Risk-Free Asset Holdings in a Black–Scholes Call Hedge

Summary

The document asks how a call writer’s risk-free asset position changes over time in a Black–Scholes delta hedge. It states the stock position as the call’s delta, given as the cumulative normal distribution evaluated at d1, and proposes an initial risk-free holding equal to the option premium minus the value of the stock hedge. The question is how to determine the corresponding holding at a later time.

The document supplies no answer or derivation, so it does not establish a rebalancing rule. In the standard frictionless Black–Scholes replication argument, the cash or bond position is inferred from the option value less delta times the stock price at the current time; the position must be updated as the option value, delta, and underlying price change. This relies on model assumptions, including continuous trading and the specified risk-free rate, and the source itself does not discuss those assumptions or practical hedging costs.

Key ideas

• A call writer’s delta hedge holds the underlying in an amount equal to the option delta.
• The initial risk-free asset position is posed as option value minus the value of the stock position.
• The question concerns how that risk-free holding is recalculated as time and market prices change.
• The document provides no worked derivation or answer to its question.
• Practical replication depends on model assumptions and rebalancing conditions.

Tags

# Quantity of risk-free asset in Black Scholes model


# Quantity of risk-free asset in Black Scholes model












When the seller of a Call option hedges themselves, we know that they should buy $\Delta(t) = \mathcal{N}(d_1(t))$ amounts of the risky asset at time $t$.

But what about the riskless asset? My understanding is that they should buy $V(0) - \Delta(0)S_0$ at time $t = 0$ where $V(0)$ is the price of the option (ie the premium given by the buyer). What should they buy at time $t$?

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