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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?

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  • $\begingroup$ Hi: take calls. You can always create a portfolio of stock and riskless asset that mmimics the return of the call, if you know the volatility, time to expiration, stock price, etc Therefore, the value of the call can't depend on the stock return. Cox and rubenstein is old but, IMHO, provides an excellent discussion of this material. There may be new texts that are better because I have not kept up with this area. $\endgroup$ – mark leeds Dec 25 '18 at 13:25
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  1. 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.

  2. 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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