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When Derivatives Can Be Replicated with Stock and a Risk-Free Asset

Article Quant Q&A · Author: Question Anxiety
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Summary

The document explains the claim that a derivative can be represented as a portfolio of its underlying asset and a risk-free asset. It connects this idea to continuous delta hedging in the Black–Scholes framework: adjusting the stock and bond positions can replicate the option’s payoff, supporting risk-neutral pricing without including the stock’s expected return as a separate input.

The answer stresses that this result depends on the model’s assumptions. In practice, discrete rebalancing, volatility changes, and jumps can leave residual risk; models with additional risk factors, such as stochastic volatility, may require other hedging instruments. The discussion is conceptual and provides no derivation or empirical evidence, so it should be read as a framework-specific explanation rather than a universal statement about all derivatives or markets.

Key ideas

  • In Black–Scholes, continuous delta hedging replicates an option using the underlying asset and a risk-free instrument.
  • Risk-neutral pricing follows from replication under the model’s assumptions, rather than from explicitly including the stock’s expected return.
  • Volatility changes and price jumps can create residual risks in practical hedging.
  • Models with additional risk factors may require additional instruments to replicate or hedge derivatives.

Tags

OptionsDerivatives pricingRisk management
Full text
# Why can derivatives be viewed as a portfolio of the underlying and the riskless asset?


# Why can derivatives be viewed as a portfolio of the underlying and the riskless asset?












I am struggling with the statement:

> "Every derivative of the underlying can be viewed as a portfolio of the underlying asset and the riskless asset."

Is this based on the put-call parity?

Also I came across this statement in Hull (2006) why the stocks expected returns are not included in the option pricing formula:

> "The key reason is that we are not valuing the option in absolute terms. We are calculating its value in terms of the price of the underlying stock. The probabilities of future up or down movements are already incorporated into the stock price."

Is this w.r.t. the delta hedging argument, that the replication of the riskless portfolio is in relative measures w.r.t. the long/short positions?

## Answer by alexprice (score 1)

https://quant.stackexchange.com/a/43299

- This is generally not true. It is true in Black-scholes framework with continuous hedging. In practice there's usually residual risks coming from other factors than underlying asset and risk-free instrument.(volatility changes, jumps). In other models, say Heston ,we would need to add vanilla options to portfolio.

- This is actually based on continuous delta hedging in Black-Scholes world where as we can perfectly hedge movement (up,down) of stock with underlying and bond, this movement (up/down) does not matter for pricing. (risk neutral pricing)

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