Thanks list for quick comment. So far I am following Hulls book. Hull's book says that 'expected return on all securities is the risk free intereat rate in risk-neutral-world'. This will coincide to my experience that to verify BS by MCSimuilation, I have to use risk free rate as expected return. This also link to the fact that spot-future is in risk free rate relationship. In option market, we have market bid/offer first hand. Then volatility is adjusted to make model fit to market. In risk neutral framework expected return is untouchable ( should be kept same as risk free rate ), so only vol will be the adjustment term that makes implied vol very much deviate from historical vol and also the source of SMILE I believe. ( Nevertheless I strongly feel that I need to understand more on measure conversion, Gilzanov, (Omega,F,P), Sigma-Algebra,etc. Then I may understand what you mean. )

You right that Hull's book suggests that 'all securities is the risk free interest rate in risk-neutral-world' and he teaches that " to use risk free rate as expected return." It stems of course from misunderstanding math and finance. The latter goes from BS interpretation of the option price. It contradicts formal logic according to which in theory to get stock at maturity with -100% expected return and +100% expected return is impossible for the same price. But BS introduce arbitrage free interpretation of the option price that in theory possible.

Thanks list for reply again. I just bought 7th edition of Hull but it still has same line as before, i.e. 'expected return on all securities is the risk free intereat rate in risk-neutral-world'. If this line is not correct, somebody should tell him to correct this. In any case I think I need to study more to be a part of any further math based discussion. Thank you for all help. I will be back.

You could try to solve the BS puzzle yourself by asking yourself. I am afraid that no one with probably a few exceptions will not be interested either professors or officials. This based on my own experience.

Re the topic of this FAQ, I have heard:"Risk Neutral pricing underprice options because risk free rate is definitely lower than expected returns.""The BS formula and the dynamic hedging ensures that at every moment your portfolio has no risk"

QuoteOriginally posted by: joeykRe the topic of this FAQ, I have heard:"Risk Neutral pricing underprice options because risk free rate is definitely lower than expected returns.""The BS formula and the dynamic hedging ensures that at every moment your portfolio has no risk"Anyone reading my pontifications on this subject in this thread and others would know that my position has been that1. Risk-neutral valuation works because the market will have already priced in any risk premium.2. Any future expected risk premium will affect an option value only through higher moments (e.g. volatility).It's now time for me to revise my position. I am now of the opinion that the market does is not anything like efficient enough to price in the risk premium and therefore that risk-neutral valuation will not enable the profits that can come from having a better model of expected value. However, if you are a market maker and are tightly limited in your bid/ask spread then you had better continue to use risk-neutral pricing or you will get creamed. Those of us who are not market-makers have the luxury of choosing whether we want to buy or sell or neither and can therefore make use of better models. If we want to make a quote it will be max[risk-neutral,ourvalue] to sell and min[risk-neutral,ourvalue] to buy. But then, of course, we will be lucky to find a counter-party, so we may have to settle for the best price we can get thereby exposing ourselves to an inability to close our position at an appropriate price later. As regards joeyk's quote, I would say this is likely correct, under most circumstances, for any market where the underlyings have a positive risk premium (e.g. stocks). If the current market underprices the underlying, then risk-neutrality will underprice the call and, by put-call parity it will also underprice the put. If the current market overprices the underlying then risk-neutrality will underprice the put and, by put-call parity, it will also underprice the call. However, there is exposure in this due to market friction (e.g. bid/ask spread) because this can cause a violation of put-call parity between buying and selling. So any expectation to profit from this (other than by sheer luck) will depend on a sufficiently small bid/ask spread relative to the pricing error from risk-neutrality both when entering a position and when exiting (unless holding until expiration). Edit: I should have added that underpricing of options using risk-neutrality depends on the same volatility being used. In practice, of course, risk-neutral pricing can restore value by increasing implied volatility. So joeyk's quote would better be expressed as "risk-neutral pricing requires an over-estimate of volatility in order to avoid under-pricing when a risk premium is present".

Last edited by Fermion on October 22nd, 2009, 10:00 pm, edited 1 time in total.

- taneururer
**Posts:**16**Joined:**

Hopefully my brain is working correctly, but isn't this statement incorrect?"If the current market underprices the underlying, then risk-neutrality will underprice the call and, by put-call parity it will also underprice the put."If the market is truly underpricing the underlying, wouldn't this make puts overpriced (forgetting RN pricing and pricing based on utility)? If the SP500 is at 1,000 and you knew that it worth 1,100 you would sell puts & buy calls.In other words, if the underlying is underpriced then the forward must be underpriced.

QuoteOriginally posted by: taneururerHopefully my brain is working correctly, but isn't this statement incorrect?"If the current market underprices the underlying, then risk-neutrality will underprice the call and, by put-call parity it will also underprice the put."If the market is truly underpricing the underlying, wouldn't this make puts overpriced (forgetting RN pricing and pricing based on utility)? If the SP500 is at 1,000 and you knew that it worth 1,100 you would sell puts & buy calls.In other words, if the underlying is underpriced then the forward must be underpriced.Context. The above statement was assuming that the risk-neutral density used to price the call has the same higher moments as the "real" density. As you say, the over-priced put argument is just as valid, but then this violates put-call parity unless the call is also over-priced. In reality, of course, this demonstrates that the conversion real density --> risk-neutral density cannot be achieved merely by changing the first moment. It also demonstrates to me, yet again, how easy it is to tie myself up in knots with this stuff....

The stupidest thing i ever heard on risk-neutrality:A "strategist" from some stupid bank telling us (the clients) that "The drift in the stochastic model of the stock price is positive because stocks have historically tended to go up"

- chaoticrambler
**Posts:**60**Joined:**

QuoteOriginally posted by: Gmike2000The stupidest thing i ever heard on risk-neutrality:A "strategist" from some stupid bank telling us (the clients) that "The drift in the stochastic model of the stock price is positive because stocks have historically tended to go up"I think I have seen it written in more than one book (or something similar).So what is the reason ? Inflation must be one, right ?

That's a good question. Some people think like this: all information about a company is already factored into today's price. If a company has better prospects than its competition, and it is likely to increase its market share, then the price should reflect that, and should not go up when the market share actually increases. So positive drift seems to be a conundrum. The answer is of course risk aversion. But isn't the risk itself already priced in? The simplest example is of a risky zero coupon bond. The riskless rate is zero. A company wants to sell you a bond which will pay $100 in one year, and the chance that the company goes under is 1%. Are you willing to buy this bond for $99? Of course not. You require compensation for your risk. You are willing to buy it for 89.5. So on average, one year from now you make some money. With equities it is the same. The company may have a good quarter, or a bad one. The dividend may go up or down, or the company may even go bust. The price of the stock will reflect that, but with a haircut for risk. When time actually passes, on average the price will go up. Best,V.

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