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PROTOCOL

Position opening

Pricing with margin

Formulas

The formulas below present the price at which a trader could open a long position at a price

$P_{O,L}$

or a short position at a price $P_{O,S}$

with an initial margin $M$

(other notations have been introduced in theoretical pricing). Side

Price to open a position

Long

β

$P_{O,L}=S_{L}*{ \bigg( \dfrac{1+r_{Q,b}}{1+r_{B,l}} \bigg) }^T - M *[{(1+r_{Q ,b })}^T -1]$

βShort

β

$P_{O,S}=S_{S}*{ \bigg( \dfrac{1+r_{Q,l}}{1+r_{B,b}} \bigg) }^T + M *[{(1+r_{Q , l})}^T -1]$

βContango protocol provides a price improvement compared to the theoretical formulas presented in theoretical pricing:

- Since$M *[{(1+r_{Q ,b })}^T -1] > 0$, the price to open a long position is at a lower price, i.e. more favourable to the trader.
- Since$M *[{(1+r_{Q , l})}^T -1] > 0$, the price to open a short position is at a higher price, i.e. more favourable to the trader.

Example

Let's consider a contract on ETHDAI expiring in 3 months (

$T=1$

) and where the traders posts $50 \:DAI$

as margin:- Given one could borrow DAI at a yearly fixed rate of$r_{Q,b}=10.10\%$β, lend ETH at a yearly fixed rate$r_{B,l}=2.90\%$and buy ETH on the spot market at$S_{L}=100.10\:DAI$then the price at which a trader could open a long a position is:

β

$P_{O,L}=100.10*{ \bigg( \dfrac{1+0.1010}{1+0.0290} \bigg) }^{0.25} - 50 *[{(1+{0.1010})}^{0.25} -1] = 100.59 \: DAI$

β- Given one could borrow ETH at a yearly fixed rate of$r_{B,b}=3.10\%$β, lend DAI at a yearly fixed rate of$r_{Q,l}=9.90\%$and sell ETH on the spot market at$S_{S}=99.90\:DAI$then the price at which a trader could open a short position is:

β

$P_{O,S}=99.90*{ \bigg( \dfrac{1+0.0990}{1+0.0310} \bigg) }^{0.25} + 50 *[{(1+{0.0990})}^{0.25} -1] = 102.70\: DAI$

βDemonstration

Long

Let's consider that a trader wants to buy 1 futures (the numerical values are taken from the above example). In this demonstration, we will present the steps to replicate the cash flows of a futures position. Let's figure out the price of the futures, i.e. the DAI money needed, to get 1 ETH at expiry:

1. To receive 1 ETH at expiry, the trader needs to lend

$\dfrac{1}{(1+r_{B,l})^T} \: ETH$

, i.e. $0.9929 \: ETH$

β2. To get that ETH, the trader first swaps

$\dfrac{S_{L}}{(1+r_{B,l})^T} \: DAI$

, i.e. $99.39 \: DAI$

.3. Since the trader has already some margin, she only needs to borrow

$\dfrac{S_{L}}{(1+r_{B,l})^T} - M \: DAI$

, i.e. $49.39 \: DAI$

.4. The debt

$D$

the trader owes at expiry (principal + interest) is: $D = [\dfrac{S_{L}}{(1+r_{B,l})^T} - M] * {(1+r_{Q , b})}^T \: DAI$

, i.e. $D=50.59 \: DAI$

.5. Hence, the money needed to receive 1 ETH at expiry is the sum of the debt and the margin provided:

$[\dfrac{S_{L}}{(1+r_{B,l})^T} - M] * {(1+r_{Q , b})}^T + M \: DAI$

or $S_{L}*{ \bigg( \dfrac{1+r_{Q,b}}{1+r_{B,l}} \bigg) }^T - M *[{(1+r_{Q ,b })}^T -1] \: DAI$

, i.e. $100.59 \: DAI$

.Short

Let's consider a trader who wants to sell 1 futures (the numerical values are taken from the example above). This means she would give 1 ETH at expiry, let's figure out the steps and how much money she would need to receive at expiry:

1. The trader will give 1 ETH at expiry to reimburse a debt. Hence the trader borrows

$\dfrac{1}{(1+r_{B,b})^T} \: ETH$

, i.e. $0.9924 \: ETH$

.2. The trader swaps the ETH to get

$\dfrac{S_{S}}{(1+r_{B,b})^T} \: DAI$

, i.e. $99.14 \: DAI$

.3. The trader lends the DAI from the swap and her margin. At expiry the trader receives an amount

$L=[\dfrac{S_{S}}{(1+r_{B,b})^T} + M] *{(1+r_{Q , l})}^T \: DAI$

, i.e. $L=152.70 \: DAI$

.4. The amount of money the trader will receive at expiry, which is also the price of the futures, is the difference between the amount L and the margin:

$[\dfrac{S_{S}}{(1+r_{B,b})^T} + M] *{(1+r_{Q , l})}^T - M \: DAI$

or $S_{S}*{ \bigg( \dfrac{1+r_{Q,l}}{1+r_{B,b}} \bigg) }^T + M *[{(1+r_{Q , l})}^T -1] \: DAI$

, i.e. $102.70 \: DAI$

.Pricing with margin ratio

Formulas

- the margin for a long position could be expressed as$M = MR*P_{O,L}$, e.g. if the price to open a long position is$P_{O,L}=100 \:DAI$and if the trader wants a margin ratio of 50%, then the required margin is$M=50 \:DAI$β
- the margin for a short position could be expressed as$M = MR*P_{O,S}$, e.g. if the price to open a short position is$P_{O,S}=100 \:DAI$and if the trader wants a margin ratio of 50%, then the required margin is$M=50 \:DAI$β

Replacing the margin

$M$

in the main pricing formula to open a position, we find new expressions depending on the collaterisation ratio $MR$

: Side

Price to open a position

Long

β

$P_{O,L}=S_{L}*{ \bigg( \dfrac{1+r_{Q,b}}{1+r_{B,l}} \bigg) }^T * \dfrac{1}{1+MR*[{(1+r_{Q ,b })}^T -1]}$

βShort

β

$P_{O,S}=S_{S}*{ \bigg( \dfrac{1+r_{Q,l}}{1+r_{B,b}} \bigg) }^T * \dfrac{1}{1-MR*[{(1+r_{Q ,l })}^T -1]}$

βExample

Let's consider a contract on ETHDAI expiring in 3 months (

$T=1$

) where the trader puts a $50\%$

MR:- Given one could borrow DAI at a yearly fixed rate of$r_{Q,b}=10.10\%$, lend ETH at a yearly fixed rate$r_{B,l}=2.90\%$and buy ETH on the spot market at$S_{L}=100.10\:DAI$then the price at which a trader could open a long a position is:

β

$P_{O,L}=100.10*{ \bigg( \dfrac{1.1010}{1.0290} \bigg) }^{0.25} * \dfrac{1}{1+0.5*[{(1.1010)}^{0.25} -1]}=100.68\:DAI$

β- Given one could borrow ETH at a yearly fixed rate of$r_{B,b}=3.10\%$, lend DAI at a yearly fixed rate of$r_{Q,l}=9.90\%$, and sell ETH on the spot market at$S_{S}=99.90\:DAI$then the price at which a trader could open a short position is:

β

$P_{O,S}=99.90*{ \bigg( \dfrac{1.0990}{1.0310} \bigg) }^{0.25} * \dfrac{1}{1-0.5*[{(1.0990)}^{0.25} -1]}=100.31\:DAI$

βLast modified 2mo ago

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Pricing with margin

Formulas

Example

Demonstration

Pricing with margin ratio

Formulas

Example