Hedging Supermartingale Price Processes in Black–Scholes

Summary

The document poses a question about whether every discounted price process that is a supermartingale can be replicated or hedged in a Black–Scholes market. It assumes the underlying follows Brownian motion and that the market has the standard Black–Scholes properties. The question adds that the process starts with at least its expected future value and stays above the option’s exercise value, including zero.

No answer or construction is provided, so the document does not establish whether such a hedge exists or state conditions for one. The question touches on important distinctions between supermartingale price bounds, self-financing portfolios, and replicability, but those concepts are not worked through here. It is best read as a problem statement rather than a usable hedging method; any conclusion would require additional assumptions and a proof beyond the material supplied.

Key ideas

• The question asks whether every discounted supermartingale price process admits a hedge in the Black–Scholes setting.
• It assumes a Brownian underlying and standard Black–Scholes market conditions.
• The proposed process is constrained to stay above the option’s exercise value and zero.
• The document gives no answer, hedge construction, or conditions proving replicability.

Tags

# Existence of a hedging portfolio and martingale property


# Existence of a hedging portfolio and martingale property












Lets assume that the underlying follows a Brownian motion and the market has the standard properties of the Black Scholes setting.

Is there a way to find a hedging portfolio for every discounted price process X which is a supermartingale?

Edit: The price process X is a stochastic process such that $X_0\ge E[X_t]$ and $X_t \ge \max(V_t, 0)$ for all t. That is the process is at least self-financing an at every time it is possible to pay oft the value of the option if its exercised.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.