nice brain teaser. No math required. Just common sense.
Suppose asset S and X have GBM dynamics (with mu_s, mu_x, sigma_s, sigma_x etc )
Suppose asset S and X have GBM dynamic too.
There's a contract C paying (ST - XT) on termination. Could be a call option or any other contract.
There's a contract C paying (ST - XT) on termination.
Q: Given X vs X have the same price today (and ditto S vs S), what can we say about C vs C price today?
To be concrete, say X0 = X0 = $0.5 and S0 = S0 = $3.
A replication portfolio for contract C -- long 1 unit of S and short 1 unit of X. This portfolio has current price S0 - X0 = $2.5. Similar replication portfolio for C has current price $2.5. Tomorrow, the replication portfolios may have different prices. On expiration, they may be different in value, though each portfolio is a perfect replication.
So the GBM dynamic is irrelevant !
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