No-arbitrage is, at bottom, a path-independence statement: the cost of replicating a derivative must be the same regardless of the hedging path, or a riskless profit would exist. The mathematical translation is that the replication cost functional is a conservative vector field — one whose line integral depends only on the endpoints, not the route. This lesson establishes the complete equivalence theorem: on a simply connected domain, a vector field is conservative if and only if it is a gradient field, if and only if its curl vanishes, if and only if every closed-curve integral is zero. Each condition is a different face of the same constraint, and each arises naturally in finance.
Learning objectives. By the end of this note you should be able to:
- state and prove the equivalence: path-independent curl-free gradient field (on simply connected domain);
- apply the Fundamental Theorem for Line Integrals: ;
- recognise the role of topology (simply connected vs. multiply connected) in conservative field theory;
- identify exact differentials in thermodynamics and finance.
In this lesson
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- Path Independence
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- The Equivalence Theorem
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- Exact Differentials
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- In Three Dimensions
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- Reconstructing the Potential
Before you start
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