Bitcoin Option’s Premium and Its Delta
There is one important factor that has a significant impact on the profit and loss of options settled in Bitcoin, such as on Deribit Exchange. Few discussions have been made on this topic yet.
Let me bring up one question first.
Which Delta should we use?
Delta is an important exposure option traders would look at frequently to estimate how much profit/loss would occur to them as the underlying drifts.
One can observe THREE Deltas on Deribit’s user interface.
First, on the upper right corner of the interface, it shows that “Delta Total “ is 0.0570.
Second, on Futures’ page, check all futures positions and total Size in Bitcoin denomination, which shows that the sum of the Futures’ Delta is 7.5395.
Third, on Options’ page, check the sum of “Delta Options”, which shows -10.3410.
If we add up Futures’ Delta and Options’ Delta, the result would be +7.5395–10.3410 = -2.8015 BTC.
If we are to hedge the portfolio to Delta neutral, which method below should we use?
1. Short perpetual or future contract with size 0.0570 BTC and make “Delta Total” zero.
2. Long perpetual or future contract with size 2.8015 BTC and make Futures Delta plus Options Delta zero.
It is an Either/Or question.
So, which way should we choose?
Explanation on Deribit FAQ
On FAQ page of Deribit website, we can find the following explanation on Delta Total:
What does Delta Total in the account summary mean?
In the Account summary you will find a variable called “DeltaTotal”. This is the amount of BTC delta’s on top of your equity due to all your positions futures and options combined. It does not include your equity. Example: If you buy a call option with delta 0.50 for 0.10 BTC, your DeltaTotal will increase with 0.40. If the bitcoin price were to rise with $1, the option would gain $0.50 in value, but the 0.10BTC you payed for it would also gain $0.10 in value. Thus your total delta change due to this transaction is just 0.40. Futures delta’s are also included in DeltaTotal calculation. Equity is not. So depositing BTC in your account has no influence on DeltaTotal. Only opening/closing positions in your account will change DeltaTotal.
The formula for DeltaTotal:
DeltaTotal= Futures Deltas + Options Deltas + Futures Session PL + Cash Balance — Equity.
(or DeltaTotal= Futures Deltas + Options Deltas — Options Markprice Values.)
This explanation is self-explanatory.
So, the answer to the previous question regarding which Delta should be used if we are to adjust portfolio to Delta neutral is to use Delta Total.
Here I would like to highlight the formula below as it will be referred to later in this article.
DeltaTotal= Futures Deltas + Options Deltas — Options Mark Price Values
How would changing Options Premium affect Delta Total?
What Deribit FAQ document didn’t explain is how changing option premium would affect Delta Total.
If you buy a call option with delta 0.50 for 0.10 BTC, and the underlying moves up, delta changes to 0.70 while premium increases to 0.20. In this case, will delta total be 0.70–0.10 = 0.60 or 0.70–0.20 = 0.50?
As per the formula given by Deribit above, the answer should be the latter one. In other words, Delta Total would change along with options premium’s change.
This is counterintuitive as I only paid 0.10 BTC as premium at first and nothing more later. Why should the changing premium affect Delta Total?
I’ll give the conclusion first as the analysis below is highly mathematical. Yes, we shall use the formula given by Deribit, and changing premium will affect Delta Total dynamically.
Simplify BSM Formula
As per the Black-Scholes-Merton formula, Call option’s price (in USD/fiat money) is given as follows:
Call = S x N(d1) — K x e^-rT x N(d2)…………(1)
S, the underlying price, and in our case, the BTC price in USD/fiat money.
K, option’s strike price
r, risk free interest rate
e, natural number 2.71828
T, time left until expiry date (annualized)
d1 and d2 are as follows; σ, pronounced as sigma, refers to the volatility of the underlying, which should be an annualized number.
We can simplify this formula by setting r=0 and S=F (F means Futures price). This is the simplification taken by Deribit as well.
Call = F x N(d1) — K x N(d2)…………(2)
Calculate the derivation of Call with respect to F, and we will get:
Call Delta = N(d1)
Divide both sides of the Call price formula (2) with futures price F, thus we convert the Call price denomination from USD/fiat money into BTC.
Call_BTC = N(d1) — K x N(d2) / F…………(3)
Calculate the derivation of Call with respect to F, we will get the change rate of Call options BTC value against BTC price change as follows:
d Call_BTC / d F = K x N(d2) x F^-2…………(4)
*Detailed derivation steps are available in the Appendix at the end of this article.
Hedging by Future contracts
We now look into using Futures to hedge Options’ Delta.
The BTC denomination value of inverse contract is:
Futures_BTC = — T / F…………(5)
T, the number of inverse contracts (unit: 1 dollar).
The nature of long inverse contracts is to sell USD, thus the right side of the formula is a negative value indicated by a minus.
Take the derivative of Formula (5) with respect to futures price F, we get the change rate of Futures BTC value against BTC price change:
d Futures_BTC / d F = T x F^-2…………(6)
When change rate of Call options BTC value plus change rate of Futures BTC value equals zero, the portfolio is Delta neutral from BTC perspective, i.e.:
d Call_BTC / d F + d Futures_BTC / d F = 0…………(7)
Input Formulas (4) and (6) into (7), we will get:
K x N(d2) x F^-2 + T x F^-2 = 0…………(8)
Sort Formula (8) to good orders and we get:
T = — K x N(d2)…………(9)
It means when the number of inverse future contracts equals the above, portfolio is Delta neutral from BTC perspective.
Cross check with Delta Total formula
Let us cross check whether portfolio is Delta neutral when Delta Total equals zero.
Delta Total = Futures Deltas + Options Deltas — Options Mark Price Values = 0…………(10)
Futures Delta = T / F
The BTC denomination of futures contract, using the number of inverse contracts divided by futures price.
Options Deltas = N(d1)
Options Mark Price Values = Call_BTC = N(d1) — K x N(d2) / F
Same as Formula (3)
Input the above items into Formula (10), and we will get:
T / F + N(d1) — [N(d1) — K x N(d2) / F] = 0
T / F + K x N(d2) / F = 0
After sorting,
T = — K x N(d2)
This result is identical with Formula (9).
As the whole calculation process is reversible, we now conclude that, if the value of Delta Total reaches zero, the portfolio is delta neutral from BTC perspective.
And the part “- Options Mark Price Values” in Delta Total formula means that the impact of option premium on Delta Total is dynamic and changes in real time.
If you buy option, you need to pay premium, and you will get negative Delta on premium;
If you sell option, you will collect premium and get positive Delta on premium.
To hedge correctly, traders should look at the value of Delta Total dynamically.
Put Option and Generalization
The above analysis is done by checking Call option pricing. The same conclusion will be reached if we check Put option pricing. And a generalization could be made if we use weighted average of a portfolio consisting of both call and put options. The conclusion is the same: Hedging should be done with Delta Total.
How to utilize Delta of Options Premium
Delta of options premium is an important factor for traders, especially for those who wish to earn more Bitcoin regardless of its USD value.
An outstanding example is long put.
If you buy put, you need to pay out premium, and negative delta of premium will be generated. Along the way of Bitcoin going south, the premium itself is enlarged, leading to more negative delta of premium. This will produce considerable extra PNL in Bitcoin.
Long put is advantageous to traders wish to short underlying price.
Appendix
Derivations
1. Proof of core formula, SxN’(d1) — K x exp(-rT) x N’(d2) = 0
Preparation
N’(x) = exp(-x²/2)/sqrt(2xpai)
pai=3.14, circular constant
d2=d1 — sigma x sqrt (T)
Stat derivation
SxN’(d1) — K x exp(-rT) x N’(d2)
= (sqrt(2xpai))^-1 (S x exp (-d1²/2) — K x exp(-rT) x exp (-d2²/2))
= exp(-d1²/2) x (sqrt(2xpai))^-1 x (S — K x dxp (-rT) x exp ((d1²-d2²)/2))
Specifically,
(d1²-d2²)/2
= 1/2 x (d1² — (d1² — 2 x d1 x sigma x sqrt (T) + sigma² x T))
= d1 x sigma x sqrt (T) — 1/2 x sigma² x T
d1 = [ln (S/K) + (r+sigma²/2) x T] / sigma / sqrt(T)
(d1²-d2²)/2
= ln(S/K) + (r+sigma²/2) x T — 1/2 x sigma² x T
= ln(S/K) + rT
exp ((d1²-d2²)/2)
= exp(ln(S/K) x exp (rT)
= S/K x exp(rT)
S — K x exp (-rT) x exp ((d1²-d2²)/2)
= S — K x S/K
= 0
Thus, SxN’(d1) — K x exp(-rT) x N’(d2) = 0 stands.
Simplify it by setting r=0 and S=F
d1 = [ln (F/K) + 1/2 x sigma² x T] / sigma / sqrt(T)
d2=d1 — sigma x sqrt (T)
F x N’(d1) — K x N’(d2) = 0
2. Derivation of Call Delta
Preparation
d1 = [ln(S/K) + (r+sigma²/2)xT] / [sigma x sqrt(T)]
d2 = d1 — sigma x sqrt(T)
d d1/dS or d1' = 1 / [S x Sigma x Sqrt(T)]
d d2/dS or d2' = d d1/dS or d1'= 1 / [S x Sigma x Sqrt(T)]
So d1'=d2'
Start derivation
Call = S x N(d1) — K x exp (-rT) x N(d2)
Take the derivative of it with respect to S,
Delta = N(d1) + SxN’(d1)xd1' — K x exp(-rT) x N’(d2)xd2'
= N(d1) + d1' x [SxN’(d1) — K x exp(-rT) x N’(d2)]
Utilize core formula,
SxN’(d1) — K x exp(-rT) x N’(d2) = 0
We get:
Delta = N(d1)
3. Call Delta in BTC denomination
Call_BTC = N(d1) — K/S x exp (-rT) x N(d2)
Delta_Call_BTC = N’(d1)xd1' — K x exp (-rT) x (N’(d2)xd2'/S — N(d2)/S²)
Utilize core formula,
SxN’(d1) — K x exp(-rT) x N’(d2) = 0
We get
N’(d1) = K x exp(-rT) x N’(d2) / S
Since d1'=d2' (See Appendix 2)
Delta_Call_BTC
= d1' x K x exp(-rT) x N’(d2) / S — K x exp (-rT) x (d1'xN’(d2)/S — N(d2)/S²)
= K x exp(-rT) / S² (d1'x S x N’(d2) — d1'x S x N’(d2) + N(d2))
= K x exp(-rT) / S² x N(d2)
Simplify it with r=0 and S=F, and we will get:
Delta_Call_BTC = K x N(d2) x F^-2
Jeff Liang
Greeks.live
8th July 2021
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