Implied (Enterprise Value) Volatility

Are Private Equity Buyouts a Disguised Volatility Trade?

Hello, apologies for the delay on the Substack. I spent the last few months graduating and traveling before I start full-time work in ~1 week. This is likely my last finance-heavy piece for the foreseeable future.

I’ve greatly enjoyed writing this Substack. I started it in January 2024 and since then I’ve written 29 pieces, coded up almost 600 graphs, written almost 100,000 words (which apparently would take over 6 hours to read), and received almost 15,000 views (this feels insane as the blog started with me just sending Google Docs PDFs to a few friends). That’s all to say: thank you to everyone who has read my work. I feel very grateful and I’m happy you’ve enjoyed my writing.

Today I hope to build some intuition connecting options math to bond and equity pricing. Once we build this out, I’ll touch on why private equity buyouts can be seen as a type of volatility trade. I think these ideas are quite interesting to explore, so I hope it will be helpful to some of my readers (and I know it has for me!).

This is a <10 minute read if you exclude the Appendix. I find the Appendix very fun, but it is a bit technical and the piece can be read without it.

As always, please reach out if you have any feedback or want to discuss any of the ideas presented in this Substack (elinickoll2@gmail.com). Also, I’ve moved to NYC and would love to meet up with anyone in the area.

Roadmap

1. Implied (Enterprise Value) Volatility

   1. Options Intuitions in the Bond Market

   2. Trade Set-Up

   3. Calculating Implied Enterprise Value Volatility

2. Intuition from Implied Enterprise Value Volatility

   1. Mapping EV Volatility to Equity Volatility

   2. Using Bond Pricing to Understand Convexity in Equities

3. Is Private Equity a Disguised Volatility Trade?

   1. Volatility Arbitrage or Private Equity?

   2. Allocators Sell Puts and Buy Calls?

4. Conclusion/Takeaways

5. Appendix

   1. Debt is 2 Puts (and 2 Calls!)

Implied (Enterprise Value) Volatility

Options Intuitions in the Bond Market

Alright, let’s do something fun. Previously, I wrote two pieces connecting equities and bonds to call and put options respectively. In Options in Fixed Income Part 1, I argued that equities are equivalent to owning a call option on the enterprise value of a business, with a strike price equal to all the debt. I also said owning a bond is equivalent to selling a put option on the business at that same strike price. Then, in Options in Fixed Income Part 2, I added a corollary to this interpretation. In a capital structure with multiple tranches of debt, owning a bond is more like selling a put spread on the EV of a business. In a simple junior/senior capital structure, buying the junior bond is like selling a put struck at the total debt burden and buying one struck at the face value of all the senior debt.

Those having difficulty with this view should read the two pieces (and maybe the Appendix!). Viewing junior bonds as selling a put spread leads to some very weird conclusions and intuitions which are summarized briefly below (with charts from that previous piece).

1. When a company becomes near-insolvent, the debt becomes more sensitive to the underlying fundamentals than the equity, making it feel “equity-like.” (Figure 1)

2. When a company becomes insolvent enough (the junior bonds will take a large haircut), the junior debt, ceteris paribus, becomes “long volatility,” meaning they prefer the business fundamentals become more volatile. This is in contrast to most traditional debt investments. (Figure 2).

3. In some ways, the distressed creditor has a similar payoff to the owner of a call option. (Figure 3).

Figure 1: Sensitivity of the Junior Bonds and Stock to $1 Changes in the Enterprise Value

Figure 2: Sensitivity of the Junior Bonds and Stock to 1pp Changes in the Implied Volatility of the Enterprise Value

Figure 3: Sensitivity of the Junior Bonds Relative to Different Options Positions

I recommend reading the original piece, linked again here, if you are still confused (or interested in exploring it more).

Trade Set-Up

Now, this simple intuition can lead to a surprisingly interesting output (and possibly, a trade). Let’s imagine a very simple capital structure. The firm has $400 worth of assets (let’s assume this is the EV of the firm for simplicity) and $300 of debt ($200 junior, $100 senior). Let’s briefly analyze the potential long positions an investor could take in this capital structure.

1. An investor could buy the stock. As shown previously, this is equivalent to buying a call option on EV struck at $300. All value above $300 goes to the stockholder, but if EV < $300, the stockholder is zeroed (loses their option premium).

2. An investor could buy the senior bonds. This is equivalent to selling a put option on EV struck at $100. If the firm is worth more than $100, the secured bondholder receives their interest and principal (option premium sold). If the firm is worth less, they begin to take losses.

3. An investor could buy the junior bonds. This is equivalent to selling a put on EV struck at $300 and buying another one struck at $100. The investor collects the net premium (interest + principal) when the firm is worth more than $300 and begins to take losses when value is between $100 and $300, but their total potential losses are capped at $200—if the firm is worth less than $100 they don’t take additional losses, the secured bond does instead.

Now that we’ve transformed each potential position into a corresponding option, we can combine them in interesting ways. Let’s assume you short the junior bonds and senior bonds. What is your payoff?

1. You are long a put struck at $300 (short junior bonds)

2. You are short a put struck at $100 (short juniors bonds)

3. You are long a put struck at $100 (short senior bonds)

The net position is thus a long put position struck at $300. Assume the following:

1. Senior bonds have a coupon of 5% and the junior bonds have a coupon of 5.5%

2. Both bonds mature in exactly one year

3. There’s a 1% borrow cost to short both the bonds.

4. Dividends and the risk-free rate are 0%.

5. The junior bonds have a delta (to EV) of -.18, the senior bonds have a delta of -.02, and the stock has a delta of .8.

Calculating Implied Enterprise Value Volatility

Now, let’s do some math. For simplicity, we’re ignoring slippage and trying to make the numbers look nice. In reality, you’d do this trade at a much smaller size. Let’s say you decide to short the entire debt stack for a year.

The net cost of this trade is $19:

1. Shorting the bonds is a $300 inflow in cash immediately

2. You will have to pay $300 in 1 year (assuming the debt is paid off)

3. You will also have to pay $16 in interest in one year and $3 in borrow costs¹

We’ve now calculated that a 1-year put option struck at $300 on EV costs $19. It’s quite simple to then pull out an implied enterprise value volatility value from this. If you plug this number into Black-Scholes, you’ll see it implies a ~40% implied volatility in enterprise value. There’s a few ways to go from here, all of which I can only briefly touch on in the space of this piece.

Intuition from Implied Enterprise Value Volatility

Mapping EV Volatility to Equity Volatility Values

Since equity is levered EV, it’s possible to roughly map EV volatility to equity volatility values (although equity vol can be influenced by other sources).

If you can accurately do this, you can compare what the price of the bonds imply about the business volatility vs. what the equity options (or other derivatives) imply. There could be a trade there (perhaps equity options is pricing vol as significantly higher and you can sell vol there and buy vol through the underlying bonds/stock).² Although, I’ve thought about this briefly and it is a difficult trade. (Keep reading to see one way to do one leg of this trade).

Either way, even backing out an implied enterprise value volatility (IEVV) can give an investor a good sanity check. Does this make sense? Is the underlying value of the business this volatile? Is it this stable? This framing may sanity check some investment decisions.

Using Bond Pricing to Understand Convexity in Equities

This is perhaps complicated, but with interest rates at 0%, Put-Call Parity implies C = (S – K) + P. We know the put option is worth $19. We also know the EV right now is $400 and there is $300 in debt.³ So, the bonds actually imply a fair price for where the stock should trade now. Because stocks have convexity, and the IEVV is rather high (40%), it should trade at more than $100 (which is what the firm is worth now).

Using Put-Call Parity, the bonds imply a fair value for the stock of ($400 - $300) + $19 = $119, showing how the convexity of the stock leads it to trade at a premium to the underlying value. Even though there is $400 in firm value and $300 in debt, equity must trade above that because it benefits from upside moves while losses are limited to zero (equity is a call!). The extra $19 is the convexity premium.

As uncertainty rises, the put option should cost more (bond yield would rise) actually making the equity more valuable, ceteris paribus.⁴ This effect is largest for firms close to insolvency as their vega is largest and they benefit the most from higher implied volatility. Firms that are reliably solvent likely see no real effect from this greater uncertainty and may be hurt as uncertainty is often a negative for a business (the positive delta dominates here). Importantly, the same forces that often increase volatility also likely affect the value of the business directionally meaning it’s hard to isolate implied volatility without impacting value.

Is Private Equity a Disguised Volatility Trade?

Volatility Arbitrage or Private Equity?

You can construct a straddle struck at $300 by selling the bonds and buying the stock, making you long volatility. You are more long volatility the closer the EV of the business is to $300. You will get more long delta (care more about value) the higher the EV goes. This happens because as EV rises, the short bond position gets more and more OTM, meaning its sensitivity to value falls. The equity is more and more ITM and eventually approaches a delta of 1.

Theoretically there’s a relative value volatility trade here. If the equity options (or other derivatives) imply high EV volatility and the short bonds/long equity trade implies low EV volatility, you can do the latter and short the former. If one doesn’t want to sell options, an investor could decide to go long one leg of the volatility trade.

One potential trade would be to look at what volatility the short bonds/long equity straddle implies and decide whether that seems reasonable. If it appears like its underpricing the volatility of the business, perhaps you should buy the “synthetic EV straddle” (short the bonds and buy the stock). Furthermore, if you think the value of the firm is trading too cheaply, perhaps you should short/sell the debt to fund the purchase of the equity. In effect, this is both a volatility trade (buying puts and calls) and a value trade (betting the call is priced too cheaply).

Wait, who does this trade? Private Equity! Private Equity is in the business of selling debt (bonds, loans, etc.) to buy a business. Thus they are long equity (long calls on EV) and short bonds (long puts on EV)! They are, in effect, owning an EV straddle and long EV vol. The more equity they put down, the more ITM they are and the more delta (improving the value of the firm directionally), not vega (increasing the volatility of the business) matters. Obviously, they always want to increase EV, but sometimes it’s easier to increase vol than value. This likely is one intuition as to the aggressive and volatile nature of sponsors when they are near the point of insolvency!

In practice, most buyouts aren’t volatility trades, that was just a provocative title. Sponsors are overwhelmingly long delta—they underwrite value creation and try to improve business fundamentals to achieve superior returns. The equity they buy is usually deep ITM relative to the debt stack once the plan is working. Thus, vega is very small and the position’s P&L is dominated by delta (directional improvements in EV). The more equity they commit or the faster they delever, the deeper ITM they move and the less the long-vol component matters.

Where they are especially long volatility is near the strike. Highly leveraged, cyclical, and distressed deals where companies are close to insolvency often lead a PE investment to behave more option like. As vega rises and delta falls near the point of bankruptcy, sponsors may be more aggressive and can be tempted to increase the volatility, ceteris paribus, of a business. It’s important to note that unlike in a true straddle, sponsors are always long delta, so they don’t want to hurt value. What changes near the strike is tolerance for variance and a tilt toward moves that might buy them time and convexity, preferably without hurting EV. Theoretically, stronger covenants can and should prevent them from playing this game.

I previously argued that the payoff to multi-manager GPs can look like they’re short calls on individual PMs and long a call on the aggregate portfolio. This can create a temptation to raise correlation across their pods. However, certain constraints (stop-losses, high-water marks, pass-through costs, the threat LPs might leave, etc.) can help inhibit this motivation.

The private-equity analogue is similar near the point of insolvency. There’s a theoretical incentive to favor actions that boost volatility. But, tight covenants can provide guardrails that prevent (or increase the cost) of such (potentially value-destructive) action from occurring. With covenants stripped from many debt instruments over the last ~15 years, you’ve seen more aggressive, volatility-inducing sponsor actions, showing why they were necessary in the first place. There are softer constraints as well (lender relationships, reputation, fundraising cycles, LP relationships) that should shift the economic calculus toward value creation rather than simply maximizing vol. In effect, these constraints should make delta (value creation) a lot more important than vega (increasing volatility).

Allocators Sell Puts and Buy Calls?

So, allocators fund private credit (selling puts on EV) and fund private equity (buying calls on EV). This is kind of equivalent to a synthetic long position in the underlying business.

Would they be better off just buying unlevered businesses? If they can successfully find capable businesses for less than the fees they are paying for both sides of this trade, then perhaps, yes.

Conclusions/Takeaways

In this piece I have (hopefully) shown the following:

1. Equity is a call on EV, bonds are like selling a put on EV

2. Junior debt is equivalent to selling a put spread plus a band-activated call on EV. (see Appendix)

3. You can calculate the cost to short a bond (or return from buying) to back out an implied volatility for the enterprise value of a firm

4. This implied enterprise value volatility (IEVV) can be used to sanity-check potential investments

5. Private equity commonly does a “straddle-like trade” by selling bonds to fund equity purchases. However, they should be far more concerned about maximizing business value than maximizing volatility (in most circumstances)

6. Allocators fund both sides: buying calls (private equity) and selling puts (private credit)

Appendix

Debt is 2 Puts (and 2 Calls!)

So, I’ve said equities are equivalent to owning a call option on the enterprise value of a business, with a strike price equal to all the debt. I also said owning a junior bond is more like selling a put spread on the EV of a business.

Well, I’m going to add another corollary to this now. Owning a junior bond is not equivalent to selling a put spread on EV. Instead, it is selling a put spread AND buying a down and in barrier call option that’s only activated under certain circumstances. If the value of the business deteriorates below the face value of all the debt and the business is forced into bankruptcy (down and in), the junior bonds will get converted into equity (the call option).

That’s actually not the whole story. If the value of the business is less than the face value of the senior debt, the senior debt will convert to equity and the junior debt will get zeroed. So, the call option is only activated when the business is valued at greater than the face value of the senior debt, but less than the face value of the total debt burden. And this occurs only when the business is forced to restructure—if it has ample liquidity and the value temporarily falls to this value; the call option is likely not activated.

Perhaps more precisely, if you’ll excuse me, buying a junior bond is equivalent to selling a put spread, buying a down and in call struck at the face value of all senior debt with a barrier equal to the value of all debt, and selling a down and in call struck at the face of value of all senior debt with a barrier equal to the face value of the senior debt. This can be called a KIKO (knock-in, knock-out barrier option). Importantly, these options (debt to equity conversion) are not activated formulaically. They are dependent on a complex web of interactions including the firm’s liquidity, the timing of a bankruptcy/restructuring, the motivations of other creditors, and the bankruptcy judge among other factors. (I’ve previously written about the potential for market participants to manipulate barrier options here, perhaps some of it applies to the restructuring process!)

Rather complicated! Let’s model this briefly for fun. For the below, assume the following capital structure of:

1. $200 of Junior Debt, pays a $20 coupon (10%)

2. $100 of Senior Debt, pays at $5 coupon (5%)

So, in the simpler, put spread model, the bond returns look like Figure 4.

Figure 4: Return of Junior vs. Senior Bonds Relative to Enterprise Value at Maturity

Now, I haven’t been able to figure out how to plot a 2-D graph of how the game theory of creditor/borrower dynamics would play out (and if I could figure it out, maybe I wouldn’t write it publicly?). So, if we’re going to plot the more complicated put spread plus knock-in, knock-out call option we need to assume that bankruptcy/restructuring occurs whenever the entity becomes insolvent. At the time of debt-to-equity conversion, the knock-in, knock-out call option the junior debt holder has looks like this (Figure 5):

Figure 5: Return of KIKO Option at Equitization Relative to Enterprise Value

Figure 5 shows that if the EV is above $100, but less than $300, the senior debt gets paid out (in either cash or new bonds) and the junior debt receives equity. This conversion scales linearly with EV in this range. Now, in Figure 1 we were looking at the return of the bondholders at maturity. Restructuring and equitization of the junior debt often happens before the maturity of the bonds. So, what does the junior debts payoff look like at the original maturity of the bonds (so we’re comparing apples to apples)?

This is actually rather simple! Once the junior debt is converted to equity, the option can no longer be knocked-out. Thus, the payoff is just a call option, and its value is dependent on the EV at the original bond maturity date (which is after the restructuring). Now, we can assume two things here, either: 1) the senior debt got paid in cash in full or 2) the senior debt received new $100 senior debt. To align the charts, let’s just assume the senior debt got new, higher yielding, senior debt with a face value of $100 (Figure 6).

Figure 6: Return of Junior Bonds (Now Equity) Relative to Enterprise Value at Original Maturity Date

Thus, this additional KIKO option has some value and can be rather lucrative to the junior debt holder. The enterprising reader might add that the senior debt also has a call option, theirs just being a knock in one struck at $100. This is true and a similar exercise can be done with the senior debt, but I am trying not to bore my readers. Perhaps this dynamic, the addition of the KIKO, makes the junior debt holders volatility-loving when the entity is insolvent as their KIKO call option matters more than the put they sold on EV.

Thank you for reading until the end! I hope you enjoyed it.

-Eli

1. Technically, some can argue that you should ignore the borrow cost as this is similar to a bid-ask spread. The seller of the put doesn’t receive any of this 1% so maybe it should be irrelevant. But, it is the cost to buy the put, so I decided to keep it here.

2. You won’t be able to do this at the same strike as equity is ~0 at an EV of $300.

3. Perhaps the biggest limitation, and why this is more an intuitive exercise rather than a deep valuation process, is that the intrinsic value of the firm is inherently unknown. We are assuming concrete, known numbers for EV here, but many investors have different estimates and different bounds on these estimates.

4. Precisely, the convexity premium rises which implies the stock rises as long as value stays unchanged. In reality, value often dives when uncertainty rises which might outweigh the higher convexity premium.