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The Early Exercise Risk Premium

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Kevin Aretz discusses asset pricing implications of being able to optimally early exercise plain-vanilla puts, contrasting expected raw and delta-hedged returns across equivalent American and European puts.

Image courtesy of interviewee. January 29, 2024

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This video video provides an in-depth analysis of the profitability of options investment strategies, specifically focusing on single stock put option strategies. The study distinguishes between American options, which can be exercised at any time before expiration, and European options, which can only be exercised at maturity. The authors introduce the concept of the "early exercise risk premium," which is the expected return difference between equivalent American and European put options due to the flexibility of early exercise.

Through theoretical exploration and Monte Carlo simulations, the authors find that the early exercise risk premium is positive under reasonable market conditions. They also identify direct relationships between this premium and factors such as the financial "moneyness" of options, time to maturity, and underlying asset volatility.

Testing yielded two primary conclusions: the early exercise risk premium is indeed positive, and it correlates positively with options that are likely to be exercised early.

In examining 15 prior strategies that suggest profitable trading methods with put options, the authors replicate these strategies and find they continue to yield significant profits. However, allowing for optimal early exercise changes profitability outcomes for many strategies. Notably, nine strategies become more profitable when considering early exercise, while five lose their profitability altogether.

Overall, the study emphasizes the importance of accounting for the early exercise feature of American options in studies about option profitability. It cautions investors against assuming the profitability levels reported in earlier literature are accurate, as some strategies may be less effective than previously indicated.

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In our study, we, take a look into the profitability

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of options, investment strategies in particular

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into the profitability

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of single stock put option strategies.

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Um, a put option is a financial instrument which gives its

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owner the right to sell some underlying asset, in our case,

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a single stock at a pre-specified price.

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The strike price over a pre-specified period,

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the time to maturity.

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A good example of one

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of the better known options investment strategies comes from

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a study by, Goyal

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and Reto published in 2009 in the Journal

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of Financial Economics.

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They advised their readers to buy

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to long options written on stocks with a high realized

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to imply volatility ratio

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and to short to sell those options on stocks

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with a low ratio.

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So the, realized to implied volatility ratio here

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is taken as a measure of the expansiveness of the options.

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So they effectively advise their readers

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to buy cheap options and to sell expensive ones,

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and they show that doing so

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generates an economically sizable

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and statistically significant positive trading profit.

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Now, where we come into this literature is by recognizing

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that all the studies in,

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the prior literature looking into options,

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investment strategies effectively assume

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that there underlying options there.

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Sample options are European options.

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European options can only be exercised.

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They can only be used to buy the underlying

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to sell the underlying asset at the end

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of the time to maturity.

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Now this assumption is factually incorrect In the real

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world, in real US options exchanges,

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all single stock options are American options.

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American options are far more flexible

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than European options.

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They can be exercised, they can be used

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to sell the underlying asset at any point over

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the time to maturity.

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So, my co-author Adnan ga

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and I then immediately came up with two questions.

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The first question is, does the ability to optimally early

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exercise a put option affect the expected return of

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that option, both from a theoretical

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and an empirical perspective?

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And if the answer to this question is yes,

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then an immediate follow up question is,

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if it affects the expected option return,

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does it also affect the profitability

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of those options investment strategies

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established in the literature?

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And that's exactly what we analyze in our paper.

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In our theoretical work, we come up with this term,

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the early exercise risk premium.

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The early exercise risk premium is,

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the expected spread return between equivalent American

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and European put options.

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By equivalent, I mean the options are written on the same

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underlying asset.

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They have the same strike price

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and the same time to maturity.

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Um, intuitively speaking,

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the early exercise risk premium measures the effect

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of being able to optimally early exercise,

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put option on the expected put option return.

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Now to generate some intuition on the sign

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of the early exercise with premium,

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whether it's positive zero

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or negative, we recall,

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par at I 1992 insight

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that an American put option can be replicated

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by a portfolio containing a long equivalent European put

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option and a long exotic derivative.

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And the exotic derivative is itself a portfolio,

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a portfolio containing an in finite number of binary

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cash or nothing but options.

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We then recall two further insights.

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The first one that the expected portfolio,

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return in finance theory is evaluated average

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of the expected returns of assets in the portfolio.

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And the second that the expected return

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of a binary option is usually closer, closer

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to zero than the expected return of plain vanilla options.

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And the reason for this is that the payoff

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of a binary option is just a constant, in our case,

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the risk-free rate multiplied by the stripe price.

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So the binary option has a systematic risk closer to zero,

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and therefore an expected return closer

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to zero than a plain vanilla option.

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So let's say the European input option has,

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for the sake of argument, an expected return of minus 10%,

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the exotic derivative, this portfolio

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of binary options has a higher expected return minus 5%,

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given that the American product option can be replicated

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by a portfolio of the long European port option

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and the long exotic derivative.

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And given that the expected portfolio return is just the

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evaluated average of the expected returns

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of the assets in the portfolio, it immediately follows

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that the American port option should have an expected return

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somewhere between minus five and minus 10%.

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So the American port option has a higher,

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a less negative expected return than

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the European port option.

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Now unfortunately, this intuition is not

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mathematically rigorous.

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We cannot prove this.

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So what we next do is we conduct a large scale montecarlo

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simulation exercise.

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So, we simulate the value

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of the underlying asset using three stochastic processes,

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black and skulls, 1973 geometric in motion.

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Two more sophisticated one has since 1993,

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stochastic volatility

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and Bates 1996 stochastic volatility jump process.

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We simulate the value

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of the underlying asset using each process under the real

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world, the physical and the risk neutral measure.

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And we use the outcomes from the simulation

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to approximate the expected returns of equivalent

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American and European port options.

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Now, contrasting these two expected returns,

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we reach two important conclusions.

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The first one is yes,

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indeed the early exercise risk premium is positive

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and the reasonable model parameter values, right?

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So there is a positive spread in expected returns

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between American and European put options.

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And second, the early exercise risk premium relates

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positively to, the possibility

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of the American put being,

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early exercised over the time to maturity.

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And because this possibility is positively related

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to manness and negatively to, time to maturity

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and volatility, the early exercise risk premium itself is

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positively related to moneys

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and negatively to time to maturity

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and underlying asset volatility.

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And these are exactly the testable implications which we

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next take to our data.

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Our two main testable implications are first,

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the early exercise risk premium.

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The expected spread return between American

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and European food options is positive.

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And the second one,

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the early exercise risk premium is positively related

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to moneys and negatively to time to maturity

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and underlying asset volatility.

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We face two challenges in testing these predictions.

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The first challenge is that it is not obvious how

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to identify optimal early exercises of

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American puts in our data.

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The literature offers relatively few guidance on this.

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We found an outstanding study by Barlow

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and Whaley published in 2012 in the Journal of Finance,

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and they suggest the, what they call market rule.

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Now, the market rule says you should optimally early

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exercise an option

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when the early exercise payoff exceeds the traded

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price of the option.

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Now in practice,

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the early exercise payoff can never exceed the traded price

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of the option because if it did,

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there were an arbitrage opportunity

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and such blatant arbitrage opportunities do

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not exist in our data.

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So what we assume is, an

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American port should be optimally early exercise.

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As soon as the early exercise payoff gets sufficiently close

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to the traded option, price comes within a 1%, 2.5%,

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5% radius.

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The second obstacle, which we face,

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and I already alluded to this

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before, is that there are no

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European single stock options traded in real US

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options exchanges.

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All single stock options traded in the US in exchanges

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are American options.

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In order to circumvent this problem, we synthetically

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recreate European port options.

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How do we do? So we start from Martin's 1973 insight

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that an American coil written on an asset not paying out

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cash over the time

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to maturity should never be optimally early exercised,

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which means the early exercise feature is worthless,

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you cannot, you should never use it.

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And the American call is essentially identical

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to a European coil right now, given

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that all sample options in our data are written on stocks

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not paying our dividends, our

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American coils are effectively European coils.

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We then use, put coil parity in order

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to convert the European coils into European puts.

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So put parity as you can,

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replicate a European put option

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by investments into the equivalent European call option,

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the underlying stock and the money market account.

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So using our traded American

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and our synthetic European put option data,

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we then calculate the returns

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of optimally early exercised American puts never early

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exercised American puts for comparison with the literature

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and synthetic European puts.

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Now we draw two important lessons from

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comparing these returns.

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First, the means track return,

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between optimally early exercised American inputs

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and synthetic European inputs is positive.

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It's equal to 2.38% per month.

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That is economically huge

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and it's also hugely statistically significant.

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So it has a key statistic of around five.

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Even if we run a sophisticated bootstrap on the T statistic,

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we can easily reject the hypothesis that,

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the mean spread return is equal to zero.

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Um, so the main conclusion is in the data, in the pool data,

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the early exercise risk premium is positive,

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but then, we also use portfolio sorts

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and Pharma Mac best regressions to show

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that the spread return

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between optimally early exercise American

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and synthetic European port options is positively related

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to moneyless and negatively to time to maturity

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and underlying asset, underlying stock volatility.

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So we find support for both

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of our main testable implications

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after we established

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that there is indeed a difference in returns

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between optimally early exercise,

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American puts never early exercise American puts

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and European inputs.

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We then returned to those studies

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and the literature proposing profitable options

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investment strategies.

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We identified a total of 12 studies

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suggesting 15 of such strategies, involving

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raw put options, delta hatch pot options.

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So these are put options controlling

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for movements in the value of the underlying asset.

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And strats strats are,

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portfolios containing long positions in the equivalent coils

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and puts now each

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of these option trading strategies suggests

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some sorting variable.

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I mentioned the realized to implied volatility ratio.

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So in each case, the idea is you go long on those options

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with a high sorting variable value or perhaps a low one,

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and you go short, you sell those options which have the

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opposite value, right?

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So to begin with, we then simply,

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replicated the 15 investment strategies conducted

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in the 12 studies.

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The news is good for the studies.

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We find that even over our updated sample period ranging

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to December, 2022, we still find that all of the strategies,

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all 15 continue to be, economically, continue

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to create economically large trading profits which are

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statistically different from zero.

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We then however, introduce one small change

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to the investment strategies we allow for the possibility

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of an optimal early exercise

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in the American puts contained in the trading strategies

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where we again, allow for

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optimal early exercises using Barlow

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and Wale's 2012 trading strategies.

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And the re results from this exercise are striking.

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We find that for 14 out of 15 trading strategies allowing

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for optimal early exercises in the American puts

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significantly changes the profitability of the strategies.

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Now, whether the strategy becomes more

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or less profitable depends on whether there are more highly

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likely to be early exercised options in the long,

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in the buy lack or in the short the select.

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In nine out of 15 cases,

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there are more such options in the long lack

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that increases the return of the long lack

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and the strategy becomes more profitable.

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But in five cases,

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and that's, one third

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of all the investment strategies we look into,

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there are more highly likely

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to be early exercise options in the short lack in the salt

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lack that increases the return of the salt lack.

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And so in this case, the strategies become less profitable.

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In fact, what we show in our empirical work is

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that if once you allow for optimal early exercises,

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these five strategies no longer produce economically

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or statistically significant trading profits.

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Our recommendation is if you are a real world investor,

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definitely don't invest into these

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strategies. There are two,

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Um, main lessons to be taken away from our study.

14:53.985 --> 14:57.005
The first main lesson is that, the ability

14:57.145 --> 15:00.245
to optimally early exercise a put option,

15:00.585 --> 15:04.045
has a large effect on the expected return of the put option,

15:04.515 --> 15:08.325
both from a theoretical and an empirical perspective.

15:08.865 --> 15:11.285
So, studies looking

15:11.915 --> 15:15.525
into put options should definitely take the early

15:16.085 --> 15:19.645
exercise feature of American options into account because

15:19.795 --> 15:23.125
otherwise they produce distorted bias returns.

15:23.585 --> 15:27.125
And then because ignoring the early exercise feature

15:27.145 --> 15:29.045
of American options leads

15:29.065 --> 15:32.525
to distortions biases in their returns,

15:33.735 --> 15:37.195
one should not take at face value the profitability

15:37.535 --> 15:39.075
of the trading strategies

15:39.215 --> 15:42.275
so far adjusted in the existing literature, right?

15:42.575 --> 15:46.675
So some of these strategies nine are way more,

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profitable than suggested in the original studies.

15:49.855 --> 15:52.195
But then on the other hand there are five strategies,

15:52.335 --> 15:55.275
one third, which are actually not profitable at all.

15:55.615 --> 15:59.795
And so readers, investors need to be careful,

16:00.045 --> 16:02.515
about the profitability of these strategies.

Evaluating Profitability of Put Option Strategies

00:00:00 - In our study, we, take a look into the profitability of options, investment strategies in particular into the profitability of single stock put option strategies. a put option is a financial instrument which gives its owner the right to sell some underlying asset, in our case, a single stock at a pre-specified price. The strike price over a pre-specified period, the time to maturity. A good example of one
00:00:34 - a study by, Goyal and Reto published in 2009 in the Journal of Financial Economics. They advised their readers to buy to long options written on stocks with a high realized to imply volatility ratio and to short to sell those options on stocks with a low ratio. So the, realized to implied volatility ratio here is taken as a measure of the expansiveness of the options.
00:01:05 - to buy cheap options and to sell expensive ones, and they show that doing so generates an economically sizable and statistically significant positive trading profit. Now, where we come into this literature is by recognizing that all the studies in, the prior literature looking into options, investment strategies effectively assume that there underlying options there.
00:01:36 - European options can only be exercised. They can only be used to buy the underlying to sell the underlying asset at the end of the time to maturity. Now this assumption is factually incorrect In the real world, in real US options exchanges, all single stock options are American options. American options are far more flexible than European options. They can be exercised, they can be used to sell the underlying asset at any point over
00:02:08 - So, my co-author Adnan ga and I then immediately came up with two questions. The first question is, does the ability to optimally early exercise a put option affect the expected return of that option, both from a theoretical and an empirical perspective? And if the answer to this question is yes, then an immediate follow up question is, if it affects the expected option return,
00:02:41 - of those options investment strategies established in the literature? And that's exactly what we analyze in our paper. In our theoretical work, we come up with this term, the early exercise risk premium. The early exercise risk premium is, the expected spread return between equivalent American and European put options. By equivalent, I mean the options are written on the same underlying asset. They have the same strike price

Impact of Early Exercise on Put Option Returns

00:03:12 - intuitively speaking, the early exercise risk premium measures the effect of being able to optimally early exercise, put option on the expected put option return. Now to generate some intuition on the sign of the early exercise with premium, whether it's positive zero or negative, we recall, par at I 1992 insight that an American put option can be replicated
00:03:43 - option and a long exotic derivative. And the exotic derivative is itself a portfolio, a portfolio containing an in finite number of binary cash or nothing but options. We then recall two further insights. The first one that the expected portfolio, return in finance theory is evaluated average of the expected returns of assets in the portfolio.
00:04:12 - of a binary option is usually closer, closer to zero than the expected return of plain vanilla options. And the reason for this is that the payoff of a binary option is just a constant, in our case, the risk-free rate multiplied by the stripe price. So the binary option has a systematic risk closer to zero, and therefore an expected return closer to zero than a plain vanilla option. So let's say the European input option has,
00:04:47 - the exotic derivative, this portfolio of binary options has a higher expected return minus 5%, given that the American product option can be replicated by a portfolio of the long European port option and the long exotic derivative. And given that the expected portfolio return is just the evaluated average of the expected returns of the assets in the portfolio, it immediately follows that the American port option should have an expected return
00:05:20 - So the American port option has a higher, a less negative expected return than the European port option. Now unfortunately, this intuition is not mathematically rigorous. We cannot prove this. So what we next do is we conduct a large scale montecarlo simulation exercise. So, we simulate the value of the underlying asset using three stochastic processes, black and skulls, 1973 geometric in motion.
00:05:50 - stochastic volatility and Bates 1996 stochastic volatility jump process. We simulate the value of the underlying asset using each process under the real world, the physical and the risk neutral measure. And we use the outcomes from the simulation to approximate the expected returns of equivalent American and European port options. Now, contrasting these two expected returns, we reach two important conclusions.

Analyzing Early Exercise Risk Premium in Options

00:06:20 - indeed the early exercise risk premium is positive and the reasonable model parameter values, right? So there is a positive spread in expected returns between American and European put options. And second, the early exercise risk premium relates positively to, the possibility of the American put being, early exercised over the time to maturity. And because this possibility is positively related to manness and negatively to, time to maturity
00:06:56 - positively related to moneys and negatively to time to maturity and underlying asset volatility. And these are exactly the testable implications which we next take to our data. Our two main testable implications are first, the early exercise risk premium. The expected spread return between American and European food options is positive. And the second one, the early exercise risk premium is positively related to moneys and negatively to time to maturity
00:07:28 - We face two challenges in testing these predictions. The first challenge is that it is not obvious how to identify optimal early exercises of American puts in our data. The literature offers relatively few guidance on this. We found an outstanding study by Barlow and Whaley published in 2012 in the Journal of Finance, and they suggest the, what they call market rule. Now, the market rule says you should optimally early
00:08:00 - when the early exercise payoff exceeds the traded price of the option. Now in practice, the early exercise payoff can never exceed the traded price of the option because if it did, there were an arbitrage opportunity and such blatant arbitrage opportunities do not exist in our data. So what we assume is, an American port should be optimally early exercise. As soon as the early exercise payoff gets sufficiently close
00:08:35 - 5% radius. The second obstacle, which we face, and I already alluded to this before, is that there are no European single stock options traded in real US options exchanges. All single stock options traded in the US in exchanges are American options. In order to circumvent this problem, we synthetically
00:09:05 - How do we do? So we start from Martin's 1973 insight that an American coil written on an asset not paying out cash over the time to maturity should never be optimally early exercised, which means the early exercise feature is worthless, you cannot, you should never use it. And the American call is essentially identical to a European coil right now, given that all sample options in our data are written on stocks

Evaluating American and European Put Options

00:09:38 - American coils are effectively European coils. We then use, put coil parity in order to convert the European coils into European puts. So put parity as you can, replicate a European put option by investments into the equivalent European call option, the underlying stock and the money market account. So using our traded American and our synthetic European put option data,
00:10:10 - of optimally early exercised American puts never early exercised American puts for comparison with the literature and synthetic European puts. Now we draw two important lessons from comparing these returns. First, the means track return, between optimally early exercised American inputs and synthetic European inputs is positive. It's equal to 2.38% per month.
00:10:41 - and it's also hugely statistically significant. So it has a key statistic of around five. Even if we run a sophisticated bootstrap on the T statistic, we can easily reject the hypothesis that, the mean spread return is equal to zero. so the main conclusion is in the data, in the pool data, the early exercise risk premium is positive, but then, we also use portfolio sorts
00:11:11 - that the spread return between optimally early exercise American and synthetic European port options is positively related to moneyless and negatively to time to maturity and underlying asset, underlying stock volatility. So we find support for both of our main testable implications after we established that there is indeed a difference in returns between optimally early exercise, American puts never early exercise American puts
00:11:44 - We then returned to those studies and the literature proposing profitable options investment strategies. We identified a total of 12 studies suggesting 15 of such strategies, involving raw put options, delta hatch pot options. So these are put options controlling for movements in the value of the underlying asset. And strats strats are, portfolios containing long positions in the equivalent coils
00:12:16 - of these option trading strategies suggests some sorting variable. I mentioned the realized to implied volatility ratio. So in each case, the idea is you go long on those options with a high sorting variable value or perhaps a low one, and you go short, you sell those options which have the opposite value, right? So to begin with, we then simply, replicated the 15 investment strategies conducted in the 12 studies.

Reevaluating Options Strategies: Early Exercise Impact

00:12:47 - We find that even over our updated sample period ranging to December, 2022, we still find that all of the strategies, all 15 continue to be, economically, continue to create economically large trading profits which are statistically different from zero. We then however, introduce one small change to the investment strategies we allow for the possibility of an optimal early exercise
00:13:21 - where we again, allow for optimal early exercises using Barlow and Wale's 2012 trading strategies. And the re results from this exercise are striking. We find that for 14 out of 15 trading strategies allowing for optimal early exercises in the American puts significantly changes the profitability of the strategies.
00:13:51 - or less profitable depends on whether there are more highly likely to be early exercised options in the long, in the buy lack or in the short the select. In nine out of 15 cases, there are more such options in the long lack that increases the return of the long lack and the strategy becomes more profitable. But in five cases, and that's, one third of all the investment strategies we look into, there are more highly likely
00:14:23 - lack that increases the return of the salt lack. And so in this case, the strategies become less profitable. In fact, what we show in our empirical work is that if once you allow for optimal early exercises, these five strategies no longer produce economically or statistically significant trading profits. Our recommendation is if you are a real world investor, definitely don't invest into these strategies. There are two,
00:14:53 - The first main lesson is that, the ability to optimally early exercise a put option, has a large effect on the expected return of the put option, both from a theoretical and an empirical perspective. So, studies looking into put options should definitely take the early exercise feature of American options into account because otherwise they produce distorted bias returns.
00:15:27 - of American options leads to distortions biases in their returns, one should not take at face value the profitability of the trading strategies so far adjusted in the existing literature, right? So some of these strategies nine are way more, profitable than suggested in the original studies. But then on the other hand there are five strategies, one third, which are actually not profitable at all.