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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?

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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).

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  • $\begingroup$ Thank you. So I think I have understood the whole purpose behind the risk neutral concept. The only question I still have is why are the pricies the same in a risk neutral world and in the real (risk averse) world ? I think I dont understand the underlying hedge. It is possible to create a riskless portfolio with the stock and the option . This portfolio have to pay the riskless rate due to arbitrage arguments. But I don't exactly understand the connection. $\endgroup$
    – student
    Sep 26, 2016 at 14:49
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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.

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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!

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