**About**

This homework assignment will use arbitrage-goggles to show several dislocations that resulted from oil futures going negative

- It can be used as an interview question to test a junior or mid-level trader’s intuition about options

## The Setup

You are a trader specializing in making markets in oil futures and related derivatives. Your tradeable universe includes:

- listed oil futures

- listed oil options

- an ETF that holds front-month oil futures (like USO)

- The market closes. You observe the following facts:

The ETF share price closes at $6.00, in line with its NAV.

- You walk into the office the next day. All of the prices and facts are unchanged. However you notice:
**The 0-strike put on the futures is****$1.00****bid!**You scratch your head. The zero strike put has a bid?? Understanding that markets don’t present free money so easily, you come to the most likely explanation — oil futures can go negative!

Now the fun begins.

The following questions will slowly help you identify what opportunities might exist. I’ve provided hints that become increasingly strong so if you want to challenge yourself don’t rush for help too soon.

## Questions

The oil future can go negative. What is the next question that immediately comes to mind (besides “why would a price go negative?”)

#### Answer

You should immediately be wondering: can the ETF go negative?
The answer is “no”. There is no reason for a shareholder to sell the ETF for less than zero because a shareholder’s liability is limited. Even if the NAV went negative, a broker cannot show up to your house and say “put up more money for this share you already purchased”.

Identify the arbitrage.

## Hint #1

The familiar hockey-stick payoff diagram for the 0-strike put at expiry

## Hint #2

Since the notional value of the future is $3,000 and the ETF NAV per share is $6.00 then 500 shares of the ETF is the equivalent position to a future.
The ETF price cannot go below 0 so if you own 500 shares, this is your ETF p/l as a function of the future price:

Interesting — another hockey stick!

## Hint #3

Recall from put-call parity that calls can be constructed from puts and vice versa by taking a position in the underlying.
If you own a call option and short the underlying on a 100% delta (sometimes referred to be “short futures 1-to-1 against your long call”), you have replicated a put option on the same strike.
You can demonstrate this in 2 ways:

**The Visual Method**Via financetrain.com:

**The Cash Flow Method**- Put expires worthless. Lose your premium.
**Net p/l:****-$2** - If the stock goes to $45:
- The put is $5 in-the-money but you spent $2 premium for it.
**Net P/L =****+$3** - If the stock goes to $60:
- Call p/l = +$3
- P/L on the future = -$5
**Net p/l:****-$2** - If the stock goes to $45:
- Call p/l= -$7
- P/L on the future = +$10
**Net P/L =****+$3**

## Put option

A future is $55. You pay $2 for the 50-strike put.

If the stock goes to $60:

## Synthetic Put (ie long call, short underlying)

Long a $50 strike call for $7 and short a future at $55. Note, that you are paying $2 for the synthetic put because $2 is the extrinsic value of the in-the-money call.
The put-call parity formula says:

*Put = Intrinsic + call - carry costs We are ignoring carry costs. Intrinsic is -$5 because the put is $5 out-of-the-money and the call is $7.*

*$2 =**-$5 + $7 + 0*

#### Answer

- Since the ETF cannot go below zero but increases in value in-step with the futures then it looks like a
**call option struck at 0!**

- If we buy the ETF and short the future, we have constructed a synthetic put. Let’s compare the payoffs to the real put side by side.

#### At any terminal futures price, you lock in a $1,000 profit!

Given the answer, what needs to happen to close the arbitrage assuming the put continues to trade for $1?

#### Answer

In the arbitrage example, we saw that no matter where the futures went a strategy of buying the ETF for $6 (at its NAV) and shorting the future at $3 guaranteed us a flat or positive p/l.

We saw that the ETF, because it cannot be worth less than 0, went from being a delta-1 product on the price of oil to a call option on the price of oil.

If the ETF trades for its NAV, we are getting the 0-strike put for free…But, we know from the futures market that the 0-strike put has value. If we buy the ETF at its NAV and short the future we are synthetically buying that coveted put for free!

So the ETF needs to trade at a premium to NAV to close the gap.

How much over the NAV should the ETF trade so the real put that’s trading for $1 in the futures option market and the synthetic put trade at the same price?

We can compute this in 2 ways.

- We can simply observe from the prior question that if we bought 500 shares and shorted the future we always made $1,000 profit. If we paid an extra $2 for those 500 shares then that $1,000 profit would disappear and our p/l scenarios would match the payoff from buying the real put for $1. So the shares should trade up to $8 to close the arbitrage, representing a 33% to the $6 NAV

- We can see that to translate a futures price to an ETF price we simply multiply by 2 since a $3 future corresponds to a fair NAV of $6 for the ETF. The premium to NAV can be seen exactly as the extrinsic value of the 0-strike put. Since that put is trading for $1 in the futures market, then that extrinsic premium needs to be $2 in the ETF market. The ETF should trade for $2 over NAV.

**The big takeaway is that the premium to NAV represents the extrinsic value of the 0-strike put!**

**Bonus Question**

#### See inside

How would you expect the ETF’s premium to NAV to vary with the futures price?

## Answer

Remember, the premium to NAV represents the extrinsic value of the 0-strike put.

- Oil futures are high If the oil future is trading for $90, the 0-strike put will be worthless. It will have no delta either. If oil goes from $90 to $85 that put is not going to budge. Still worthless. If the put is so far out-of-the-money that it’s worthless, the ETF will not warrant a premium to NAV either.

- Oil futures are extremely negative If oil is trading for -$90 the 0 strike call is likely worthless. So the ETF NAV will be zero AND it will warrant no premium to NAV or extrinsic value.

The closer to zero oil futures are, the higher the extrinsic value of the 0-strike (the option premium of any out-of-the-money strike is pure extrinsic value regardless of whether that option happens to be a put or call).

#### The delta

The ETF is a call option on the price of oil.

When oil futures are high, the call option trades like a deep-in-the-money 100% delta call moving in lockstep with the futures.

When the oil future is deeply negative, that ETF has zero delta, like a way out-of-the-money call. It’s worthless whether the future moves up or down a dollar.

How about when the future is near zero?

The ETF will have a delta somewhere in-between. Think of it this way:

If the oil future goes from 0 to $50 along the way, the ETF premium to NAV will go from a large number (in our example the $2 premium was 25% of the $8 ETF non-arbitrage price) to zero as the 0 strike goes further out-of-the-money. That means as the oil futures rallied the ETF went up mechanically because it holds futures BUT the premium partially eroded. This erosion offsets part of the gain in the ETF and what you’ll experience is an ETF that has less than 100% sensitivity to the future!

Keep this in mind for the ensuing discussion.

## Advanced topic

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