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The Effect of Time on Trump’s Win Likelihood
Not everybody is an options trader, but during an election year there is at least one binary option that most of us care quite a bit about and that’s the option on the US Presidency. There are ways to trade the binary option, but my interest here is not in valuation.
People generally understand that if an option with a payoff of $1 or $0 (depending on the outcome of the event) is trading at $0.60, it means that the market is pricing a 60% chance that the event will occur. (Because 60% chance of $1 + 40% chance of $0 has an expected value, if we ignore discounting, of $0.60). But what I think many people don’t naturally understand is how the 60% chance changes over time even if nothing changes in the underlying circumstances.
If you’re not an options trader, this might be confusing. If Trump has a 60% chance of winning based on the current circumstances on July 31st, then if the exact same circumstances prevail on October 31st shouldn’t the odds of him winning still be 60% (and therefore, the price wouldn’t change)? The answer is nope, not at all.
Let’s suppose that we can summarize “the current circumstances” with one metric, that being the national polling margin.[1] Trump currently polls about 2 points ahead of Harris nationally according to the RealClearPolitics average. If that’s still true on November 5th, Trump will win (again, pretending that the winner of the popular vote automatically wins the election). The odds of him winning would be, of course, 100%. If he only drew 49% of the vote, his odds of winning would be 0%. That’s on the day of expiry, when the odds have to collapse to 100% or 0%.
Prior to the last day, time and volatility work in favor of the challenger and against the leader. If I am the leader, then I want nothing to change. Good things will help me, but won’t change the situation (I’m still expected to win), but bad things might change my expected win to an expected loss. The more volatility there is, the more crazy things happen, the more chances there are that my victory will turn to a loss. If I am behind, I want chaos, and the closer I get to ‘expiration’ the more I am willing to risk to get the chaos. Think about pulling the goalie in an elimination round in hockey or soccer…if the team is behind by 1 goal with 1 minute left, then the opposition scoring on an open goal doesn’t change anything – but having an extra man forward has a chance to change the outcome.
Key point: volatility helps the out-of-the-money option. Higher volatility raises the delta (loosely thought of as the chance of ending up in-the-money) of an out-of-the-money option. Similarly, higher volatility lowers the delta of the in-the-money option. This is why there are October surprises.
Time works on options pricing similarly to volatility (fun fact – doubling the implied volatility has the same effect as quadrupling the time to maturity, I the Black-Scholes world). As the expiry of the option approaches, the delta of an in-the-money option gradually rises until it gets to 1.0 at expiry; the delta of an out-of-the-money option gradually declines to 0.0. It’s the same reasoning. With more time to play, there are more chances for good (outcome-changing) accidents to happen to the person who is trailing, and more chances for bad accidents to happen to the person who is leading.
In the context of the election, here’s what this means. (Note: remember this is just an illustration, not a prediction. It’s a pricing model where I’ve made lots of assumptions so as to be able to show this point).
If Trump still has a similar lead in the national polls in two months, his odds of winning will rise – to something like 70%, versus 60% now. After that, it will rapidly go to 100% over the ensuing weeks. And so event contracts will behave likewise…Trump contracts should gradually rise, even if nothing changes, as the opportunities for changes in the race get narrower.
Two more caveats.
One: the option here is not really a European exercise that is determined on November 5th. Since there is such a thing as early voting, the effective expiry is closer. If Trump is leading on October 15th, he’s accumulating actual votes. So it’s really like some kind of weighted average-price exotic option. But the point is, the delta decay will happen faster than I’ve modeled, for that reason.
Two: I’ve assumed volatility is constant. It most assuredly is not constant. Implied volatility ought to rise as we get closer to expiry (October surprise!). That will tend to offset the delta decay that I have modeled.
In summary – don’t take this as trading advice, but as a hopefully useful insight into how deltas (and binary contracts) evolve over time. I hope you found this interesting.
[1] Obviously, we know that’s not right because the US elects Presidents based on the Electoral College process, which is the result of a compromise between those who wanted the President selected by Congress and those who wanted the President selected by popular vote. In recent years, the winner of the popular vote hasn’t always won the Presidency because the most-populous state, California, has overwhelmingly voted Democrat and skewed the popular vote numbers.
Thanksgiving Memories: Re-Blog of Two Goodies
Since there aren’t a lot of folks out there trading today, that also means there probably aren’t a lot of folks reading articles about markets. I could just talk about the OpenAI guy going to Microsoft and then back again (I really couldn’t care less, but it seems everyone is breathless for new episodes of the Real Housewives of Artificial Intelligence), but I thought readers would be better served by a reprise of a couple of my old articles on inflation tails.
The first one is a lightly edited re-post of “Royally Skewed,” first posted May 9, 2011. (Wow, I’ve been doing this blog for a while!) Incidentally, feel free to go to the inflationguy.blog and search for topics of interest. Sometimes you can find a nugget among the 1100 or so articles!
Royally Skewed
Although commodities do occasionally crash, in general commodity prices are positively kurtotic (fat-tailed) and positively skewed. This is in contradistinction to equity prices, which are positively kurtotic but negatively skewed. In English, that means that both stock prices and commodity prices crash more than we would expect them to if price changes were random, but while stocks tend to crash down, commodities tend to crash up.
The reason for this is simple: commodity supply curves become very inelastic (steeper) when the level of actual, current inventory is fully allocated. There are only so many soybeans available right now. But at low levels of demand and lower prices, the supply curve gets more and more elastic (flatter), which means large declines in demand don’t drop prices as sharply as large increases in demand can increase them at the other end of the curve.
The practical import of this observation is this: one must be more careful shorting commodities than shorting stocks, because while a bull market in stocks can grind you to death, a bull move in commodities can rip you to suddenly to shreds (the fact that in a limit up market there is literally no price at which you are allowed to cover, while this situation rarely exists in equities, means that market infrastructure contributes to the danger).
Skewness and kurtosis, in addition to being great cocktail-party words, are also important concepts for investors to understand. More specifically, it is important for investors to think carefully about the difference of the “higher moments” (as skewness and kurtosis are sometimes collectively called) between asset classes and particular investments. Given a choice between two investments with the same expected return and variance, a long-only investor should always choose the one with ‘fat tails’ on the upside rather than the one with ‘fat tails’ on the downside. This is true for two reasons. First, the marginal pleasure of a gain, for most investors, is lower than the marginal pain of a loss, and this is increasingly true for large gains and losses. Second, a large gain increases the bankroll, but a large loss can be a portfolio-ending experience. All of the rules about long-term investing are based on the assumption that the long term can be reached – or, as Warren Buffett has said, one “-100%” really messes up any series of portfolio returns.
Recently, in a great customer letter called “Five fallacies about inflation (and why global policy rates are too low),” Markus Heider, Jerome Saragoussi, and Francis Yared of Deutsche Bank made some very adroit observations about the risks of inflation going forward. The quick summary is that they see inflation as the greater risk than deflation because 1. The output gap is smaller than suggested by the high unemployment rate; 2. A negative output gap does not imply declining inflation [frequent readers know I harp on this a lot]; 3. EM countries are exporting inflation rather than disinflation; 4. Commodity price inflation is becoming structural and is exacerbated by low global real policy rates; and 5. Central banks’ credibility is at risk of being eroded.
But the single best part of the report, in my opinion, is the chart they created to summarize the effect of their views on the distribution of possible inflation outcomes going forward. That chart is below (reprinted with permission):
In short, the higher expected value, flatter distribution, and fat upper tail combine to make long-inflation bets worthwhile even if they are somewhat expensive right now. This is one reason that TIPS are seemingly egregiously priced. It’s all about the skew. If we don’t get inflation, we probably bounce around between 1% and 3% inflation for a while. If we do get inflation, it could get ugly. Therefore, it makes sense to give up some current return to ‘buy the tail option.’ I agree, and think their picture is truly worth a thousand words. (I still think that TIPS are too expensive for my taste even with this fact, but it is the reason I was willing to be long them when 10-year real yields were as low as 1%. It’s just a harder call at 0.65%!).
I highly recommend you contact your Deutsche Bank contact to get a copy of this report (from April 1). Honestly, while the overall state of inflation research is clearly better now than it was, just a few years ago, these guys at DB seem to me to have some of the most consistently high-quality research in the space.
The second article dovetails with that one. In this article, from December 7, 2021, I provide a guess at the value of long inflation tails. This article is cleverly titled “A Guess at the Value of Long Inflation Tails,” because “Royally Skewed” was already taken.
A Guess at the Value of Long Inflation Tails
In my last post, “You Have Not Missed It,” I promised the following:
“There is one final point that I will explain in more detail in another post. Breakevens also should embed some premium because the tails to inflation are to the upside. When you estimate the value of that tail, it’s actually fairly large.”
So, as promised, here is that explanation.
Viewing the forward inflation curve as a forecast of expected inflation (whether using “breakevens” or, more accurately, inflation swaps) is biased in a particular way. Or, at least, it should be. The “breakeven” inflation rate is the rate at which a long-only investor over the ensuing period would be roughly as well off with a nominal bond (which pays a real rate plus a premium for expected inflation) and an inflation-indexed bond (which pays a real rate, plus actual inflation realized over the period). Obviously the inflation-indexed bond is safer in real space, so arguably nominal bonds should also offer a risk premium to induce a buyer to take inflation risk.[1] Ordinarily, though, we ignore this risk and just consider breakeven inflation to be the difference between real and nominal yields. Inflation swaps are cleaner, in that if inflation is higher than the stated fixed rate, the fixed-rate payer on the swap ‘wins’ and receives a cash flow at the end, whereas if inflation turns out to be lower than the stated fixed rate, it is the fixed-rate receiver who wins. So from here on, I will talk in terms of inflation swaps, which also abstract from various bond-financing issues of the breakeven…but the reader should understand that the concept applies to other measures of expected inflation as well.
Now, suppose that you expect 10-year inflation to come in at 2% per annum. Suppose that in the inflation swap market, the 10-year rate is 2% ‘choice’ – that is, you may either buy inflation at 2% or sell inflation at 2%. Since you expect inflation to be 2%, are you indifferent about whether you should buy or sell?
The answer is no. In this case you should be much more eager to buy 2% than to sell 2%, given that your point estimate is 2%. The reason why is that the distribution of inflation outcomes is not symmetrical: you are much more likely to observe a miss far above your expectation than to observe a miss far below your expectation. Therefore, the expected value of that miss is in your favor if you buy the inflation swap (pay fixed and receive inflation) at 2%. There is, in other words, an embedded option here that means the swap market should trade above where most people expect inflation to be.
We can roughly quantify at least the order of magnitude of this effect. Consider the distribution below. This chart (Source: Enduring Investments) shows the difference, from 1956 until 2011, of 10-year inflation expectations[2] compared with subsequent 10-year actual inflation results. The blue line is at 0% – at that point, actual inflation turned out to be right where a priori expectations had it. The chart obviously only covers until 2011 since that is the last year from which we have a completed 10-year period. Recognize that I am not charting the levels of inflation, but the level of inflation relative to the original expectation.
Notice that the chart has a cluster of outcomes (and in fact, the most-probable outcomes) just to the left of zero, where expectations exceeded the actual outcome by a little bit, but that there are very few long tails to the left. However, misses to the right, where the actual outcome was above the beginning-of-period expectations, were sometimes quite large. The median point (where half of the misses are to the left, and half are to the right) is 0.21%. But this is not a symmetric distribution, so if we randomly sample points from this distribution, we find that the average of that sample is 0.59%.
So, if you buy the inflation swap at 2% when your expectations are at 2%, on average you’ll win by 59bps, at least historically. Of course, past results are no indication of future returns, and a Fed economist would argue that we have much better control of inflation now than we ever have in the past (Ha ha. I crack myself up.). And inflation volatility markets, when they can be found, don’t trade at such high implied volatilities. Noted, although the wild swings in growth and the deficit and the money supply, not to mention recent realized outcomes, might make more cynical observers question whether we should be so confident in that view right at the moment.
Moreover, a counterargument is that at the present time an investor also has the advantage of investing when expectations are fairly low, so the downside tails are not as likely. The worst outcome of that whole 1956-2011 period was an 8.75% undershoot of inflation versus expectations. This happened in the 10 years following September 1981, when expectations were for 10-year inflation of 12.70% and actual inflation was 3.95%. But with expectations at 2.50%, is it really feasible to get a -6.25% compounded inflation rate? That would imply a 50% fall in the price level (and, I should note, it would mean that investors in TIPS would win hugely in real space since they get back no worse than nominal par. But that doesn’t help the swap buyer).
To be a little more fair, then, the following chart considers only the periods where inflation expectations were 5% per annum or less at the beginning of the period. That truncates only 10% of the distribution, but as you might expect the vast majority of the truncation is on the left-hand side. This is fair because it’s naturally harder to miss far below your expectations when your expectations are very low to begin with.[3]
The value of the expected miss in this contingent view is 1.13%. So, in order for the market to be priced fairly if general expectations are for 2.5% average CPI inflation the 10-year inflation swap would have to be around 3.63%. Again, even allowing for the “policymakers are smarter now” argument (an argument quite lacking, I would argue, in empirical evidence) I would feel comfortable saying that 10-year inflation swaps, and breakevens, should embed at least a 50bps or so ‘option premium’ relative to expectations.
I don’t believe that they do. Indeed, consider that the buyer of 10-year TIPS (with breakevens at 2.50%) not only wins if 10-year inflation is above 2.50% but the average win historically (conditioned on breakevens being below 5% to start, and by construction only considering wins) has been about 2.07% per annum – a massive outperformance. Not only that, but any losses are essentially guaranteed to be small because the tails on the left-hand side are truncated: if inflation is negative (that is, if the loss would have been greater than 2.50%) it is limited by the fact that the Treasury guarantees the nominal principal.
As an aside, we do consider this sort of option in other contexts. In the Eurodollar futures market, for example, we recognize that the person who is short the Eurodollar contract (and therefore gets a positive mark-to-market when interest rates rise) is in a better situation than the long (who gets the positive mark-to-market when interest rates fall), because the short gets to invest wins at higher interest rates and borrow losses at lower interest rates, while the long must borrow to cover losses when interest rates are higher, and but gets to invest wins when interest rates are lower. As a result, Eurodollar futures trade lower than the forwards implied from the swap curve, since the buyer needs to be induced by a better-than-expected price. And there are other such examples. But I am pretty sure I have never seen an example of an embedded option like this that is priced so differently relative to history than the embedded options in the inflation market!
[1] However, since this risk is symmetric – the seller of the bond also has risk in real space, but in the opposite direction – it isn’t immediately obvious why one side should get an inducement over the other. So I will leave the ‘risk premium’ aside.
[2] For long-term inflation expectations back before the advent of TIPS, I used the Enduring model relating real yields to nominal yields, about which I’ve written previously. You can find a brief discussion of this and an illustration of the model at this link: https://inflationguy.blog/2016/12/23/a-very-long-history-of-real-interest-rates/
[3] The author’s wife has been known to make something like this observation from time to time.
A Guess at the Value of Long Inflation Tails
In my last post, “You Have Not Missed It,” I promised the following:
“There is one final point that I will explain in more detail in another post. Breakevens also should embed some premium because the tails to inflation are to the upside. When you estimate the value of that tail, it’s actually fairly large.”
So, as promised, here is that explanation.
Viewing the forward inflation curve as a forecast of expected inflation (whether using “breakevens” or, more accurately, inflation swaps) is biased in a particular way. Or, at least, it should be. The “breakeven” inflation rate is the rate at which a long-only investor over the ensuing period would be roughly as well off with a nominal bond (which pays a real rate plus a premium for expected inflation) and an inflation-indexed bond (which pays a real rate, plus actual inflation realized over the period). Obviously the inflation-indexed bond is safer in real space, so arguably nominal bonds should also offer a risk premium to induce a buyer to take inflation risk.[1] Ordinarily, though, we ignore this risk and just consider breakeven inflation to be the difference between real and nominal yields. Inflation swaps are cleaner, in that if inflation is higher than the stated fixed rate, the fixed-rate payer on the swap ‘wins’ and receives a cash flow at the end, whereas if inflation turns out to be lower than the stated fixed rate, it is the fixed-rate receiver who wins. So from here on, I will talk in terms of inflation swaps, which also abstract from various bond-financing issues of the breakeven…but the reader should understand that the concept applies to other measures of expected inflation as well.
Now, suppose that you expect 10-year inflation to come in at 2% per annum. Suppose that in the inflation swap market, the 10-year rate is 2% ‘choice’ – that is, you may either buy inflation at 2% or sell inflation at 2%. Since you expect inflation to be 2%, are you indifferent about whether you should buy or sell?
The answer is no. In this case you should be much more eager to buy 2% than to sell 2%, given that your point estimate is 2%. The reason why is that the distribution of inflation outcomes is not symmetrical: you are much more likely to observe a miss far above your expectation than to observe a miss far below your expectation. Therefore, the expected value of that miss is in your favor if you buy the inflation swap (pay fixed and receive inflation) at 2%. There is, in other words, an embedded option here that means the swap market should trade above where most people expect inflation to be.
We can roughly quantify at least the order of magnitude of this effect. Consider the distribution below. This chart (Source: Enduring Investments) shows the difference, from 1956 until 2011, of 10-year inflation expectations[2] compared with subsequent 10-year actual inflation results. The blue line is at 0% – at that point, actual inflation turned out to be right where a priori expectations had it. The chart obviously only covers until 2011 since that is the last year from which we have a completed 10-year period. Recognize that I am not charting the levels of inflation, but the level of inflation relative to the original expectation.
Notice that the chart has a cluster of outcomes (and in fact, the most-probable outcomes) just to the left of zero, where expectations exceeded the actual outcome by a little bit, but that there are very few long tails to the left. However, misses to the right, where the actual outcome was above the beginning-of-period expectations, were sometimes quite large. The median point (where half of the misses are to the left, and half are to the right) is 0.21%. But this is not a symmetric distribution, so if we randomly sample points from this distribution, we find that the average of that sample is 0.59%.
So, if you buy the inflation swap at 2% when your expectations are at 2%, on average you’ll win by 59bps, at least historically. Of course, past results are no indication of future returns, and a Fed economist would argue that we have much better control of inflation now than we ever have in the past (Ha ha. I crack myself up.). And inflation volatility markets, when they can be found, don’t trade at such high implied volatilities. Noted, although the wild swings in growth and the deficit and the money supply, not to mention recent realized outcomes, might make more cynical observers question whether we should be so confident in that view right at the moment.
Moreover, a counterargument is that at the present time an investor also has the advantage of investing when expectations are fairly low, so the downside tails are not as likely. The worst outcome of that whole 1956-2011 period was an 8.75% undershoot of inflation versus expectations. This happened in the 10 years following September 1981, when expectations were for 10-year inflation of 12.70% and actual inflation was 3.95%. But with expectations at 2.50%, is it really feasible to get a -6.25% compounded inflation rate? That would imply a 50% fall in the price level (and, I should note, it would mean that investors in TIPS would win hugely in real space since they get back no worse than nominal par. But that doesn’t help the swap buyer).
To be a little more fair, then, the following chart considers only the periods where inflation expectations were 5% per annum or less at the beginning of the period. That truncates only 10% of the distribution, but as you might expect the vast majority of the truncation is on the left-hand side. This is fair because it’s naturally harder to miss far below your expectations when your expectations are very low to begin with.[3]
The value of the expected miss in this contingent view is 1.13%. So, in order for the market to be priced fairly if general expectations are for 2.5% average CPI inflation the 10-year inflation swap would have to be around 3.63%. Again, even allowing for the “policymakers are smarter now” argument (an argument quite lacking, I would argue, in empirical evidence) I would feel comfortable saying that 10-year inflation swaps, and breakevens, should embed at least a 50bps or so ‘option premium’ relative to expectations.
I don’t believe that they do. Indeed, consider that the buyer of 10-year TIPS (with breakevens at 2.50%) not only wins if 10-year inflation is above 2.50% but the average win historically (conditioned on breakevens being below 5% to start, and by construction only considering wins) has been about 2.07% per annum – a massive outperformance. Not only that, but any losses are essentially guaranteed to be small because the tails on the left-hand side are truncated: if inflation is negative (that is, if the loss would have been greater than 2.50%) it is limited by the fact that the Treasury guarantees the nominal principal.
As an aside, we do consider this sort of option in other contexts. In the Eurodollar futures market, for example, we recognize that the person who is short the Eurodollar contract (and therefore gets a positive mark-to-market when interest rates rise) is in a better situation than the long (who gets the positive mark-to-market when interest rates fall), because the short gets to invest wins at higher interest rates and borrow losses at lower interest rates, while the long must borrow to cover losses when interest rates are higher, and but gets to invest wins when interest rates are lower. As a result, Eurodollar futures trade lower than the forwards implied from the swap curve, since the buyer needs to be induced by a better-than-expected price. And there are other such examples. But I am pretty sure I have never seen an example of an embedded option like this that is priced so differently relative to history than the embedded options in the inflation market!
[1] However, since this risk is symmetric – the seller of the bond also has risk in real space, but in the opposite direction – it isn’t immediately obvious why one side should get an inducement over the other. So I will leave the ‘risk premium’ aside.
[2] For long-term inflation expectations back before the advent of TIPS, I used the Enduring model relating real yields to nominal yields, about which I’ve written previously. You can find a brief discussion of this and an illustration of the model at this link: https://inflationguy.blog/2016/12/23/a-very-long-history-of-real-interest-rates/
[3] The author’s wife has been known to make something like this observation from time to time.
