Risk-Neutral Valuation Versus Discounted Cash Flow for Options
Summary
The document compares discounted cash flow intuition for stocks and bonds with derivative pricing. For an option, the future payoff is uncertain and depends on an underlying asset, so a discounted-cash-flow approach would first require a probability distribution for that payoff and then a discount rate consistent with the risks in that distribution. Under assumptions such as a chosen physical distribution and CAPM-based discount rate, this route can lead to the Black–Scholes result.
The response explains why practitioners commonly prefer risk-neutral valuation: it uses a risk-neutral probability measure paired with discounting at the risk-free rate, producing an equivalent price under stronger theoretical foundations. The choice of a physical distribution and a CAPM risk premium is difficult to justify objectively, and CAPM has empirical limitations. The explanation is conceptual and does not derive the model or discuss how to calibrate it to market prices.
Key ideas
- Option valuation requires modeling uncertain future payoffs as well as choosing a discount rate.
- A physical-measure discounted-cash-flow approach needs a specified payoff distribution and a matching risk premium.
- With assumptions such as CAPM, this approach can yield the Black–Scholes model.
- Risk-neutral valuation pairs a risk-neutral distribution with the risk-free discount rate.
- The response favors the theoretical basis of risk-neutral pricing while noting that physical-distribution and CAPM assumptions are difficult to establish.
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Full text
# Fundamental difference between stocks/bonds and options that require different pricing method # Fundamental difference between stocks/bonds and options that require different pricing method As for the pricing of stock, we can use the DCF method. For the pricing of bonds, we can also use the DCF method. As i understand, for the pricing of stocks and bonds, to use the DCF method, we must know the "risk premium" in order to discount the future cashflows. I am wondering what difference between stocks/bonds and options that prevent us from using pricing method for stocks/bonds (like DCF) to price options ? Moreover, i often see that they use DCF method to price stock and bond, which means that they can determine which "risk premium" to use for stocks and bonds. Why can't they do the same thing (i.e. determine the risk premium) as for options ? Thank you for your help! ## Answer by Adam N. (score 2) https://quant.stackexchange.com/a/68346 If we were to price an opion with the DCF method, we would have to contend with 2 problems: - We don't know the cash flow. So we would have to make an assumption that it comes from a specific probability distribution, say $\mathbb P$, and then take an expectation of that. - We need to know the discount factor. Since we made an assumption about $\mathbb P$, we know the level of risk embedded in it, so we could use, say, CAPM, to obtain the appropriate discount rate. If we do the above, we arrive at precisely the Black-Scholes model, see this answer and the paper linked there, or the original BS papers. People just don't like to think about valuation of derivatives in these terms, because it's difficult to find objective grounds on which to choose $\mathbb P$, and CAPM is also a very strong assumption, with empirical shortcomings. So the preferred approach is to use the risk-free valuation, which imposes a specific risk-free distribution $\mathbb Q$ and dictates the use of the risk-free rate for the discount factor to go along with it. In the end, we arrive at an equivalent result, on stronger theoretical grounds.
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