Futures trading 101: Pricing of futures and arbitrage (Excel) transcript

hello everyone and welcome again to nettle the best platform around for distance learning in business finance economics and much much more please don't forget to subscribe to our channel and click that bell notification button below so that you never miss fresh videos and tutorials you might be interested in many thanks to our current patreon supporters for making this video possible and would also greatly appreciate if you consider supporting us as well so please check the link in description for more details my name is seva and today we are going to investigate arbitrage on futures markets and how you can identify potential arbitration opportunities between sport and futures prices and make yourself some riskless profits and we will be heavily building up on the two cases of long and short futures trading that we have investigated in previous videos so if you are mainly interested in margining with futures trading and how to calculate payoffs and all of these sort of things please check those videos out first but without further ado let's start investigating the arbitraging model on futures markets our first case involves a hedger that is a manufacturing company that is heavily exposed to oil oil price being a key component in their costs and they decided to reduce their exposure to oil or to eliminate it perhaps even in the best case scenario by entering a long futures contract on the 30th of september 2020 maturing at the end of november 2020 and they have eliminated their exposure for 10 000 barrels of oil meaning that as the contract size is 10 000 barrels they bought 10 futures contracts and it's all fair and square and obviously we could see how the futures price evolves throughout the time and calculate contract payoffs using the simple formula relating the current futures price to the start futures price and multiplying it by the total size of our position and here obviously it is important to note that as you are long on futures you gain if the price increases beyond what you have agreed upon at the start and you lose when the price falls down so here for example we can see that uh on the 1st of october 2020 we are notionally losing 13 000 because of foregone upside as we have fixed our oil price at which we would effectively buy it at the end of the period at 42.75 per barrel and we could have agreed for a better deal buying it for less and this is how the contract payoff can be calculated straight away and we can see that our ultimate contract payoff is actually positive and it's forty eight thousand four hundred dollars as the futures price at the end and the price that we would have paid if we were not entering any futures contracts if we were not using derivatives would have been much higher the price that we would have paid so it means that overall we have guaranteed ourselves some decent upside given how prices evolved throughout the lifetime of the futures contract but can we do any better could we have perhaps altered our hedge somehow in the middle of the period arbitraged on the market and guaranteed us a better payoff well to investigate that we have to first learn what this equation means and this equation is a key component of the identification of whether an arbitraging opportunity exists and on the left hand side we have got the spot price plus the present value of net holding costs for the underlying asset here it would be our commodity brand crude isn't it and on the right hand side we have got the present value of the futures contract so the value of the money we would have paid for the futures or would have received if we shorted the futures accounted for time value of money considerations and uh this equation uh straightaway tells you why it's not as easy as just comparing the spot price to the futures price today and um seeing that if they're different there is an arbitrage opportunity because obviously as we have already spotted during our previous videos there is a notable gap between sport prices and futures prices that starts quite wide before uh the futures mature and slowly narrows down as the maturity date approaches and the reason for this gap existing is simply because uh holding spot until the very end could be quite costly as first of all you need to pay up your cash straight away on the spot market and not at the end of the period when uh the commodity is to be delivered to you as per the futures contract but perhaps most importantly if you own this commodity as you bought it from the sport market previously you have to account for storage costs insurance costs and so on and so forth or perhaps even transportation costs that type of thing so this consideration is what accounts for this gap first and uh as time goes on and as the maturity date approaches the impact of holding costs and the impact of the time value of money considerations decreased steadily with time and this is why the futures price and the sport prices converge almost perfectly at the end of the period when the maturity date is upon us but how to express it mathematically and most importantly which assumptions are we using to explain the arbitration opportunities using this equation well first of all we need to make sure that there is liquid trading in both the sport market and the futures markets so we can trade both without any concerns second we need to make sure that short selling is enabled and not restricted or banned outright for either the sport or the futures and this is mainly revolving around some legal or regulatory frameworks that might be in place for example if you look at stock markets in the uk quite notably short selling of stocks was banned from the heyday of the 2007-2009 financial crisis and this ban has been lifted only in so you need to make sure that you actually can short both futures and the sport for this arbitraging identity holding true also some futures contracts also for some futures contracts the underlying might not be tradable at all for example weather's futures has been quite prominent throughout history for hedging the risk of agricultural companies and you can buy and sell weather futures but you cannot really buy or sell weather on the spot that's not how it works so for these types of futures arbitrage would also be impossible what we also assume is that holding costs are clearly measurable and we can know them in advance so they are not uncertain that is and we also require the assumption of perfect capital markets perfect capital markets is quite an unrealistic and stringent but very common assumption that is made in a wide range of finance models this one included to simplify the derivation and calculations the logic of the perfect capital markets assumption requires anyone any agent to be able to either borrow or land money indefinitely at the risk-free rate so the interest rate for everyone is equal and is equal to the risk for rate and you can borrow or lend money indefinitely so there are no such liquidity or default risk concerns that would prevent that uh the easiest way to explain how unrealistic this assumption might be is that if the perfect capital markets assumption holds true you could go to your bank and ask them to lend you 1 billion dollars at 0.89 per annum and the bank would agree which obviously is not the case in the real world but again even though this assumption seems really unrealistic it doesn't make much difference turns out and that is why it's quite heavily used in all areas of finance and finally we need to account for the fact that the impact of transaction costs so brokerage fees and bid ask spreads is negligible so this equation would hold true without any fraction so if we identify a mismatch between the two sides of the equation arbitrage can be executed while in reality if you account for brokerage fees or bid ask spreads there is some margin of error that this equation can allow before arbitraging opportunities will exist this out of the way the idea the main concept that allows you to generate such an equation is the idea of a replication strategy well what is the outcome the ultimate goal of longing a futures contract well it is to let be left with a certain amount of barrels of oil in that case at the end of the period end of october 2020. how you might achieve the same ultimate outcome without using the futures markets well it's quite easy isn't it you can't just buy oil at the spot straight away and just hold it until the maturity date and then pretend that the maturity date is actually when you have received the oil so you buy the oil on the spot straight away forget about it for some time and then at the end of the period you just take it out and imagine that you've just got it from your futures contour counterparty so both of these approaches would yield equivalent results at the end and that means that this sport strategy by an oil at the spot and holding it until the maturity date is the replication strategy and that's what allows us to assume that the payoffs of both strategies should be equivalent and if they're not and provided that all of the assumptions i was discussing before hold true we can use this equation to detect arbitrary opportunities so without further ado let's do that first of all we have to account for time value of money considerations and here we can calculate the discount factor by using the simple logic of continuous compound interest rates we first of all check how much time would have passed from the start date until the date we're interested in so this difference shows you how many days uh are passing between 30th of september and the present date then we divided by to convert it to annual frequency and then we just convert the annual risk free rate of 0.89 percent which is uh quite a representative risk rate for present times into this frequency and then we divide one by this one plus discount rate to get our discount factor quite simple and easy and we can see that at the start our discount factor is one because no time has passed yet but later on our discount factor slowly decreases accommodating the time value of money considerations for example at the very end uh at 30th of november 2020 our discount factor is 0.9985 and obviously the time value of money considerations would be more prevalent if the risk rate was higher right now the risk rate is quite low so this concern is not that important especially for short-term futures trading but if we imagine that it was for example 20 then the discount factor even over two months would be quite noticeable and material um yielding 0.97 of a discount factor 30th of november 2020 meaning that one dollar uh 30th of september 2020 would have been equivalent to only 97 cents at the end of november but returning back to reality turning our risk rate into 0.89 again we can still see how money in the future is less valuable than money in the present with 0.99 85 30th of november 2020 being equivalent to 1 30th of september 2020. then we need to account for holding costs somehow and here we can make an assumption that we already know what the holding costs are for example rent for the storage of oil and we assume that first of all the holding costs the storage fees for example are payable upfront and they are 0.05 dollars per day per barrel so 5 cents per day per barrel of storage and then we can calculate the holding cost if we have bought the spot at a particular day by using this simple logic first of all we figure out how many calendar days we would have needed to store oil for from the date when we buy oil 30th of september 2020 here until the very end date until the maturity date of 30th of november 2020. then we need to multiply by how many barrels do we need to store 10 000 in that case and multiplied by the fee per day per barrel 5 cents per day per barrel in this case so 0.05 dollars and here we can see that if we want to store our oil for two months we would have needed to pay more than thirty thousand dollars up front for storage and this slowly decreases as we need less and less time to store it as we go further and further into the lifetime of our futures contract then we can calculate the left-hand side of our equation of our futures valuation model by accounting for both the spot price and the net present value of holding costs per barrel so here we add up this sport price in dollars per barrel and our holding cost per barrel so holding cost over the number of barrels and multiplied by the adequate discount factor that is prevalent at the current time and this would be the left-hand side of the equation now we can consider the right-hand side of the equation and here we just need to adjust the futures price by the discount factor at the very end and this would not change as for futures trading you are always exchanging cash flows at the maturity date because for sport prices you need to account for time value of money considerations as of the date of transaction but for futures trading you always exchange cash flows at the very end so the discount factor is always this and now we can calculate the difference between the two figuring out whether the spot is are overvalued or the futures is overvalued and uh given the fact that here our difference is positive meaning that the spot is overvalued in relation to the futures you could have arbitraged if you were just and playing arbitrage or approaching the futures markets and the support markets from this angle you could have short sold the sport and longed the futures and locked in uh quite decent risk-free profit this corresponding to the net present value of the risk-free profit per barrel of oil but if we recall that our case involves a hedging counterparty initially and they have longed the futures contract already we need to figure out at which point in time if any we can beneficially do the transaction the other way around if we can do the arbitrage based on shorting the futures and long in the sport given the fact that this accounts for our business needs and this accounts for the hedges we have already got in place we have already got a long hedge in place so we need to figure out whether we can profitably unwind this hedge by just canceling it out with a short futures contract and then just log in the oil we need at the spot and holding it until the date we need it the maturity date 30th of november 2020. so let's see if this difference becomes negative at some point negative difference being the identification of an opportunity to profitably short sell the futures and along the spot so here we can see that the first day when this difference becomes negative is 9th of november 2020. here we can see that the spot is undervalued with relation to the futures price so at that stage we can unwind our current hedge the hedge beam the long features in oil so we just short sell the futures and oil at the current prevalent market price and immediately we also buy oil at the sport and hold it until the very end so let's account for this and calculate all of the cash flows that would be associated with this arbitraging strategy so we need to long the spot so it means that we need to buy 10 000 barrels of oil at the market price at the sport price on the 9th of november 2020 meaning that we need to pay up and this would be immediately on the 9th of november isn't it uh 412 300 we also need to pay the holding costs up front referring to the holding costs calculated as of 9th of november but here the problem is that we need to get this cash somewhere well because this is the concern that we do not have in the futures contracts in the futures trading we would have needed to pay uh for the oil at the very end of the period so here we need to provide for both the spot price so buy novel in the spot and the storage costs uh immediately uh on the 9th of november 2020. so the night on the 9th of november 2020 we need to take out a loan that corresponds to 422 800 and we need to pay interest for this loan taking into account the loan principal and the maturity of the loan and here the loan would be quite short term it would just be 21 calendar days isn't it because we take it out 9th of november and repay it 30th of november but still to account for present value of money considerations and that's where it comes into picture physically as payoffs as cash flows we can calculate the amount of interest that we would have needed to pay to facilitate that loan and it would be of interest that would be accounted with a negative sign with a minus sign as this is something we need to pay up then at the very end we would have accounted for our initial hedge so we would have honored our initial contractual obligation by buying uh 10 000 barrels um on the futures markets so paying uh 427 500 dollars for the our initial uh futures contract but given the fact that we shorted futures on the 9th of november would have also received 42.4 dollars per barrel for the 10 000 barrels meaning that this effectual effectively cancels our initial hedge and we can stick with the physical oil we have bought at the sport 9th of november and here we can just sum up all of the payoffs and arrive at uh 426 thousand five hundred and sixteen dollars that we uh would have needed to pay to guarantee ten thousand barrels of oil uh at the end of the period and if we compare it to the payoff of the initial strategy just long in the futures as of 30th of september we can see that such a payoff would be smaller we would have gained 984.4 dollars for the whole contract uh if we arbitraged uh compared to the instance where we didn't arbitrage reinforcing the same logic for the short futures trading case we can first of all see that we need to look at the inverse relationship between the left-hand side of the equation and the right-hand side of the equation why is that because in that case as we are an agricultural company that tries to reduce or mitigate its exposure to wheat prices we shorted the futures for wheat initially meaning that our standing hedge is a short futures so we need to figure out whether there is an opportunity to profitably unwind our hedge by longing the futures and short selling our wheat at the spot meaning that the assumption here is quite a little bit more stringent than in the previous case we also need to assume that first of all our commodity is available to short at the sport market throughout this period which might not be realistic so we need to account for that and also that the holding cost component of our commodity uh evolves like that so there is some storage costs uh payable per day per bushel just as it was with oil and obviously for oil it is very realistic but for wheat for longer time periods it might not be as realistic because if you store wheat for 20 years for example you won't have much of a usable product left at the end of the period perhaps because well agricultural products do spoil they do expire they do rot if you speak plainly but across two or even three months periods you can assume that wheat would not rot or expire or nothing terrible would happen with it so we can stick with our initial assumptions and here we can see how this difference evolves through time we can see that to start with the sport market is undervalued uh with regards to the futures markets so we made a good decision initially i guess to short uh the futures and not short the sport but as the situation evolves we can see that here on the 2nd of december 2020 we have got our first case when the difference flips when the spot price becomes overvalued with regards to the futures price when holding costs and time value of money considerations are accounted for and that means that 2nd of december 2020 we can start arbitraging and here it just goes the other way around because as we sell the sport second of december 20 2020 so means that we receive that much for our 70 000 bushels of wheat we also save up on the holding costs so we actually cancel our storage and get back the money that we have paid so here we can see that this 840 dollars that we would have needed to pay if we stuck with the futures contract are now free for us to use um any way we want and it means that we can deposit the proceeds from uh shortening the sport and saving on holding costs within the bank and earn the risk for rate for our deposit period and the deposit period is again quite short it's just 12 calendar days isn't it from uh 2nd of december 2020 until 14th of december 2020 but still will make 118 of interest then we need to account for the fact that we have unwinded our initial hedge meaning that we short our initial futures at 5.78 and we also long the futures on the 2nd of december and here is just a coincidence that futures prices at the start of the period 30th of september and uh at our arbitrage date 2nd of december were identical so these payoffs are the same in magnitude but different in science obviously and here we can account for the fact that we have got a very uh complicated payoff structure in this arbitrage case so we add up all of our payoff components so the short selling of the spot savings on holding costs interest that we have earned from depositing the proceeds and also the payoffs from two futures transactions long in the futures on the arbitrage date and shorten the futures on the start date 30th of september and here we can say that our total payoff so the amount of money we have generated from selling 70 000 bushels of wheat is 406 258 and if we compare it to the proceeds we would have received if we stuck with the original uh short futures contract we would have just received 404 600 for the whole bulk of wheat meaning that our arbitrage has generated us 1 658 of profit and this profit is risk-free given that at 2nd of december 2020 we already know all of the prices that we would have paid and that we would have received and we also know the risk-free rate that we would have been uh rewarded for depositing our proceeds obviously these are the two cases that are built up on initial hedging considerations so hedges can become arbitrage wars on futures markets when there is a favorable condition plainly if this relationship breaks down significantly in the direction that favors the hajja so they can beneficially unwind their existing hedges but if you approach uh this issue as a pure arbitration so if you have no hedges to start with and you just look at the dynamics of sport and futures markets you can estimate uh the possibilities of beneficial arbitrage either way so for example um a pure arbitrary would have arbitraged at that stage given the fact that they had no uh initial hedges while here an initial arbitration would have actually shorted this spot and long diffuses at that case given that this difference um is quite significant and uh in the direction that a pure arbitrage can exploit unlike our counterparty that had initially alone hedge and that's all there is for arbitrage on futures markets and how hedges can become arbitrals if it is profitable please leave a like on this video if you found it helpful in the comments below i make to see any further suggestions on videos on business economics or finance topics you would like me to record and please don't forget to subscribe to our channel and consider supporting us on patreon thank you very much and stay tuned