BitMEX Quanto, Linear & Inverse Futures Contracts

This formula can tell you how much “bang for your buck” you’re getting after the dust settles. Now, let’s go over this first without going into the maths of the formula and explain it.

If the Euro appreciates against the dollar, you’re in luck. You’re making extra money on top of whatever the property is making.

If your European property’s value jumps up by 3% while the Euro appreciates against the dollar by 5%, you’re not just up by 3%, but you’re up 8% when you convert those euros back to dollars.

Considering the property’s performance and the Euro, this shows you how much cash you could pocket in dollars.

Dealing with international property investing is more than just putting up that “For Sale” sign; it’s also about the currency game.

Now, Let’s Get into the Details of the Maths

This equation estimates the return an investor can expect from an investment. It takes into account the change in the asset’s value in local currency (r) and the change in the exchange rate (f𝑥).

To explain it in our example:

Let’s assume a US investor is interested in a European property valued at €100 million. In this scenario, the investor is concerned with the changes in the property’s value and the Euro’s value against the US dollar. The European property is appreciated from €100 million to €103 million.

In academic jargon, this phrase would be included in research papers on finance.

For example, to the US investor, a property with a current worth, V_0, of €100 million can be viewed as having a return R, based on the next period value V_1, as shown in this equation:

(V_1- V_0)/V_0 = {[V_0 * [1 + r] * [1 + fx]] – V_0 ≈ r + fx

I will make the equation more readable with LaTeX.

Equation 40.1

Note phrase:

“V_0, of €100 million can be viewed as having a return R, based on the next period value V_1”

V_0 is €100 million, and V_1 is €103 million, yet we don’t know “r” yet. So, let’s figure that out.

The total percentage gain or loss in the value of the total investment is denoted by “r,” which tells us how much our investment is worth compared to when we first invested.

We need to find out how much the property’s value changed if we bought it for €100 million and it’s now worth €103 million. The change in value (€103M — 100M) is €3 million.

€103 million — € 100 million = €3 million

Next, we want to see how this change is relative to what we paid, so we divide the €3 million by the original €100 million.

(103 million — 100 million) / 100 million

In this case, dividing €3 million by €100 million gives us 0.03, but to turn this into a percentage, we multiply by 100, which gives us 3%.

Did you waste my time reading to give me a basic explanation of how to calculate the percentage gains/losses? Yes, yes, I did.

We have confirmed that our gain on the property is 3%.

Suppose we are a US investor buying European property at €75 million with our dollars. We are hoping for a 4% increase in the property’s dollar value over the year.

Let’s simplify the math and assume €75 million is worth $100 million. We purchased this European property worth €75 million and paid $100 million.

What happens to our dollars if the Euro appreciates or depreciates against the dollar?

Scenario 1 (Euro Appreciates):

Property value up 4%
Property value: 75 * (1 + 0.04) = €78M
Euro appreciation: 2%

Now, to find the “Next-period property value in dollars” denoted as P_next:

In our case:

By plugging in the numbers, we get:

P_next = (75 × (1 + 0.04)) × ((100÷75))×(1+0.02) = $106.08M

Scenario 2 (Euro Depreciates):

Property value increase: 4%
Next-period property value: (€75M * 1.04) * 0.98 = €76.44M
Euro depreciation: -2%
Next-period property value in dollars:

Now, to find the “Next-period property value in dollars” denoted as P_next:

In our case:

By plugging in the numbers:

P_next = (75×(1+0.04)×(100÷75)×(1−0.02)) = $101.92M

So, you managed to purchase property overseas, and even if the local price of the property doesn’t move, the value back home can swing just due to the currency exchange rates.

You’ve bought your property in Europe, and the Euro might fall against the dollar (EUR/USD down), and suddenly, your property isn’t worth as many dollars as before.

Your investment worth in your “home currency” (let’s say dollars) can jump around even if the price of your property in the “local currency” (let’s say Euro) has stayed the same.

It’s not just about property value. It’s also about the foreign exchange game you’re participating in. Without heading, you’re pretty much partly gambling.

Key traditional currency risk assumptions of cross-border investing

When investing in real estate across borders, it’s important to be aware of currency risk since you’re indirectly playing the foreign exchange market.

A common belief in currency risk management is that fluctuations in the value of the investor’s home currency can influence the value of their foreign investment.

This means that if the currency in the country where you have invested (for example, the Euro) strengthens or weakens against your home currency (for example, the US dollar), it will affect your returns.

For instance, if an American investor’s European asset appreciates by 4% in USD terms, traditional wisdom suggests this is independent of the USD’s market performance. That viewpoint implies that owning foreign assets inherently involves currency risk.

However, this might not hold in significant currency devaluation or inflation scenarios.

In such situations, the nominal value of real assets, when calculated in the inflating currency, tends to mirror the high inflation rate.

begin{align*}
frac{(V_1 – V_0)}{V_0} &= left{ frac{[V_0 times (1 + r)] times [1 + fx] – V_0}{V_0} right} approx r + fx, \
text{where} quad & V_0 text{ is the current value of the property in euros,} \
& V_1 text{ is the value of the property in the next period,} \
& r text{ is the percentage change in the property value in euros,} \
& fx text{ is the percentage change in the value of euros expressed in terms of US dollars.} \
& text{The equation is an approximation because it ignores the cross-product } r times fx. \
end{align*}

Remember the equation above, which helps us understand the combined effect of the property’s value change and the currency exchange rate on your investment’s overall value.

So picture this: you’ve got a property where the local currency, say the Euro, is taking a nosedive compared to the US dollar. In finance, we’d expect the return of the property (that’s our friend “r” in Equation 40.1) to mirror this.

When investing in properties across different countries, you’re playing a dual game of real estate and foreign exchange markets. Here’s where Equation 10.1 becomes relevant.

Equation 10.1

begin{equation}
sigma_p = frac{partial sigma_p}{partial w_1} w_1 + ldots + frac{partial sigma_p}{partial w_N} w_N
end{equation}
Where:
begin{itemize}
item $sigma_p$ is the standard deviation of the portfolio’s returns, representing total risk.
item $w_i$ is the weight of the (i^{th}) asset in the portfolio.
item $frac{partial sigma_p}{partial w_i}$ is the partial derivative of the portfolio’s standard deviation with respect to the weight of the (i^{th}) asset, representing how sensitive the portfolio’s risk is to changes in the weight of the (i^{th}) asset.
item The summation runs from 1 to (N), where (N) is the total number of assets in the portfolio.
end{itemize}

This equation helps us determine how our whole investment is based on the risk of each investment (like properties and currency holdings) and how much of our money is in each part.

In real case scenarios, if the currency in which an investment is made devalues significantly, say by 25%, the property’s value in that currency may increase substantially.

This helps to maintain the property’s real value. Traditional models often overlook this possibility.

It’s not just about the property’s price tag; it’s also about how currency values and economic conditions in the investment country can affect your returns.

Accessing foreign assets with futures and Quanto futures

In finance, equity futures contracts are financial instruments where investors agree to buy or sell a specific stock index or individual stock at a predetermined price on a future date.

I assume the reader has a basic understanding, but let’s review it just in case.

Equity futures are contracts you make with someone to buy or sell stocks at a set price on a future date. You don’t actually get the stocks; you settle the bet with cash. Traders use these contracts to protect against risks or to try to make money from expected price changes. They’re useful because you can trade a lot with a little bit of cash with leverage.

So, at expiration, the buyer and seller settle the difference between the contract and market prices in cash. There’s no physical exchange of the actual stocks or indices. On the other hand, oil futures often involve physical delivery.

(Remember during the COVID pandemic, when oil futures ended up trading negatively because almost nobody wanted physical oil delivery at their door).

These contracts are generally linked to the performance of indexes such as the SPX500. They are traded in the currency of the underlying assets. For instance, if the futures contract is based on the SPX500 index in the U.S., it would be traded in U.S. dollars.

How does that work? It’s quite straightforward.

Each contract has a specific dollar value assigned to each point movement of the index. Consider a contract valued at $250 per point. If the SPX500 index rises by 2 points, a market participant holding a “long” position (bet on the index rising) would gain $500.

That profit is calculated as:

contract value * points = PNL
$250 * 2 = $500

A Quanto derivative takes this concept a step further.

Who coined the term “Quanto derivatives”, and where did this “Quanto” originate from?

Now, it’s worth asking, “Who came up with this genius idea?” Well, as far as I know, it’s a mystery. The term “quanto” is believed to have first appeared among a team of traders at Salomon Brothers. Salomon Brothers was famous for creating new ways to manage currency risk. At Salamon, they used “quanto” to describe derivatives linked to foreign currency indexes.

The story originated from a hopeful interviewee drenched from a 6-block walk and walked into Cooper Neff’s office in Philadelphia. Cooper Neff was a renowned proprietary trading firm known for its innovative strategies in options trading and other financial derivatives. They were also known for their grueling, rigorous interviews.

During the interview, the candidate was presented with a challenging question that included Siegel’s Paradox.

The question was something like this: The USD/DM (U.S. Dollar/German Deutsche Mark) exchange rate is 1. It has a chance of rising to 1.50 or dropping to 0.50 with equal probability. What’s the fair value?

Simple math suggests the fair value is 1.00 USD.

Now, let’s flip or invert it to DM/USD. The rate could either hit 0.67 (1/1.5 = 0.67 ) or soar to 2.00 (1/0.5 = 2). When you average these out, you get an expected value of 1.335 DM, suggesting a fair value of 0.75.

What gives?

The fair value is the average of the two possible future rates. Since the average (expected value) is 1 DM per USD, the fair value under the given assumptions and market conditions is also 1 DM per USD.

Let’s go over the math a bit.

USD/DM
Current rate: 1 USD = 1 DM
Possible future rate: 1.50 DM
Possible future rate: 0.50 DM
USD/DM Expected value: (1.5 DM + 0.5 DM) / 2 = 1 (straight forward)
Fair value: (1.50 + 0.50) / 2 = 1

Note: Fair value is average of the two possible futures rate

DM/USD (inverted)
1 USD = 1.5 DM
Possible future rate: 1 / 1.50 ≈ 0.67 DM/USD
Possible future rate: 1 / 0.5 ​≈ 2 DM/USD
DM/USD Expected value: (0.67 USD + 2 USD) / 2 = 1.335 USD

Note: To determine the fair value of USD/DM
Calculate the expected value of DM/USD by inverting the 1.335 USD

Fair value DM/USD: 1/1.335 ≈ 0.75

Now, someone new to trading may wonder, “What? How can the same currency pair have two different values?”

I understand where the insight came from. This is not a paradox but just a matter of perspective. USD/DM and DM/USD represent the same relationship differently. They are not separate but just different units of measurement. This means that the rates are priced in different units. Therefore, there is no actual inconsistency or trading opportunity.

There is no financial opportunity for purchasing power arbitrage, so there is no real paradox; the difference comes from how we mathematically interact with these rates, and there is no “free lunch.”

When looking at currency rates, we found that complex math is sometimes misused in finance. This includes misusing concepts like Jensen’s inequality, which can lead to misunderstandings in financial theories and practices. In our case, when we averaged rates like USD/DM and DM/USD and converted them, we applied a non-linear function.

We calculated the expected value by taking the average of future rates and then finding its inverse through division, which involves non-linear transformation.

Fast-forward a few years. The Chicago Mercantile Exchange (CME) introduced innovative currency cross-rate contracts priced in USD, like DM/JPY. This was an interesting development but didn’t quite take off as hoped.

However, we are familiar with the Base and Quote currency.

What are the Two Main Types of Quanto?

There are two main types of quantos: two-currency quantos and single-currency quantos.

The Two Currency Quanto: Imagine you’re investing in something priced in euros, but you want your return in dollars. This type is initially valued in euros. When it’s time to cash in, it’s converted into dollars. For this type, where conversion into the domestic currency happens, the pricing tends to be more straightforward. That’s because the conversion rate is known in advance, making it easier to calculate the payoff.

The Single-Currency Quanto: This one is a bit different. Let’s say you’re in the Japanese stock market and eyeing the Nikkei Index. With a single-currency quanto, even though your investment tracks the Nikkei, you’ll get your payout in your good old dollars. No conversion is needed. This type of pricing is more complex, where the payout remains in the domestic currency without any conversion. In this case, the pricing model takes into account any potential fluctuations in exchange rates during the contract’s life.

Quanto options basic

Take quanto options, for example. This is a special type of option contract where the payoff is converted into a different currency at a predetermined exchange rate. Unlike regular options, which pay off in the currency of the underlying asset, quanto options allow investors to receive payoffs in a currency of their choice, using a fixed exchange rate agreed upon at the contract’s inception.

Currency conversion: Quanto options are an effective way for market participants to gain exposure to foreign markets without being exposed to the associated currency risk. These options provide payoffs in a currency different from the underlying asset’s currency.
Fixed exchange rate: When entering into a contract, the exchange rate used to convert the payoff from the underlying asset’s currency to the chosen currency is determined and fixed. This fixed rate remains the same regardless of any changes in the market exchange rates at the time of payout.
Floating-rate quanto options have no predetermined exchange rate. They offer investors flexibility but also expose them to currency risk. Instead, the exchange rate used to convert the payoff is determined at the time of settlement.
Hedging against currency risk: Quanto options allow investors to hedge against fluctuations in currency exchange rates by locking in the exchange rate.
Payoffs in preferred currency: Investors have the flexibility to receive payoffs in their preferred currency, which can be more convenient as it aligns with their risk management.

A quanto option is different from a regular option in that it converts the payoff into a different currency using a fixed exchange rate set at the contract’s start. In contrast, a regular option has a straightforward payoff in the underlying asset’s currency.

Imagine a regular option that pays out $4 at the end of its term. Imagine a quanto option with the same underlying asset and conditions but a different way of handling the payoff.

In a quanto option, the $4 payout is converted to euros, but not at the current market exchange rate. Instead, it is adjusted to a predetermined exchange rate, for example, $1.25 per Euro.

In a quanto option, the payout of $4 is converted to euros, but not at the current market exchange rate.

Instead, it is adjusted to a predetermined exchange rate, for example, $1.25 per Euro. It is important to note that the $4 payout is not directly converted to euros at the $1.25 per euro rate stated in the terms.

Rather, the payout is converted to an equivalent value in euros, meaning the investor receives 4 euros. If you convert 4 euros to dollars using the predetermined exchange rate of $1.25 per Euro, it would equal $5.

4 * 1.25 = 5

Instead of a $4 return, it now equals a $5 return.

The math for quanto options is more complex than for regular options because it involves two currencies and an FX risk premium.

This mechanism offers a unique advantage. It shields the investor from the unpredictability of exchange rate fluctuations.

In another case, imagine the dollar depreciates 10% against the Euro, meaning that if the current exchange rate were $1.10 per Euro, it would shift to $1.21 per Euro.

1.10 * 1.1 = 1.21

If an investor holds a non-quanto fixed-rate investment at $1.25 per Euro, any dollar depreciation will affect the returns when converting to euros.

Despite the exchange rate fluctuating to $1.21 per Euro, the Quanto derivative with a fixed rate remains anchored at the predetermined $1.25 per Euro, making it unique. This means the Quanto contract’s investor is protected against unfavorable market fluctuations caused by foreign exchange movements.

Let’s look at a practical example to understand this better. If the quanto derivative results in a payoff of 10 euros with a floating rate under normal circumstances, this would be converted into dollars at the prevailing market rate.

Given the dollar’s 10% depreciation, the investor would normally receive $12.10 (10 euros * 1.21 per Euro) instead of $11 (10 euros * $1.10 per Euro) before the depreciation.

However, with the quanto contract’s fixed rate of $1.25 per Euro, the investor still receives $12.50 (10 euros * $1.15) despite the shift in foreign exchange.

By locking in the exchange rate from the outset, the investor’s return is influenced by the underlying asset’s performance and the predetermined exchange rate.

Quanto derivatives offer a sophisticated way to hedge currency risk while allowing investors to tap into foreign asset markets.

How Do Quanto Futures Contracts Work?

We are all familiar with futures contracts. Futures contracts are commonly known agreements that allow parties to buy or sell a stock index at a predetermined price on specific dates. These contracts can be utilized to hedge or speculate on the price of a particular asset.

Futures contracts allow investors to manage risk and maximize returns. They can hedge investments to protect against price fluctuations or speculate on an asset’s price to profit from them.

In a Quanto futures contract, the underlying asset is usually priced differently than the investor’s home currency. For example, a Quanto future contract in the U.S. might be based on the Euro Stoxx 50 index, which is denominated in euros, but the contract’s payouts are in U.S. dollars.

In a Quanto futures contract, the exchange rate is determined and locked in when the contract is created. This rate stays the same throughout the contract’s life.

This fixed rate calculates the value of gains or losses in the underlying asset, such as a stock index, in the currency used for contract settlement.

Imagine you’re in the U.S. and enter a Quanto futures contract based on a European stock index like the DAX, measured in euros.

The fixed exchange rate for this contract is set at 1.2 USD for every euro.

If the DAX goes up by 100 euros, you would normally expect to get the equivalent of that increase in your own currency (USD). Still, with a quanto futures contract, this 100 euro gain is automatically converted into USD at the fixed rate of 1.2, regardless of the current market exchange rate. So, you’d get $120 (100 euro * 1.2)

If you opt for a quanto future trade, you can lock in the exchange rate at 1.2 and receive payment in dollars, even if the euro decreases in value against the dollar.

For example, if the EUR/USD exchange rate falls by 10%, you will still earn $120 on your trade. However, if you were trading a non-quanto future, you would still earn 100 euros, but the dollar value would be 10% less due to the decline in the EUR/USD exchange rate. So, instead of earning $120, the 100 euros would only be worth $108.

By trading quanto futures, you know exactly how foreign gains or losses will translate into your own currency, eliminating the uncertainty of exchange rate fluctuations.

So, in a regular scenario, you face two risks if you trade the Euro Stoxx 50 index, DAX, or Nikkei 225. The index performance risk and the currency exchange risk. The fixed exchange rate neutralizes the currency risk in Quanto futures. Your payoff is still linked to the foreign asset but converted into your local currency using this fixed rate.

You’re no longer exposed to the risk of currency fluctuations; your risk is now purely based on the asset’s performance, adjusted by a fixed currency rate.

How Does a Quanto Futures Multiplier Work?

Notably, most futures contracts are “quantity-adjusted” and use “multipliers” to adjust for the quantity.

A contract’s settlement has a multiplier. For example, a wheat future contract might represent 5000 bushes of what, or an S&P500 future contract might be 250 times the index value.

These multipliers merely scale the size of a single contract to a convenient level. In simpler terms, multipliers make the size of the contract more practical and sizable for trading. They don’t change the fundamental nature of the risk.

Quanto futures contracts also use a form of quantity adjustment, but it’s different. Instead of adjusting the physical quantity of size, they adjust the payoff using a currency exchange rate.

For instance, in a Quanto futures contract based on the Nikkei 225 index, the gains or losses in the index are converted into another currency, like USD, using a predefined exchange rate.

This rate converts the payoff from the underlying (yen) currency to the settlement currency (USD).

For example, if a Quanto futures contract is based on the Nikkei 225 index but settles in USD, and the fixed exchange rate is 0.009 (meaning 1 Japanese Yen equals 0.009 USD), then 0.009 acts as a multiplier that determines how much a gain or loss in Yen is worth in USD.

As explained earlier, Quanto Futures contracts take it a few steps further. Quanto futures are a type of financial derivative that combines elements of futures contracts with currency exchange rates.

Quanto Futures stands out from other futures contracts due to its exceptional feature of including currency exchange rates in the contract. This means that the contract’s performance is not solely dependent on the underlying asset, such as a stock index, but also on the fluctuations in an exchange rate.

To Quanto or Not to Quanto? Does the Perfect Hedge Exist?

The idea of the “perfect hedge” is to protect your investment from losses by managing your position size and protecting yourself against price or currency changes.

However, this ideal is not the reality for most portfolio managers, who tend to keep the share quantity fixed. Instead, they only adjust their hedge against currency changes.