimum number of symbols possible. There is a math result that says if
you have "Z >= a" and "Z >= b" then you have "Z >= max(a,b)" where
"max(a,b)" means the maximum of a and b.
If you think that one over, you'll see that it actually makes sense. For
example, if I tell you that from my memory, the temperature at 5pm
yesterday was greater than or equal to 40F, and after consulting my
records that the temperature at 5pm yesterday was in fact greater than
or equal to 50F, then you know it was greater than or equal to the max or
the two: 50F.
In our put option example I tell you the put is worth at least zero, and
it is worth at least the exercise value: X-S. It follows that it is worth
at least the max of the two:
"P >= max(0, X-S)", as stated.
We deduced that it must cost money up front for two reasons: because a
longer maturity (American) call costs more, and a lower strike call is
also more expensive. I correctly drew the payoff diagram for the
position. Here are the profit diagrams for the
volatility spread (different diagrams depending upon the intial stock
price in February when you enter the position).
I was wondering why there is no arbitrage opportunity with copper
futures. I understand why it has a negative cost of carry. What I don't
understand is why a cash and carry strategy for an individual with no
interest in using the copper would not produce an arbitrage opportunity
the same way it would if gold or silver were trading at a forward
discount. Please let me know when you have a chance.
Reader: Note that at this time the futures price of copper was
lower than the spot price - a forward discount.
1) The forward price of copper is lower than the spot
price.
2) If you enter (i.e. go long) a cash-and-carry strategy, you borrow
money to buy the copper in the cash (i.e. spot) market, and you store it.
this means you end up owing the amount you borrowed plus interest plus
storage costs, but you will own the copper.
3) The amount owed in 2) is more than the spot price of copper
(because of
the storage and interest costs). If you have no other interest in copper,
then you do not get any "convenience yield."
4) If you seek an arbitrage profit, then you must offset your
cash-and-carry with an opposing position to lock in your profit. The only
position that seems to make sense here is a short futures (agreeing
to sell the copper you stored).
5) However, with a forward price below the spot, and the amount owed
above the spot, you end up getting less money from the forward than you
need to pay off the borrowing and storage costs.
forward price < spot < amount owed
So you have locked in an arbitrage LOSS, not an arbitrage profit.
What is you next move.........? This must generate a question from
you....
Let me know.
After thinking about this I would want to do a reverse cash and carry. I
would want to short copper in the spot market and go long copper in the
futures market. If this is correct then what is the catch because
arbitrage would not exist for this long a time period. Please let me know
when you have a chance.
The people who *could* benefit are those who hold physical
copper in inventory waiting to put it into production. They could benefit
because they could sell their copper in the spot market at the high price
and save (i.e. benefit) by not having to pay the storage costs. However,
they will not do this because they *need* physical copper available to go
into production. If they sell the copper and go long a futures on copper
they lose the "consumption benefit" or "convenience yield" that goes with
owning physical copper. So no one can/will move to take advantage of the
forward discount on copper.
Now what do you do....?
In fact nothing can be done. The copper sells at a forward discount
because the only people who can take advantage of this pricing do not
wish to do so because they derive a consumption benefit - or convenience
yield - from holding the physical commodity ready to put it into production.
Just for my information, I was wondering what the difference between a
warrant and a call? They seem a lot alike (perhaps the difference being
that the issuing company is more "in control" of a warrant).
Just wondering.
An IBM warrant and a standard IBM call are both call options on the
stock of
IBM. However, the warrant is issued (i.e written) by IBM corp, while the call
is written by an individual investor (like you or I). Since both are
calls, the naming is a little confusing - let me call them warrants and
calls.
IBM warrants are normally given only to employees (or sometimes to
stockholders) of IBM. They are a form of compensation. IBM calls are
held by anyone at all - including you or I if we wish.
Exercise of an IBM warrant leads to the company issuing more shares and
therefore dilutes existing equity stakes. Exercise of an IBM call has no
effect on number of shares outstanding.
Pricing of warrants is similar to pricing of calls, but you need to worry
about possible dilution of the equity stake if the number of warrants
outstanding is large.
I was a little brief. Do I need to clarify any point that I made? I would
expect this to raise other questions in your mind.
Postscript: In some countries the nomenclature is reversed - a
warrant is traded on an exchange, and an option is issued by a company.
Yes, correct! However, the delta is the sensitivity of derivative
value to
*small* changes in the underlying, so although you have *exactly* the right
idea, I would have used smaller price changes in the example.
And therefore, the gamma of the second derivative is bigger than the
first one?
Hmmm. No. Gamma is more difficult to understand. The gamma is a measure
of how the delta changes with changing stock price. Let me try some
examples:
Suppose we have a stock with price S, and two derivatives with values V1,
and V2.
EXAMPLE 1:
EXAMPLE 2:
B) At the new prices of (S,V1,V2)=(11,5.5,5.5) we can ask again, how
sensititve is each derivative to changes in stock price. Let us suppose
it is now the case that if S goes up by one more dollar (to $12), then
V1
goes up by 60 cents (to 6.10), but V2 goes up by 80 cents (to 6.30).
That
is, the sensitivity (i.e. delta) of each of the derivatives has
increased.
Gamma is a measure of the rate of change of delta as you move through
higher and higher sotck prices. In my example, the sensitivity (i.e.
delta) of the second derivative increased more rapidly than the
sensitivity of the first derivtive (going from EXAMPLE 2A to EXAMPLE 2B
its response moved from 50cents to 80cents for a dollar increase in
stock
price, while the first derivative's response moved from 50cents to 60
cents for a dollar increase in stock price). Thus the second derivative
has a higher gamma.
My other question is what exactly is the underlying asset in the put
futures option (same as "option on futures" where the underlying asset is
the futures contract???), and are we responsible for knowing how to value
it using the Black Scholes Formula? I know these are really long
questions and I hope you can understand what I am saying.
The underlying is the same for the put and the call, it is the futures
price and the futures contract. The put lets you sell (i.e short) a
futures, the call lets you buy (i.e go long) a futures.
Black's (1976) futures options formula -- just stick
the futures price in place of S, the futures volatility in place of the
stock volatility, and the interest rate in place of the dividend yield.
An aside: is their anything like the uptick rule in the futures market?
I understand in the stock market you can only take a short position after
an uptick. In futures you can enter any position that someone else is
willing to take an offsetting position to, right?
Hmmm. The uptick rule for stocks is to reduce volatility. There are
rules
in the futures markets also with the same purpose, but they are a little
different. For example, each contract has a "daily price limit." The future
price cannot change by more than this during the day. If the limit is
breached, the market is shut down until someone wants to trade within
the bounds, or until the next day's opening, or until a certain period of
time has elapsed, or until a certain number of minutes prior to the close
(depending upon which contract you are looking at). For example, the
daily price limit is 1.5 c/lb on live cattle futures, and 2 c/lb on lean
hogs. In the currency futures, there is typically no price limit in the
first 15 minutes of trade, and then there are poted price limits that can
change according to certain rules. These price limits are typically
either lifted completely, or dramatically expanded (eg x2) if thecontract
is in the last month of its life.
Bottom line. You cannot sell if the futures price has already "moved
limit," just as you cannot short stock on a downtick. Both rules restrict
behaviour after price moves and should restrict volatility.
Do short squeezes generally only occur at the expiration of a contract?
Hmmm. I do not know for sure. I think this would make most sense
because
its the time when you *must* get out of a contract if you are short and
do not want to make delivery.
Long Oct 50 Call
Short May 60 Call.
If S goes from $10 --> $11
and V1 goes from $5 --> $5.50 and V2 goes from $5 --> $5.60, then the
second
derivative has a higher delta -- EXACTLY as you described above.
A) Now suppose *instead* that when S goes from $10 --> $11
and V1 goes from $5 --> $5.50, V2 goes from $5 --> $5.50 also -- so the
derivatives have the *same* initial delta (sensitivity to change in
underlying).
