# 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
$P_{O,L}=S_{L}*{ \bigg( \dfrac{1+r_{Q,b}}{1+r_{B,l}} \bigg) }^T - M *[{(1+r_{Q ,b })}^T -1]$
Short
$P_{O,S}=S_{S}*{ \bigg( \dfrac{1+r_{Q,l}}{1+r_{B,b}} \bigg) }^T + M *[{(1+r_{Q , l})}^T -1]$
Contango 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
$r_{Q,b}=10.10\%$
​, 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:
$P_{O,L}=100.10*{ \bigg( \dfrac{1+0.1010}{1+0.0290} \bigg) }^{0.25} - 50 *[{(1+{0.1010})}^{0.25} -1] = 100.59 \: DAI$
• 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:
$P_{O,S}=99.90*{ \bigg( \dfrac{1+0.0990}{1+0.0310} \bigg) }^{0.25} + 50 *[{(1+{0.0990})}^{0.25} -1] = 102.70\: DAI$

### Demonstration

#### Long

Let's consider that a trader wants to buy 1 expirable (the numerical values are taken from the above example). In this demonstration, we will present the steps to replicate the cash flows of a expirable position. Let's figure out the price of the expirable, 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.
$0.9929 \: ETH$
2. 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 expirable (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 expirable, 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

Given the margin ratio
$MR$
:
• 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
$M=50 \:DAI$
• 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
$M=50 \:DAI$
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
$P_{O,L}=S_{L}*{ \bigg( \dfrac{1+r_{Q,b}}{1+r_{B,l}} \bigg) }^T * \dfrac{1}{1+MR*[{(1+r_{Q ,b })}^T -1]}$
Short
$P_{O,S}=S_{S}*{ \bigg( \dfrac{1+r_{Q,l}}{1+r_{B,b}} \bigg) }^T * \dfrac{1}{1-MR*[{(1+r_{Q ,l })}^T -1]}$

### Example

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:
$P_{O,L}=100.10*{ \bigg( \dfrac{1.1010}{1.0290} \bigg) }^{0.25} * \dfrac{1}{1+0.5*[{(1.1010)}^{0.25} -1]}=100.68\:DAI$
• 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:
$P_{O,S}=99.90*{ \bigg( \dfrac{1.0990}{1.0310} \bigg) }^{0.25} * \dfrac{1}{1-0.5*[{(1.0990)}^{0.25} -1]}=100.31\:DAI$