> For the complete documentation index, see [llms.txt](https://contango-2.gitbook.io/exchange/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://contango-2.gitbook.io/exchange/protocol/protocol-pricing/position-opening.md). # 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](/exchange/protocol/theoretical-pricing.md)).
SidePrice to open a position
LongP_{O,L}=S_{L}*{ \bigg( \dfrac{1+r_{Q,b}}{1+r_{B,l}} \bigg) }^T - M *[{(1+r_{Q ,b })}^T -1]
ShortP_{O,S}=S_{S}*{ \bigg( \dfrac{1+r_{Q,l}}{1+r_{B,b}} \bigg) }^T + M *[{(1+r_{Q , l})}^T -1]
{% hint style="info" %} Contango protocol provides a price improvement compared to the theoretical formulas presented in [theoretical pricing](/exchange/protocol/theoretical-pricing.md): * 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$$, the price to open a short position is at a higher price, i.e. more favourable to the trader. {% endhint %} ### Example Let's consider a contract on ETHDAI expiring in 3 months ($$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$$. ## Pricing with margin ratio ### Formulas Given the [margin ratio](/exchange/resources/glossary.md#margin-ratio) $$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$$ in the main [pricing formula](/exchange/protocol/protocol-pricing/position-opening.md) to open a position, we find new expressions depending on the collaterisation ratio $$MR$$ :
SidePrice to open a position
LongP_{O,L}=S_{L}*{ \bigg( \dfrac{1+r_{Q,b}}{1+r_{B,l}} \bigg) }^T * \dfrac{1}{1+MR*[{(1+r_{Q ,b })}^T -1]}
ShortP_{O,S}=S_{S}*{ \bigg( \dfrac{1+r_{Q,l}}{1+r_{B,b}} \bigg) }^T * \dfrac{1}{1-MR*[{(1+r_{Q ,l })}^T -1]}
### Example Let's consider a contract on ETHDAI expiring in 3 months ($$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$$