Startups often issue stock options or warrants. Typically they give the owner the right to purchase stocks of the company at a set price, usually the same price paid by investors in a recent or upcoming funding round.

There are different reasons why it would be useful to know how much those options are worth at the time of issuing. For instance, if the options are granted as part of compensating an advisor or an employee, to give something which is fair to both parties.

Options are a quite standard financial instrument these days but mostly in the traded/liquid assets world. Options belong to a broader set of financial instruments called derivatives, that give their name to the fact that they are contracts derived from another asset, called the “underlying”. Given a set of assumptions around the contract, but mainly about the underlying, there is a well known formula to assign a value to an option at any point in time, the Black and Scholes formula (“B-S”). Despite the controversy regarding the assumptions vs. reality, in practice, clearly the formula is present within professionals working with derivatives.

So, the first attempt to value startup warrants could be to apply B-S. But there are several theoretical and practical issues with this. B-S assumes the underlying assets behaves like a lognormal random variable. Even for traded stocks this has been challenged, as empirical results show “larger tails” than predicted by such a model. But for startups, the lognormal assumption is far from true. The consensus is that startups behave more like a “power law” distribution.

For liquid stocks, whenever there is a market for options, what is typically done is to use the observed prices as inputs, and then assume that the formula is right, and obtain the value for the least observable parameter in the formula (the volatility), and obtain the “implied volatility”. So in practice, to value a given option, one would look for equivalent options in the market, and if there is no equivalent one (same term, exercise price, etc.), to interpolate the value of the implied volatility and calculate the price according to Black and Scholes. But for startups, there is no market for the stock and even less for options, so this is not an option. There are other assumptions that B-S relies on to come up with the results, but the two mentioned above are enough to show this is not the right method.

So, an alternative method is proposed here, that is more adept to startups. Of course it’s also not perfect, has several assumptions behind, and the lack of liquidity of both the underlying shares and the options themselves make the result of this calculation not very useful after the instruments are issued, but it can help guide certain discussions.

The idea is to get a value for the warrants as a % of the value of the share. So the first assumption is that the relevant pricing of the share is a fair value. And to do so, we’ll evaluate the difference in payoff between holding 1 unit of the share vs. holding 1 unit of the option. The second assumption is that the value of the underlying stock can be modeled by a power law distribution, in particular we’ll use the Pareto distribution with the key parameter alpha. Here we are not aiming to model an entire stochastic process as it’s done in B-S related analysis, but we are aiming to model an exit value. Here we are following a very useful article from Jerry Neumann (“JN”) that discusses modeling startup investments and funds of startup investments with power laws. To keep it simple, we’ll be modeling the exit multiple of an investment of 1 (your favourite currency) in the company. For example, if someone invested $100.000\$$ in a company, and after a sale obtained $400.000\$$, we’ll say the multiple was 4x, or in our model, just 4. In this case we are not going to worry about when this potential exit happens, as this element will be equivalent for warrants and the underlying asset.

Pareto distribution requires a minimum value, and following JN, we’ll use this distribution to model returns over 1x. Of course, the returns can be lower than 1, and often they will be 0. For now, we’ll assume that there will be a probability mass of the return being 0: $p_0$, and then, assume the returns for the investor in the other scenarios will be 1 or higher (there are arguments in JN’s piece, like standard practices of investors using “liquidation preferences” try to protect their initial investment). In a second iteration we’ll add a second probability mass of returns being 1x for equity investors.

Warrants are typically issued with an exercise or strike price equal to the value of the round, and that means the holder needs to pay for the shares at a fixed price whenever a liquidity opportunity arises. For instance, if the company is being sold a few months or years after issuance, at a relevant price that gives the shareholders of the associated initial equity round of 4x, in this case the warrant holders will exercise their options paying 1x, get the shares and sell them for 4x, netting 3x. So basically, the warrant holders get M-1, where M is the cash multiple of the equity investors.

With this assumptions and facts, we can try to create a model. We’ll do it two ways, one more mathematical and one more financial: 1- Analytically we can calculate the relationship between the payoff of the equity investor vs the warrant owner:

$E[S] = p_0 \cdot 0 + (1 - p0) \cdot E[S^*] \text{, where } S^* \sim Pareto (\alpha, x_{min} = 1)$ $E[W] = p0 \cdot 0 + (1 - p0) \cdot E[(S^*-1)_+] \text{ (the } (X)_+ \text{ notation means } Max(X,0) \text{)}$ $E[S^*] = \dfrac{\alpha}{(\alpha - 1)} \quad \href{https://en.m.wikipedia.org/wiki/Pareto_distribution#Moments_and_characteristic_function}{\text{See Wikipedia}}$ $E[(S^*-1)_+] = \dfrac{1}{\alpha - 1}$This means that $\dfrac{E[W]}{E[S]} = \dfrac{1}{\alpha}$

What we are saying here is that the lower the alpha, the higher the value of the warrant with respect to the value of the share (this is why the assumption above regarding the fairness of the pricing of the equity was relevant).

So this problems ends up being a problem of estimating the alphas for companies at different stages. Without going into further detail in this post, it can be argued that the more mature the company, the higher the alpha. And intuitively it makes sense when you see the behaviour of a power law distribution. Basically companies at pre-seed stage could have multiples of 100.000x (think Google or Apple if you invested when they were starting), but a series C company, probably can “only” aim for a 1.000x maximum multiple.

For the sake of illustrate the results of the model, we could assign different alphas for different company stages (as proxied by funding round):

- Pre-seed: 1.2
- Seed: 1.4
- Series A: 1.6
- Series B: 1.8

With these values, the relative values of the warrants are 83%, 71%, 63% and 56%. For instance, if a series A company, being valued at 30M $\$$ pre-money, issues warrants for 1% pre-money of the company, the appropriate valuation should be $1 \% \cdot 30 M \$ \cdot 63 \% = 189.000 \$$. This values are generally higher that what you would see in traded options calculated via B-S. The main reason for this, is that the value of a startup share is mostly due to its potential at the positive tail, and in that tail, the proceeds from owning warrants is almost as large (ie., 20x - 1x is still 19x, so the warrant owner gets only 5% less than the share owner).

It’s worth noting that the value of $p_0$ is irrelevant for this calculation, so regardless of the odds of the company going bust doesn’t affect this relationship. This is probably not very accurate, due to what was mentioned above regarding liquidation preferences (ie., there is a non-negligible set of scenarios where the stock holder at least keeps the original value of the investment, while the warrant holder gets 0), and that will be modeled in an upcoming post. It won’t change the direction of the argument or the methodology.