Do cross-asset lightning nodes do not offer premium-free American call options?
I would argue that cross-asset lightning nodes do not offer premium-free
American call options for the following reasons.
Say I wanted to set up to purchase 1 WJT for P bitcoins at time t < T where t
is the time I close the contract and T is the expiry time.
In order to set up the contract I must pay P bitcoins to the contract,
incurring an opportunity cost of x_i1. Assuming we set up the contract at time
t_0=0, this will be equivalent to the money I could have earned by loaning the
currency at interest during the period t.
I must also pay the issuer of the contract a premium x_p (in the case where I
am both recipient and issuer, see further down).
If S(t) is the spot price at time t and K = S(t) - P then the payoff for me is
S(t) > P: K - x_p - x_i1
S(t) < P: -x_i1 - x_p
If x_i2 is the opportunity cost of paying 1 WJT to the contract for time t then
the payoff for the other party (issuer) is as follows:
S(t) > P: -K + x_p - x_i2
S(t) < P: x_p - x_i2
If x_p = 0 then the issuer is guaranteed a loss. Therefore no rational contract
issuer will issue an American call option for free.
In the case where I am both recipient and issuer of the contract, to get the
payoff we add the above payoffs:
S(t) > P: -x_i1 - x_i2
S(t) < P: -x_i1 - x_i2
This is a guaranteed loss.
Lightning nodes do not offer premium-free American call options because whether
or not the contract and issuer are the same person, setting up a premium free
American call option using a HTLC guarantees a loss for one or both parties.
Even if the opportunity costs were 0, then setting up a contract with myself
would have a guaranteed 0 payoff.
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