Understanding Pseudo Delta Neutral
Last updated
Last updated
Introduction
Pseudo Delta Neutral (PDN) Liquidity Provisioning (LP) is a technique to initially take no ‘price’ exposure to a specific token while still providing liquidity and earning ‘yield’ from providing liquidity on a Decentralized Exchange (DEX).
TLDR
Max profit on PDN LP strategies occur when the SOL price is exactly at the price we entered our position. As long as the expected yield earned from trading fees is greater than the borrow rate for our leverage/debt, we expect positive profits at or near that starting price. However, profits can become losses when the price of SOL moves away from our starting price, especially when it does so quickly, before we have had time to earn yield from fees on our LP position.
No yield is free of risk.
Introduction
Delta is a common risk metric when valuing options within finance. Delta refers to the price change of the option when the price of the underlying asset changes by 1 dollar. For example, when an option on the S&P 500 has a 0.5 delta, when the S&P 500 goes up (down) by 1 dollar, the option price will go up (down) by roughly, but not exactly, 50 cents.
The reason we call this ‘pseudo’ delta neutral, is because when providing liquidity (just like options!) delta will change as the price changes (this is called ‘gamma’). As you will see in the stylized example below, the only time the delta is truly neutral (0 delta=0 price exposure), is when the price of the liquidity pool is the same price as when you entered the position. Said another way, the only time you will not have price exposure to the risky asset is when the risky asset’s price is the same as when you entered the position.
Also, an extremely important point is…. The price of max profit for your position is also the same price as when you entered the position.
For a detailed description of PDN, see . This article goes through PDN on a Constant Product AMM (CPMM), not a Concentrated Liquidity AMM (CLMM/CLAMM) like the example below, but the logic is similar. For a primer on CLAMM Versus CPMM look .
Example Setup
Let’s go through a stylized example of a PDN LP position and see our expected profit/loss, risky token exposure, and empirical delta and gamma.
We set up a position when the price of SOL is 200 USDC. We start with 100 USDC and no SOL as our equity position, so our initial equity is 100 USDC.
We take a 3x levered position, so we borrow 2x our initial 100 USDC, for a total borrow of 200 USDC and total position size of 300 USDC. We borrow 0.75 SOL (150 USDC worth of SOL) and 50 USDC. We then LP that 0.75 SOL (150 USDC worth of SOL) and 150 USDC. We are using a CLAMM like Orca or Raydium. We set our upper and lower bounds to roughly 5% above and below the starting 200 price. The lower bound is roughly 190.5 and the upper bound is 210.
We also need to make assumptions on the borrow rate of our debt and the expected yield earned from LPing on the CLAMM. We assume a 40% APY borrow rate that translates to 0.0922% Daily borrow rate and 1.8% daily yield from LPing in our chosen band. Note it's important that the expected yield from fees/LPing is greater than the expected borrow rate, or you can have negative returns on a PDN LP strategy.
We simulate what the position will look like at SOL prices of 180 to 220, or -10% to +10% from the starting 200 price.
Finally, we look at holding this position for 1 day and 7 days.
Below is a graph of the PnL as a % on the y-axis versus the % change in SOL price on the x-axis. We see all 3 graphs peak at the SOL=200 point. However, we see that as time passes, our max profits increase, our losses at the -10% and +10% disappear, and our breakeven points (SOL % change that we make no money but also lose no money) widen from the 0% change point. There is also a graph showing what your PnL would be if you just bought 100 USDC worth of SOL (0.5 SOL tokens) instead of LPing (the blue line).
When we hold for 1 day, our breakeven is close to +/- 5.5-6%, at 7 days the breakeven is beyond the +/- 10% we are showing here.
Focusing on the 7 day data, when the SOL price is 200, we earn 38.6% on our equity, or 38.6 USDC on the 100 USDC starting equity position. However, when the price is 190, we only earn 33%, and when the price is 210, we only earn 35%.
This is why some people call this a ‘short volatility’, ‘short gamma’ or crab strategy. This strategy performs well when the price oscillates around the starting price (we need trading to occur to earn fees, but we also don’t want the price to move too much or we lose money due to the convexity of LPing).
180
1.52
273.33
0
0
273.33
0.75
0.75
50
135
185
88.33
309.69
185.65
124.04
0.77
24.04
180
-10
0.97
0
185
1.52
280.93
0
0
280.93
0.75
0.75
50
138.75
188.75
92.18
318.29
189.41
128.88
0.77
28.88
185
-7.5
0.97
0
190
1.52
288.52
0
0
288.52
0.75
0.75
50
142.5
192.5
96.02
326.9
193.17
133.72
0.77
33.72
190
-5
0.97
0
195
1.15
223.56
71.71
71.71
295.27
0.75
0.75
50
146.25
196.25
99.02
334.54
196.94
137.6
0.4
37.6
195
-2.5
0.55
-0.09
200
0.75
150
150
150
300
0.75
0.75
50
150
200
100.0
339.9
200.7
139.2
0
39.2
200
0
0.1
-0.09
205
0.37
75.47
227.32
227.32
302.79
0.75
0.75
50
153.75
203.75
99.04
343.06
204.46
138.6
-0.38
38.6
205
2.5
-0.34
-0.08
210
0
0
303.7
303.7
303.7
0.75
0.75
50
157.5
207.5
96.2
344.1
208.23
135.87
-0.75
35.87
210
5
-0.75
-0.04
Also below we can see the same graph and table if the borrow rate is 20% not 40%. We can see that the max profit increases, as well as the breakevens widen. When we hold for 1 day, our breakeven is close to +/- 1.55%, at 7 days the breakeven is closer to +/-4.5-4.75%, and at 30 days the breakeven is closer to 11-12%.
180
1.52
273.33
0
0
273.33
0.75
0.75
50
135
185
88.33
309.69
186.2
123.49
0.77
23.49
90
-10
0.97
0
185
1.52
280.93
0
0
280.93
0.75
0.75
50
138.75
188.75
92.18
318.29
189.97
128.32
0.77
28.32
92.5
-7.5
0.97
0
190
1.52
288.52
0
0
288.52
0.75
0.75
50
142.5
192.5
96.02
326.9
193.75
133.15
0.77
33.15
95
-5
0.97
0
195
1.15
223.56
71.71
71.71
295.27
0.75
0.75
50
146.25
196.25
99.02
334.54
197.52
137.02
0.4
37.02
97.5
-2.5
0.54
-0.09
200
0.75
150
150
150
300
0.75
0.75
50
150
200
100.0
339.9
201.29
138.6
0
38.61
100
0
0.1
-0.09
205
0.37
75.47
227.32
227.32
302.79
0.75
0.75
50
153.75
203.75
99.04
343.06
205.07
138.0
-0.38
38.6
102.5
2.5
-0.34
-0.08
210
0
0
303.7
303.7
303.7
0.75
0.75
50
157.5
207.5
96.2
344.1
208.84
135.26
-0.75
35.87
105
5
-0.75
-0.04
Net SOL Position
Next we can see what our net SOL token position is at different price points. Remember we borrowed 0.75 SOL in debt (a liability). We immediately used that SOL as part of our LP (an asset). So our initial equity/token exposure (asset – liability) for SOL is 0, which we can see on the graph below. This graph shows our total SOL token exposure on the y-axis and the % change in SOL price on the x-axis. As the price moves below 200, we start to have positive token exposure to SOL. This is because in a CLAMM, as the price moves towards the lower bound of your chosen range, you get more exposure to the asset losing in price (SOL in this case) and less token exposure to the asset gaining in price (USDC in this case). And when the price moves towards the upper bound, the opposite occurs. However, we always have 0.75 SOL Debt (negative SOL position). So as the price moves below 200, we actually start to have positive SOL token exposure (our LP increases in SOL but our short SOL continues to be static). And when the price moves up, we start to have negative SOL token exposure (our LP decreases in SOL while our short SOL continues to be static).
Once we hit our lower bound, our positive SOL token exposure flatlines (at 0.77) because our LP is out of range and is statically 100% SOL. And once we hit our upper bound, our negative SOL token exposure flatlines (at -0.75) because our LP is out of range and is 100% USDC. You can see a couple examples in the table above, in the ‘Net SOL Tokens’ column.
Empirical Dollar Delta
We can also look at our dollar delta (empirically derived). This graph shows how an increase in SOL price of 1 USDC affects our equity position. This dollar change in equity position for 1 USDC change is SOL price is on the y-axis and the % change in SOL price on the x-axis. The reason this graph looks extremely similar to the one above is because our delta is positive when we have SOL token exposure over 0 (i.e. our position is positively exposed to the price of the SOL token, so SOL price increasing is good for our position). The opposite can be said when the price of SOL is above our 200 starting point. In this range (when price above 200), we have negative SOL exposure (our debt in SOL is greater than the SOL in our LP, so we are net short SOL). So when the price of SOL goes up, our debt goes up more than our asset value.
Empirical Dollar Gamma
Our last graph shows how our dollar delta changes with the change in SOL price. This dollar change in dollar delta position for 1 USDC change is SOL price is on the y-axis and the % change in SOL price on the x-axis. This graph looks like an upside down “U” because we don’t have any change in dollar delta when we are outside of our range. Said another way, when our SOL token exposure is static (either positive or negative), we don’t have any change in the delta, so gamma is 0. When the SOL price is inside of the range, our dollar gamma is negative because our dollar delta has a decreasing slope. What this means is, we get more long SOL as the SOL price goes down, and we get more short SOL as the SOL price goes up. This ‘negative convexity’ is the opposite of what you want in trending markets. However, no convexity is free. We are compensated for taking on this ‘negative convexity’ risk through the trading fees we earn. This is very similar to a short straddle position in options (short a call and short a put at the same strike price). That short straddle option position also has negative gamma, and you earn ‘option premium’ if the underlying asset expires at/near that strike price.
Summary
We have seen that our max profit on this stylized PDN LP example is when the SOL price is exactly at the price we entered our position. As long as the expected yield earned from trading fees is greater than the borrow rate for our leverage/debt, we expect positive profits at or near that starting price. Our breakeven SOL prices widen and our profits increase as time goes on, more trades occur, and especially when price oscillates near the starting price. We are taking on a short volatility, short gamma, crab strategy position with this trade. We are short convexity with positive expected yield. But yield does not equal total returns. We need to account for the fact that the assets underlying our LP position change as the price of SOL relative to USDC changes! Once we have all the facts, we can see the shape of our total expected returns and how they increase over time.
Written by Marco_112358
