Risk-Neutral Pricing, No-Arbitrage Hedging, and the Physical Measure
Summary
The document discusses why derivatives are often valued under a risk-neutral measure and how this connects to no-arbitrage pricing and hedging. One response frames risk-neutral valuation as an abstract link between a replicable payoff and the cost of the hedge. Another emphasizes that market prices of hedging instruments inform the measure used by a bank seeking to manage its exposure. Under continuous hedging and a model that captures the relevant dynamics, the choice of measure can support equivalent hedge valuation, though the explanation is conceptual rather than a derivation.
The discussion also distinguishes risk-neutral pricing from forecasting real-world returns: removing a predictable drift in the pricing model does not show that actual assets grow at the risk-free rate. It offers informal explanations for stocks and bonds as well, but these should not be treated as a complete account of asset pricing. The document supplies no worked numerical example and gives limited detail on conditions such as market completeness or American-option exercise.
Key ideas
- Risk-neutral valuation connects no-arbitrage pricing with the cost of hedging a payoff.
- Market prices of instruments used to offset risk inform derivative valuation from a hedging perspective.
- A risk-neutral measure is a pricing framework, not a claim that real-world expected returns equal the risk-free rate.
- The discussion assumes suitable hedging and a model that captures the relevant market dynamics.
- The explanations are conceptual and do not fully establish the conditions for pricing every derivative.
Tags
Full text
# Still confused : risk neutral measure/world
# Still confused : risk neutral measure/world
I know that this is probably the most asked question in finance but I still can’t get my head around how everything belongs together (risk neutral measure, risk neutral word,real word, arbitrage). Especially in the continuous time. Why can I price any option (even American) in a risk neutral world under the risk neutral measure? Why can I assume that I can model the underlying stock price with the risk free rate? And why can I only do this when pricing option? What about stock pricing in the risk neutral world?
## Answer by SmallChess (score 3)
https://quant.stackexchange.com/a/30219
I have an answer (no mathematics!) in About the definition of a complete market.
Basically, risk neutral is an abstract concept that relates no-arbitrage and hedging. In quantitative finance, if you can show how you can hedge an instrument (eg: option) via the no-arbitrage argument, you can price it. To understand why you'd need risk neutral, you should start from the discrete model (it's in my answer).
## Answer by lehalle (score 2)
https://quant.stackexchange.com/a/30223
For hedging purpose and when you are a bank (i.e. you are not supposed to take inventory risk), the risk neutral measure has nice properties
- it is built using market prices: as soon as an instrument that can lower your exposure (i.e. can be used to sell your hedging book) is available on the market, you can do it with no bad surprise on your hedging costs. This is really valuable: the capability to net your risk in the market is priced when you use the risk neutral measure.
- if you hedge continuously (and if all the market dynamics singularities are in you model --I mean, if you use a semi martingale--) it is equivalent to hedging using any other "more realistic" measure... Is it a good news ;) ?
You seem surprise the trend term is no more present under the risk neutral measure. If it really shocks you, just imagine you put its realizations in the Brownian term. Applied mathematicians persist in naming this term a Brownian or a "noise term", but economists call it the "innovation". In this case it can help you to see the Brownian term (and the jump one if any) under the risk neutral measure like economists see it: all that is unexpected. And for sure a deterministic trend that can be expected at $t$ (i.e. that belongs to ${\cal F}_t$) is unrealistic.
Of course a good point would be: "ok but I know how to predict the trend". If it is the case you should go in an hedge fund, not in a bank. The role of a bank is not to take this kind of risks on an inventory that can be as huge as their hedging books. They would have to put an incredible amount of capital in front of this risk; it is not worth to do that.
## Answer by user3264325 (score 1)
https://quant.stackexchange.com/a/33786
> Why can I price any option (even American) in a risk neutral world under the risk neutral measure?
Imagine calculating a price from the point of view of people who are 1) risk averse or 2) risk seeking? You would come up with 2 valuations, one slightly lower than the other respectively. The risk neutral approach, takes subjective feelings about risk out of the equation
> Why can I assume that I can model the underlying stock price with the risk free rate?
If the market mostly takes a risk neutral view of pricing and there is not much opportunity for arbitrage, then we would expect most assets to have the same payoff (i.e. stocks and t-bills)
> And why can I only do this when pricing option?
Options are seen as illiquid - i.e. their market prices are sticky. Much of finance is about breaking down illiquid assets into more liquid ones in order to find more accurate prices. In order to do this, we need no arbitrage, and in order to have no arbitrage we need to have precise non-subjective prices (i.e. risk neutral ones)
> What about stock pricing in the risk neutral world?
You can do the same with stocks and bonds. Stocks can be seen as long call options on the underlying value of the company, and bonds as short put options on the underlying value of the company!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.