Introduction to Regulated Derivatives: Futures and Options Contracts

PART 3: REGULATED DERIVATIVES FUTURES & OPTIONS REGULATED DERIVATIVES: FUTURES AND OPTIONS CONTRACTS ON STOCKS, STOCK INDEXES, COMMODITIES, CURRENCIES

Introduction to regulated derivatives • Derivatives are quite complicated instruments. They are called that way because the behaviour of a derivative contract derives more or less closely (depending from case to case) from that of another asset, named underlying. • Trading in derivatives attracts many investors thanks to the high intrinsic leverage of these instruments, allowing to reach high gains with small money amounts. And since the leverage works in the same way in both directions, amplifying both profits and losses, these instruments demand for a deep knowledge, both in terms of risk/reward profiles and the ways to manage open positions. • Difficulties in the field of derivatives arise from seven sources (which we’ll see in details): • they have a maturity, at which they cease to exist and are replaced by new contracts; • at maturity they can involve cash settlements or physic settlements; • the purpose of trades held till maturity (hedging of risks of some sort) is totally different from that of short term trades (just speculation); • their intrinsic leverage demands for a strong risk management, based on a daily regulation of profits and losses, according to rules we’ll see; • every open contract is made of two active counterparties, having opposed expectations; • pricing of derivatives changes according to the nature of the underlying; • prices often follow dynamics not completely explained by market variables (especially in the field of options).

Buyers Vs Writers For any derivative contract active in the market there is always someone long (buyer) and someone short (writer). The buyer of a futures contract, for instance, expects that the price will increase, the writer expects it will decrease. Things are different in the field of options, as we’ll see, and yet for each open contract there are still two counterparties active at the same time, with different (not exactly opposed) expectations. Derivatives always need two active counterparties to exist. They can be traded exactly like stocks, on the long, mid, short or very short term. They can be purchased and sold also hundreds times a day, and transfered among traders just with electronic transactions. Short selling of derivatives does not require borrowing or lending. There are no special commissions and costs for short selling. Profits and losses of buyers and writers are symmetrical: the derivatives game is a zero-sum one.

Transfers of contracts among traders Derivatives can be trasnfered from a trader to another at any time, symply clicking on the mouse. A buyer and a writer who created a contract in the first place are not bound one to the other: in the moment one of them wants to get out of the contract he just need to transfer his side of the contract to someone else willing to take the same side. Buyers and writers are not bound each other, then. Let’s consider the case of investors A and B: A thinks the price of an asset is going to increase, so he buys a derivative contract written on that asset. B has opposed feelings, hence he decides to be the writer. Let’s use the symbol + for long positions and – for short positions: Imagine that after some time B changes his mind and wants to get out: he needs to buy to cover, so he needs someone else willing to sell. Be C this person: A +1 B -1 A +1 B -1+1=0 C -1

Transfers of contracts among traders The short side of the contract has been transfered from B to C. Now imagine that A wants to get out: he has to sell, so he needs someone willing to buy and takeover the long side of the contract. Be D this guy: What happens, then, after a while is this: The existing contract is still the same created in the beginning by A and B: it just changed hands. Should, in the end, C and D decide to get out at the same time, then they could meet in the market and close their opposed positions, becoming flat both at the same time. In that situation the contract would disappear. But if E and F showed up, buying the first, writing the second, a new contract would be created out of the blue. A +1-1=0 C -1 D +1 C -1 D +1 A flat B flat

Futures DEFINITION a futures contract is an agreement between two parties to buy or sell an asset (that can be financial or physic) at some date in the future, at some specific price. It’s a derivative contract, that is, its behaviour is linked to the one of another asset, named underlying. This is the classic definition of futures contract, but to be stringent it applies only in a specific context: when the two parties keep the contract opened till its maturity, the good is physic, and it can be delivered. This behaviour, indeed, is the answer to a specific need: the need for the underlying at some date in the future at some price decided in advance; this is a typical need for companies willing to hedge a risk of some sort (a concept we will see in full details forward). To the contrary, in most of the cirumstances a futures contract is just a way to speculate on the trend of an asset, on a short or very short term, with a huge leverage. Apart from some exceptions which we’ll see at the right moment, the futures price moves generally together with the spot price (the price of the underlying) but most of the time it is different: it can be higher, equal or lower, depending on the contingent situation and the peculiar features of the underlying asset (read forward for details).

Futures Futures are standardized contracts, quoted on regulated markets. This means that for any contract everything is set in advance: the asset to be traded, the quantity, the quality (in case oh physic goods), delivery dates, minimum price change (tick), maximum daily price ranges, and so on. A parte of the regulation is dedicated to the management of the high intrinsic leverage of these assets, as we’ll see in details. Futures are given of a maturity, at which they are closed, regulated and retired from negotiation. If the underlying is physic and can be delivered (like cereals, metals, stocks) there happens to be a physic delivery of the underlying; in many cases though there is just a cash settlement. The physic trading of goods can happen only between institutions and individuals (named commercials) registered in a specific registry. Private investors are not allowed to hold futures contracts till the maturity if there is a physic settlement: in case of need their positions are closed by force by their intermediaries.

An example: a stock index futures Futures contract written on the FTSE MIB index: • underlying: 1 FTSE Mib index • quotes: in index points • value of a point: 5€ • minimum price variation (tick): 5 points (25€) • contract size: price of the futures times the value of a point (for example, if the index quotes 20000 points, its value is 100000€) • maturities: every quarter of the first year, then every six months for two more years, then every year for up to five years • expiration day: third Friday of the expiration month • settlement price at maturity: opening price of the expiration day • settlement: cash • initial margin (see slide 10): usually between 10 and 15%, self-adaptive to market volatility The variation field of the initial margin, which, as we’ll see in next slide, is a measure of risk, is the answer to the need for a risk management policy istantaneously adaptable to changes in the risk of the underlying: the higher the risk, the higher the initial margin, automatically. What does all of this mean?

An example: a stock futures Futures contract written on the stock Eni: • underlying (lot): 500 Eni shares • quotes: in euros and eurocents • minimum price variation (tick) : 0.0001 euros • contract size: futures price times lot; for instance, being 15€ per share the futures price, the contract size is 7500 euros • maturities: two closest months, four closest quarters, then every six months for several years • settlement day: third Friday of expiration month • settlement price at maturity: opening price of expiration day • settlement type at maturity: cash or physic (both isted) • initial margin: still related to the risk of the underlying, generally much higher than that of the index The main difference between stock index futures and stock futures is the lot, equal to the number of units of underlying controlled with the contract. The lot changes from stock to stock. The full list is available on the internet site of Borsa Italiana, in the derivatives section  contract specifications.

Initial margin and leverage The initial margin is the money needed to open a position on the FTSE Mib futures, and as we’ll see it is not an expense, but just a cash deposit: a money amount deposited as a trade is opened, for the coverage of its risk. In the moment the trade is closed the initial margin is restituted. If the margin requested by the market is 10%, this means that to open a position valued 100k€ we just need 10k€. The point is that we use 10k but we are exposed for 100k, that is profits and losses are referred to 100k, not 10k. It takes, then, just a +10% do double the money, or a -10% to lose all the money. This is the main peculiar feature of futures, then: they have a huge intrinsic leverage. In this case the need for capital is just a tenth of the value of the position on the market (there are futures with leverages higher than 100…). The leverage is what makes futures very attractive for speculators. The initial margin is a safe deposit asked for the coverage of a risk. It is then integrated and managed according to some clear rules having the purpose to protect the market from insolvency risk. In order to fully understand this issue we have to introduce the role of the clearing house. But we need to understand the nature of the problem first.

Profit or loss Payoff +1000€ 18€ ABC price 17€ 19€ -1000€ Risk/reward profile (payoff) The payoff is a chart that shows the result – in terms of money – of an investment, depending on the price of the asset at some date. In the field of derivatives, the payoff is used to visualize the profit or loss of a trade at the maturity of contracts for several prices of the underlying. It’s a way to see what happens in case of some possible outcomes. On the horizontal axis we place the prices of the underlying on arbitrary intervals, on the vertical axis we place the economic result of the position. Let’s first consider the case of a stock: suppose To buy 1000 shares of the stock ABC at 18€ each. The payoff of this position is a straight line, with a 45° slope, growing from bottom left to top right: profits and losses are straight proportional to the movement (up or down) of the stock: In case of futures things are slightly different, since the futures’ price is not equal to the underlying’s price (spot price), for reasons that we’ll see.

Potential profits and losses In passing from stocks to futures, to understand the payoff we have to first understand a key difference in terms of potential profits and losses and real profits and losses. Here we need to focus on a paramount concept: on the stock market, till a trade is not closed, profits and losses are only nominal. This happens because the profit or loss is the difference between what we paid on purchase and what we get back selling. If we buy a stock at 10€ and the price rises to 11€, if we do not sell those shares the profit is not real; at the same time, if the price drops to 9€, the loss is not real, if we do not close the trade. This means that trades that generate nominal profits or losses do not have any impact on the capital of the investor while they are opened: only when they are closed they generate a real profit or loss. Moreover, in stock trading there is no insolvency risk: the value of a position is totally paid in the moment of purchasing, so the only individual exposed to a risk is who holds the stock in his hands. His counter party – who sold the stock to him – already got paid of the whole amount, so he doesn’t need to care about that stock anymore. In other words, if a stock holder loses money it is just his business: no one else is at risk in that case.

Buyers Vs writers From the point of view of the holder of the stock, the maximum risk is given by the loss of the entire capital invested, due to the bankruptcy of the company. Under normal conditions, the entire value of the stock is paid on purchase with money owned by the buyer. If the trader uses a leverage he has a debt with the bank, and that bank asked for collaterals as an insurance for the money lending. And in any case the entire value of the stock is paid, so there is not a counter party somewhere out there at risk to lose money because someone is not going to pay him. In futures trading, as we already know, things are different: who loses money in a trade loses that money in favour of a counterparty set on the other side of the contract, who gains the same money amount. Since the initial margin covers just a fraction of the value of the contract, the risk here is that the loss could be higher than the money deposited: what if I decided not to pay my dues in case of losses higher than the initial margin? Here insolvency risk could arise: one of the two parties in the contract could not be able (or willing) to pay his debts.

Insolvency risk A strong source of risk in derivatives trading is given by the leverage, that puts the market in front of the insolvency risk, originated by the possibility for losses to be higher than the moneydeposited in the first place. The problem is that if, for instance, the buyer faces a loss higher than the money deposited and has no money to pay for those losses, the writer could not be paid the money he gains. This issue involves the need of some sort of risk management policies. There is a paramount difference between buying a stock and buying a futures written on that stock: to buy 1000 shares of ABC at 18€ each we have to pay 18000€ cash immediately, and this provides a full coverage of the maximum potential risk of the trade (the stock goes bankrupt and its price falls to zero). Now let’s consider a futures written on ABC, assuming it’s lot is 1000 shares and the initial margin is 10%. The investment is valued 18000€, but to open it we just need 1800€. This means that the risk is covered only up to a 10%. This generates a risk for both parties: the buyer faces the risk to be not paid if the price rises more than a 10%; the writer faces the risk to be not paid if the price drops more than a10%. Generally speaking: both the buyer and the writer are exposed to insolvency risk.

Marking to market The management of insolvency risks requires the use of a specific methodology, called marking to market. Marking to market is applied to all regulated derivatives and is based on a daily regulation of profits and losses: as the market closes, every day, nominal profits are immediately credited, nominal losses have to be immediately paid. To be stringent, intermediaries today apply this mechanism in real time, but to make things easier let’s take it as an end of day procedure Daily profits and losses are managed by the clearing house, that acts as counter-party for any open trade: the clearing house pays who has to be paid and asks money from the banks whose clients are losing. Banks then take money out from the accounts of clients losing money. This way the clearing house always ensures the good ending of any transaction, always paying who has to be paid.

How do margins work The initial margin serves as intial safe deposit and has to cover a risk calculated on a statistical basis: it expresses the maximum expected loss from a day to another on the underlying asset. This calculation accounts for the impact that a significant increase or decrease in the value of the underlying would have on the trader’s account. To explain it in a few words, today the systems account for the worst of the worst case scenarios on a 1, or 2 or 3 days basis, according to the Normal model and the interval included in the mean of past returns plus and minus three standard deviations. Since both counter-parties face insolvency risk they are both required to deposit the initial margin, calculated as a percentage of the value of the contract deriving from some kind of “stress test” on the underlying, as decided by the clearing house. Doing this, risk is secured till the next day only. Then, depending on how the trend of the underlying evolves and so to always keep under strict control risk, positions are monitored every day. At the end of any day the closing price is compared to that of the previous day, and the initial margin is adjusted to a new value (see also next slide). Then profits are immediately credited, losses debited. These money transfers are named variation margins (see an example in slide 19).

Initial margin adjustments The initial margin is set as a fraction of the contract value; this implies that it changes according to changes in the value of the futures. Every time the closing price of a day is different from that of the previous, then, the amount of initial margin changes, hence the value already deposited has to be adjusted. According to what written in slide 8, initial margins can change both because of a change in the value of the futures and in the percentage automatically set by the IT systems according to the volatility of the market. The two effects could compensate or amplify each other. In other words, initial margin can change from day to day both due to the trend in the underlying and the risk of the underlying, measured by its volatility: the systems automatically adapt the percentage of the initial margin as a consequence of the latter and the monetary value of that percentage according to the former. Every night, then, initial margin is adjusted according to the evoultion of the market: if it reduces, some money amount is released on the account, and viceversa.

Summing all up To conclude, initial margin is just a safe deposit, adjusted every day depending on the evolution in price and risk. When the position is closed all the initial margin (including any adjustment) is entirely restituted. Variation margins are the real profit or loss of the trade: they are credits or debits collected or paid on a daily basis so to leave untouched the initial margin, so that it could always keep protecting the two counter-parties from insolvency risks.

Initial margin adjustm. Example 1 – A long trade . Note: for the sake of this example it has been considered a 10.25% fixed initial margin, day after day

Initial margin adjustm. Example 2 – a short trade . Note: for the sake of this example it has been considered a 10.25% fixed initial margin, day after day

SIZABLE LOSSES In some specific situations it can happen that positions in futures evolve very unfavorably, so that every day huge margin integrations are required. Think back, for instance, to September 11, 2001: in that day (and in the following ones as well) investors who had long positions on stock futures and stock index futures had to face sizable variation margins. The deposit of margins is required by the market regulation, hence it is mandatory. If, on a specific day, as the markets close, an investor has not enough money on his account to face variation margins the intermediary has to warn him to close his positions the next day on open; if he doesn’t, the intermediary has the power (by law) to close positions by force. Some intermediaries, indeed, do not wait for the market close to intervene: being trading continuously monitored, at 5.15 p.m. they check positions and if things are getting bad they warn clients to close positions before 5.30, otherwise the bank will do it in their behalf. It has been proven that the meticulous application of marking to maket prevents insolvency risk in almost a 100% of cases, and in any case uncovered losses can be very reduced. Unfortunately, not all the intermediaries apply correctly these “well oiled” risk management policies.

Average daily potential profits on futures To understand what it means to trade on futures we just need to do some basic calculations on market data. Let’s look at the italian market. The average daily variation field (distance between minimum and maximum) for the price of the futures FTSE Mib over the 2469 days between June the 1st 2007 and March the 31st 2017 is 430 points. If we had a trading system able to catch every day average profits equal even just to a 25% of the daily range, we could make 107.5 points of gain every day, that is 537.5 euros, trading only one contract every time. With which capital? Hard to tell exactly, but we can make a precautionary estimate: being always 15% the initial margin, the average daily margin over the period of time above mentioned would have been about 16100 euros. Using the mini-sized futures, valued 1 euro per point, the initial margin would be a fifth of that amount, that is about 3220 euros. Profits would be 107.5 euros per day, on average. Either way, the average daily return would be 3.33%...

Futures pricing: introduction Previously it has been stated that the futures price generally differs from the price of the underlying: it can be equal, higher or lower; generally speaking, the gap between the spot and the futures price depends on the physic characteristics of the underlying good and the equilibrium between supply and demand. The futures price can be derived with some math reasoning under a clear purpose: to eliminate any arbitrage opportunity. An arbitrage is a trade that restitues a sure profit without any risk; such situations usually happen when the same product is quoted on two different markets with two different prices: if an investor can have access to both markets, and on those markets there are no limits to short selling, then he can buy the good where it’s cheaper and short it where it is more expensive. Sooner or later the two prices will align; in the moment the two prices go back in line, the arbitrager can close both the positions, gaining for sure without any risk. A preliminary note: the futures price is generally different from the spot price during the lifetime of the contract, but it is exactly equal to the spot price in the moment of expiration (why, will be clear going on). Keep this in mind in order to understand everything that follows.

Futures pricing: stock futures Suppose to have growing expectations on a listed stock, on which are written futures; given a specific capital amount, if we buy the stock we need some money amount, and if we buy the futures we can use a capital amount surely lower, thanks to the leverage. This makes the futures advantageous in any case: we can use it to save liquidity to be used elsewhere. Being, just to say, 10% the initial margin on the futures (and assuming there are no variation margins, to make it easier), the remainder 90% can be invested in risk free assets (a government bond, for example), rising the global result on the whole capital whatever happens. Be, in fact, one year the timeline of the investment, 5% the interest rate on 1-year maturity government bonds, 10% the initial margin requested to buy a futures written on the stock we want to buy, 10€ the price of that stock, and 1000 shares the lot of the futures written on it. Finally, be 11€ the price of the stock at the end of the year. Buying 1000 shares of the stock means to invest 10000 euros, and at the end of the year it restitutes 11000 euros, that is, a 10% profit.

Futures pricing: stock futures Assuming the futures price is equal to the spot price (e.g. the price of the underlying), buying a futures means to deposit 1000 euros in terms of initial margin, and to get a 100% profit at the end of the year on that initial margin; this because in futures trading profits and losses are on the real value of the position, not on the money used. And since to get that result we used only a 10% of the available money we can use the remainder 90% to buy a 1-year maturity government bond, that will restitute a 5% gain at the end of the period, with no risk. Summing up, on the 10% of the capital we gain a 100%, and on the remainder 90% we gain a 5%. The global return is the weighted average of the two returns: total return = 100% (return) * 10% (ratio of capital) + 5% (return) * 90% (ratio) = 14.5% It is clear, then, that thanks to the leverage it is always possible to enhance profits, no matter the direction of the price of the underlying(see next slide).

Futures pricing: stock futures Suppose, in fact, the stok loses a 10% in the current year: in this case the profit on the futures is a 100% loss, but the gain on the BOT is still achieved. The global profit is again the weighted average of the two results: total profit = -100% * 0.1 + 5% * 0.9 = -5.5% It should be clear then that the leverage allows investors to gain more (or to lose less) no matter the direction of the price: it amplifies profits if things go well, it reduces losses if things go bad. The consequence is that stock futures need to have a cost of some sort, linked to the interest rate and the residual life of the contract: that cost has to erase the interest rate achievable on the money we do not use. In other words, the price of a stock futures (for different underlying assets the matter gets a bit more complicated) has to be equal to the spot price only in the moment the futures expires; in any other moment it has to be higher than the spot price. This applies to stock futures (just in part for stock index futures as well). In case of other underlying assets, as we’ll see, other variables come into play.

Stock futures price: basic hypothesis • The futures pricing procedure requires the assumption of some specific initial conditions: • there are no limitations to short selling and to the use of the money collected from it • money can be lended and borrowed at the same interest rate • it is not possible to make arbitrages • there are no commissions on trades • Under those conditions (whether they are reasonable or not), the futures price can be achieved after the analysis of two limit situations: that the futures price was way under the spot price or way over it. • We shouldn’t even take into consideration the first, according to what seen in previous slides, but it’s time to give mathematical proof that in that case arbitrages would come into play.

First scenario: futures < spot Suppose a stock quotes 10€ and the price of the futures with a one year maturity written on it is lower than the spot price (we know it couldn’t but we still have to prove it); let’s say 9€. In that case it is possible to make an arbitrage: we can short sell the underlying, collecting 10€ a share; a small part of that amount can be used to buy the futures at 9€, and the remainder liquidity can be lended at the risk free interest rate (buying a BOT, for instance). The final result is a profit on money we don’t even have! Suppose, indeed, for instance, that at maturity the stock price is 7€; in that case the short position on the stock restitutes a profit of 3€ a share: sold at 10€, bought to cover at 7€, 3€ are in our pocket; the long position on the futures restitutes a loss of 2€; bought at 9€, sold at maturity at 7€, 2€ are gone: summing up the two positions we have a net profit of 1€. If at maturity the stock price is 13€, just to say, the short position on the stock restitutes a loss of 3€, but the long position on the futures restitutes a profit of 4€; a net gain of 1€ again, then. In both scenarios, to the 1€ net profit we can add the profit coming out of the money lending. No matter the direction of the market, then, we gain. And moreover we gain on someone else’s money!

Second scenario: futures >> spot Now suppose the stock quotes 10€ and the price of the futures expiring in one year is much higher than the spot price, let’s say 11€. An arbitrage can be done in this situation too: we buy the underlying at 10€, sell the futures at 11€ and get a 1€ profit for sure after one year. At the end of the year, indeed, the futures expires and its price coincides with the underlying, and it is easy to prove that for whatever price of the underlying we have a 1€ profit, that is, a 10% profit (in this case there is no initial margin, since the position is risk free, hence the capital invested is 10€ per share) without any risk, since losses on one asset are always compensated by the profits on the other, in both directions. Be, for example, 8€ the price of the underlying at the maturity of the futures: we lose 2€ on the underlying, we gain 3€ on the futures, so the net result is +1€; if the price at maturity is 16, just to say, we gain 6€ on the underlying, we lose 5€ on the futures, so the net result is still +1€. Conceptually speaking, then, the futures price has to be higher than the spot price, so to erase the advantage of the leverage (in terms of extra gains, like in slides 25-26), but not too much, otherwise it generates arbitrages’ opportunities. The solution is quite simple: the futures price has to capitalize the advantage of the leverage, in order to erase it: you pay the advantage in the futures price.

The fair valure of a stock futures It is quite easy to prove, then, that the correct futures price is equal to the capitalization of the spot price at the risk free intrerest rate on a number of days equal to the residual life of the contract; in maths: Fut = S * exp(r * T) where S is the spot price, exp is the exponential function, r is the yearly gross risk free interest rate, T is the residual life on an annual basis, that is in days to maturity (including the present day) divided by 365 Note that in the formula there is not any component related to the actions of supply and demand: it is just a mathematical formula, related to the risk free interest rate and the maturity of the contract. There are no expectations in the futures price: it is not the expected price at maturity. The futures price is just the price of the advantage of not holding the underlying today. A question arises, then: is this formula valid for any kind of futures?

Validity of the formula The formula on previous slide is the more accurate one for the calculation of stock futures’ prices If, on one hand, it doesn’t take care of the dynamics in supply and demand, on the other hand the price of a stock futures cannot be different from the one that comes out from the formula, otherwise it would leave room for arbitrages. Once again: there is not any kind of expectation in the stock futures price; if it were, arbitragers would immediately take advantage from it. This because long and short positions can be opened both on stocks and futures, so arbitragers can combine them in any way so to take advantage of disalignments in prices. All of this under the hypothesis seen in slide 27, of course (lthe imitation to the use of money collected from short selling of stock, indeed, is sufficient to nullify all the reasoning seen in previus slides). More or less the formula on previous slide can be used to price stock index futures too, but here arbitrages are not so easy to take, since to buy or sell an index in order to make an arbitrage is not as simple as to buy or sell a single stock: indexes are not negotiable, so it often happens that stock index futures do not respect the formula (arbitragers find it much harder to take advantage of prices’ disalignments).

Example The stock abc quotes 10€ a share; calculate the futures price with a 130 days maturity, given a 1% risk free interest rate. Fut = 10 * exp(1% * 130 / 365) = 10.0357 To prove that this price is correct is quite easy. Suppose for example to try and make an arbitrage, buying the stock at 10 and selling the futures at the price just calculated. After 130 days the profit is 0.0357, exactly equal to the interest that can be collected depositing the value of the position on the stock on an account remunerated at a 1% annual interest rate continuously compounded. Looking at the formula we can state that: 1. the futures price is a crescent function of the risk free interest rate; 2. the futures price is a crescent function of the time to maturity. The higher the interest rate, the longer the maturity, the higher the stock futures’ price.

Futures Vs spot Comparing the charts of the spot price of a stock and the futures price, we can see that the futures price moves always atop of the spot price, but the gap between them reduces day after day; at maturity they coincide:

Profit / loss Profit / loss underlying Long position Short position Futures Spot price Spot price Futures underlying Futures payoff: cash settlement We can now finally give an explanation to a sentence previously stated: the payoff of a futures at maturity and the payoff of an equivalent position on the underlying are slightly different. For the buyer, indeed, the extra-cost due to risk free interest rates and time is eroded day after day, and is zero at maturity. Referring to data in slide 32, then, if at maturity the spot price is still 10€ the buyer lost 0.0357€ per share. If the settlement price is 12€ he gains 2€ minus 0.0357€ per share. If the settlement price is 8€ he loses 2€ plus 0.0357€ per share. For the short seller it’s the opposite: the extra-cost paid by the buyer is a little profit, no matter the final result of the trade.

Futures payoff: physic delivery What seen on previous slide is what happens buying or selling a futures at some time and keeping it till its maturity, when the underlying cannot be delivered, so there is just a cash settlement. When the underlying is a stock, at the maturity there is a physic delivery of the underlying: the holder of the futures has the obligation to buy the stock, the writer of the futures has the obligation to sell the stock to the buyer. The delivery happens at the settlement price, the opening of the third Friday of the expiration month (on the italian futures market). So the payoff at maturity is conceptually different, because the trade changes nature: from a position in a futures the two parties pass into a position on shares, with different implications on both sides, depending also on what happened between the moment the futures was opened and its maturity. We then have to distinguish between three possible outcomes: price up, price stable, price down. In the first case the holder of the futures was right about the future trend of the spot price. He got paid the difference between the spot price at maturity and the futures price in time zero, as a sum of daily variation margins; then, the day of the maturity of the futures he buys the stock for the settlement price. This has a clear consequence: no more leverage, because in the moment of purchase of the stock the entire value has to be paid. Since the value of the position is the value of the settlement price, all the gain achieved on the futures plus the money needed to buy the stock in time zero is invested in the new position, that after that moment can move in any direction.

Futures payoff: physic delivery The conclusion is that speculators do never hold positions in futures till the maturity, because it’s a contraddiction in terms to use futures for the leverage and keep them till the maturity to lose the leverage! On the side of the writer of the futures, he paid the distance between the futures price in time zero and the spot price at maturity to the holder, and he has the obligation to sell shares to the holder at the settlement price; if he does not possess shares he has the obligation to sell them anyway, and it means he has to open a short selling position. All of this happens also if the price of the stock at maturity is equal to the price in time zero or below that level: in the first case the buyer buys the stock for the same price as in time zero and lost the gap between the futures price in time zero and the spot price at maturity. The seller sells the stock for the same price and collected the money. In the second case the buyer buys for a reduced price but lost money day after day in terms of variation margins; the seller has to sell shares at a reduced price but he collected the difference between the final and the initial price in cash. In any case, the bottom line is that shares are bought and sold practically for the exact initial price! That’s the reason why no matter where the underlying goes a futures intended to be held till maturity allows both parties to set in advance the price for a trade delayed in time (as from the definition of futures contract).

Futures strategies: time spreads In the world of futures there are not many strategies that can be made: basically there is nothing more than a long or a short trade. The only combination that can be done is the so called time spread, made of a long futures on a maturity and a short futures on a different maturity, with the same underlying. The goal is to take advantage from a change in interest rates between the day the trades are opened and the first maturity. Example: short FTSE Mib futures Dec14 – long FTSE Mib futures Mar15. The idea is that if interest rates rise before the end of 2014, the futures expiring on December 2014 will appreciate less that the one expiring on March 2015, and this would provide a gain no matter the direction of the price of the underlying

Dividends and stock futures price As from contract specifications the buyer of a futures is not entitled to collect dividends paid during the life of the contract (if the dividend is paid after the maturity it is a nonsense to consider its potential effects on the futures). Now, since the dividend causes a correspondent drop in the price of a stock in the moment it is paid, its value has to be somehow cut from the futures price, otherwise this would leave room for arbitrages: knowing that from a day to the next one the price will drop for sure due to the payment of the dividend, some investors could short sell the futures on closing of the day before the payment and take a sure profit the next day on opening. Here how it works in words: the value of the dividend the day of the payment has to be discounted to the present day, in order to get to its present value; this value has to be subtracted from the spot price before calculating the futures price.

Dividends and stock futures price Example Be 3.75% the risk free interest rate; consider a stock that quotes 10€ and pays a 0.5€ dividend 20 days from today. Calculate the fair value of the futures that expires 30 days from now. The present value (PV) of the dividend that will be paid in time T (DT) – that is, the money amount that invested today at the risk free interest rate on the next 20 days would restitute the amount of the dividend that will be paid in T – is equal to: D = PV(DT) = 0.5 * exp (-3.75% * 20/365) = 0.498974 The futures price is then: Fut = (10 – 0.498974) * exp (3.75% * 30 / 365) = 9.53 Generally speaking, then, the formula for the pricing of stock futures in case of dividends is the following: Fut = (S – D) * exp(r * T) Where D is the present value of the dividend and all the other parameters are already known.

Futures Vs spot It’s interesting now to see what happens to the gap between spot and futures price when there are dividends paid during the lifetime of the futures:

Dividends and stock index futures price • A complication arises when we have to calculate the fair value of a stock index futures when there are dividends going to be paid between the current date and the maturity of the futures. • Here the problem is dual: on one hand it happens that the stocks included in the index pay dividends at multiple dates between the present day and the maturity of the futures. On the other hand we have to cumulate all the dividends according to their dates of payment. • The procedure is quite complicated, but basically we need to follow these steps: • take the composition file of the index (to know the weights of the stocks in the index) • take the dividends for each stock and calculate their incidence on the price of the stock • weigh that incidence on the weight of the stock on the whole index • sum up all the incidences for each payment date • calculate the value of the cumulated incidence on each date on the value of the index day after day • In this way, day after day, ww have the total value of the dividends paid at some date, to discount to the pressent day and then subtract from the spot price. • If there are more dividends paid during the lifetime of the futures we have to account for them all, discounting all of them at their respective time to payment.

Commodity futures In case of physic goods, like cereals, metals, or oil, the stock exchanges that issue futures written on those goods have warehouses for the storing of goods that are traded on the markets as underlying assets for futures contracts. Storing of those goods causes costs that have to be somehow included in the futures price (transportation is an additional cost that the buyer of the goods have to pay separately). As a matter of fact, storing costs act like negative dividends, hence their present value has to be added to the spot price before capitalizing it to the maturity at the risk free interest rate. Being X the present value of storing costs, then, the futures price is the following: Fut = (S + X) * exp(r * T) EXAMPLE Suppose the price of an oil barrel is 90$ and that storing costs are 5$ per barrel per year. Calculate the price of the oil futures expiring 3 months from now, given a 2% annual interest rate. The storing cost for a quarter of a year is 1.25$, and its present value is the following: The futures price is then: In order to discuss the validity of this formula we need to introduce some more issues.

Contango and backwardation What is not considered in the formula on previous slide is the dynamics acting on demand and supply, which could disalign prices among different maturities, depending on several events. Indeed, the commodity futures price depends also on expectations in terms of production and consumption. Sometimes, prices are driven by climatic events that occur in specific regions of the world (think of hurricanes and their effects on agricultural production in the eastern american states) or by other factors that can significantly impact on the balance between supply and demand. All of this often leads to commodity futures prices not following math formulas. In the end it is the market who decides the right price, according to facts and expectations. An important issue here is that many events impact on prices just for limited time periods: it often happens that some factors affect shorter maturities only, leaving untouched the longer ones (see forward for some examples). Under normal circumstances – when supply and demand are not affected by peculiar situations – longer maturities quote higher prices due to the capitalization of the same spot price to a longer time and with higher storing costs. This “natural” situation is called contango. The longer the maturity, the higher the contango in respect to shorter maturities.

Contango and backwardation The equilibrium between supply and demand can disalign in two opposite directions, with opposite effects. And here expectations come into play. When there is an excess in the immediate supply in respect with the current demand, while the expected future supply and demand are in equilibrium, the contango amplifies. At the end of an unexpected mild winter, for example, oil producers might have a lot of oil barrels ready to be sold; demand decreases because spring is coming, so current prices decrease in respect with the prices that are expected to show up in late summer, waiting for the new cold season. Short term maturities’ futures prices are much lower than longer term maturities’ futures prices. Sometimes the opposite situation, called backwardation, can show up: shorter maturities quote prices higher than longer maturities. Such situations show up, for example, when in the middle of an unexpected very cold winter the (poor) oil reserves accumulated in fall on the basis of the weather forecasts reduce much faster than expected and an immediate need for more oil shows up; buyers need the oil immediately, so they are willing to pay more just to be sure to have it: short term maturities have much higher prices than longer ones.

Currency futures In case of exchange rates between currencies there are two interest rates to take care of in the calculation of the futures price: the interest rates of the two countries involved. The dynamics in pricing here have to neutralize the opportunity to get higher risk free returns just moving money from a country to another, making the two investments absolutely equal. In maths, the need above mentioned is accomplished capitalizing the spot exchange rate to the difference between the interest rates of the two countries, in a specific way. The formula is the following: Fut = S * exp [(Ri – Rf)*T] Where i and f are for internal and foreign. The internal rate of interest is that of the currency at the denominator of the exchange rate, the foreign one is that of the currency on the numerator. Example: be 1.0675 the spot exchange rate EUR Vs USD (euros on the numerator, US dollars on the denominator), 0.5% the american interest rate, 0.1% the european one, one year the maturity. The futures price is then: Fut = 1.0675 * exp [(0.5% - 0.1%) * 1] = 1.07178

Currency futures Why is – and it is meant to be – the futures price higher than the spot? The answer is that if I am a european citizen, I have money in an account in any european country and I buy a european bond, I get a 0.1% rate of return (risk free interest). Looking at the american market and seeing it returns 0.5% I could change my euros in US dollars and buy an american bond, getting a higher risk free return. But I am obviously exposed to an exchange rate risk, which I have to protect myself against. If I didn’t, the extra-return on the US bond could be erased partially or totally by a loss in terms of exchange rates between currencies. To hedge that risk I can look at the futures market, so to decide right now the exhange rate for the conversion back from US dollars to euros (see next slide for numbers). Doing this, if the future exchange rate is equal to the current one I can get a higher return with no risks of any sort. If, on the other hand, the future exchange rate is higher than the current, then the advantage of the extra-return of the american bond is reduced by the cost of the hedging. Finally, If this cost is exactly equal to the gap between the two interest rates the advantage is fully neutralized, making the two investments absolutely replaceable.

Currency futures: an example Let’s see it in numbers. Suppose to have 100k Euros in time 0 and to invest them in the american bond. We change euros and we get 106750 US dollars, that we invest for a 0.5% yearly interest rate. In time T (after 1 year) we get: Final value US bond = 106750 * 1.005 = 107283.75$ Changing back this money amount into euros for the exchange rate we had in time 0 it would be: Final value US bond in euros = 107283.75 / 1.0675 = 100500€ In this case I would keep all the benefit of the extra-return. If, instead, the exchange rate in time T is 1.07178, then we have: Final value US bond in euros = 107283.75 / 1.07178 = 100100€ Given the higher value of the exchange rate, the real return on the american bond is exactly the same of the european bond: the futures price neutralized the convenience of the american bond, cutting away any arbitrage opportunity.

Corporate hedging: what derivatives are for • To do hedging means to apply strategies in order to erase or at least minimize a series of risks companies may be exposed to in their activity. • Risks can be of different types: • price: supplies to produce goods, energy to move plants, oil to make airplanes fly, cereals to produce pasta, oranges to produce orange juice are only some examples • interest rates on loans and mortgages • exchange rates: if a company exports goods or imports supplies it can happen it is exposed to exchange rates’ risks • unpredictable events (think for example at the effects of bad weather conditions on agriculture!)

Conditions for the need of hedging • Risks need to be hedged when we have at least one of the following conditions: • volatility: if prices are stable, then we do not need to make any kind of hedging; viceversa, if the volatility of the risk is high, then hedging is necessary • exposition: if the company can charge the volatility of costs on the price of the goods it sells, then there is no need of hedging; utilities companies, for example, can increase prices of their services if their costs increase; but if the company cannot charge higher costs on market prices, then hedging is necessary • incidence of costs on the final price: if the price of a good needed for production incides only a few on the price of the output, then hedging is not necessary; and viceversa if the incidence is high

The hedging process The first step consists in the costs analysis: we first need to analyse each single entry in the costs’ chain Then we have to evaluate the exposition And finally the volatility Once we have identified the components that need to be neutralized the final step consists in searching assets to build the hedging strategy

Introduction to Regulated Derivatives: Futures and Options Contracts