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Topic Title: Risk neutral valuation vs Real world valuation
Created On Thu Mar 12, 09 01:15 PM
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sarty06711
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Thu Mar 12, 09 01:15 PM
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How does the generation of scenarios in a monte-carlo framework differ between the "risk-neutral mode" vs "real-world mode"? It would be great if somebody can take a simple swaption example and explain how the scenario generation/pricing would be different between these two modes. Also, would like to know which method is used when?


 
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choenix1312
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Fri Mar 13, 09 08:36 AM
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Hi,
when it comes to pricing, risk-neutral world instead of real-world is assumed. Because all of cash flows can be discounted by risk-free rate under risk-neutral world wheareas cash flows under real world should be discounted by risk-free rate plus the risk premium. Since the risk premium is hard to determeine in reality, the risk-neutral approach becomes popular. Therefore, if the purpose of Monte Carlo simulation is pricing, it is done under risk-neutral world.

Nonetheless, if you'd like to price swaption or interest rate derivatives, the Monte Carlo simulation may be done under the "forward risk-neutral world" because some interest rate models formulate their stochastic process under forward-risk-neutral world such as BGM and some specify their process under risk-neutral world such as Vasicek. So, it depends on the interest rate model you used.
 
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list
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Fri Mar 13, 09 11:13 AM
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Let you have 2 stochastic cash flows. The transactions actually might occurred at different moments of time. If one use definition that two flows are equal when their PV are equal then to find spread you do not need the notion of risk neutrality. If you think that this is incorrect then try to apply to 2 cash flows risk neutral construction. There are 2 versions of it. One used by French mathematicians along with 2 Polish that is mathematically correct but do not suitable to finance. The start point is the risk neutral probability space with probability measure (roh)*dP. On this risk neutral space they present stochastic cash flows. If you return to original probability space then one could discover that commonly used benchmark processes that used for approximation instrument prices will change their forms. These forms will unusual and say irrelevant to financial modeling while remain mathematically correct. Other approach to the risk neutrality is provided by financials professors. They used benchmark processes on original probability space and think that making Girsanov measure change to arrive at Risk-Neutral World. This approach was first applied to present probabilistic representation of the solution of the BS equation. This approach is mathematically incorrect and simply reduces to replacing real stochastic cash flow with real return mu by the other having the same volatility and return r. So you have a variety of choices.
 
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quantmeh
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Sat Mar 14, 09 03:41 PM
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not all pricing is done in risk-neutral way, i can assure you in that. you're lucky if you have a risk-neutral model for your problem. it's not always the case. you can use actuarial credit risk model, and the real world probabilities. the difference with RN boils down to discounting - you can't use risk-free rates in this case, must add up the premium.

Edited: Sat Mar 14, 09 at 03:42 PM by quantmeh
 
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list
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Sun Mar 15, 09 12:41 PM
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All problems in finance are real word problems. Transition to the risk neutral setting from the real world is a mathematical one. Each step must be accurately defined to avoid ambiguity. In particular all parameters involved in definition of the risk neutral world should be written explicitly in correct mathematical formulas. Otherwise this world is undefined and reminded "ponzi" mode in which clients or students are think that they get something good though this good based on earlier delusions.
 
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crmorcom
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Mon Mar 16, 09 02:31 PM
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sarty, the general idea is this. If you are pricing a derivative of another tradeable, financial asset, then you tend to use a risk-netrual pricing framework, and these prices are OK (assuming your model is not too far off reality) so long as you can hedge all sources of risk (i.e. delta-hedging in a BS world). In fact, if you don't use risk-neutral pricing, your prices are arbitrageable where they disagree. That's why it's called risk-neutral: you can hedge to remove all the risk. So, if you are pricing exotic equity derivatives, and your model is right (a big if) and you hedge correctly, you don't care what happens in the equity markets - you make the bid/ask spread that you booked up-front when you put the trade on.

Of course, this is in theory: nothing ever gets risk-neutral in practice. Remember the first rule of economics - all models are wrong.

So: if you are pricing an IR swaption, you almost certainly want to be risk-neutral, at least to start with. You may well be hedging it with IR swaps. If you have valued the swaption in real-world, you will be pricing the derivative and the hedges differently, and you will get very peculiar answers. You should almost certainly be using risk-neutral pricing, and you should calibrate so that your model gives you the correct market swap rates.

From a technical perspective, so long as all forms of risk are fully priced, the only difference in a diffusion model between risk-neutral and real-world pricing is the drift of discounted asset prices. They will be r in risk-neutral world, and something else in the real world (usually higher: this is the risk-premium). You get this by using Girsanov's theorem or one of it's variants/equivalent methods. You can get this by using more heuristic, intuitive reasoning, too: if the risk-neutral drift is NOT r, then it's arbitrageable. Read Black-Scholes original paper, or any of the later papers by Merton, Kreps, Pliska, Harrison etc.

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"Where so many hours have been spent in convincing myself that I am right, is there not some reason to fear I may be wrong?" - Jane Austen
 
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list
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Mon Mar 16, 09 03:14 PM
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To avoid formulas and take a look what does it mean the idea of risk-neutral world one could see that:
Take original probability space { O , F , P }. Chose density roh such that the log normal security price with parameters ( mu , sigma ) on risk neutral world { O , F , roh *dP } is equivalent log normal security price with parameters ( r , sigma ) on { O , F , P }. This procedure shows how to transform the real parameters ( mu , sigma ) to ( r , sigma ). There is no mathematical mistake in this transformations.
This risk neutral approach serves to explain how we replace original ( mu , sigma )-process by ( r , sigma ) –process. The ( r , sigma ) –process is the underlying of the parabolic Cauchy problem. In this transformation there is no finance.
The drawbacks come from the observation: Why we actually need to determine ( mu , sigma )-process on risk neutral world? As we could consider ( r , sigma )-process on { O , F , P } and present probabilistic representation of the Cauchy problem for parabolic ( BS ) equation. It contradicts formal logic. The idea to explain this phenomena comes from the fact that BS derivatives pricing replaced the real stock by neutral that itself means that BS derivatives takes their values not from the stock but from a heuristic security that does not exist in reality but only in our mind. That of course contradicts the general interpretation of the derivatives which should takes value from underlyings.

 
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coolwindsg
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Tue Mar 17, 09 03:16 AM
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I am not very sure about your definition of "real-world mode". But I think it might be explained using a CDS or corporate bond as an example.

When you hold a position, you have to think how to exit. When you exit, you may have a loss. Here there is a difference between "trading" and "investment".

Trading is to hold a position in very short term and to speculate. In this case for a CDS, the company really default or not during your holding period is not so important but other people "THINK"ing the company's default risk is getting higher or lower is more important. As long as there is an idiot in the market willing to take over your position at a more favorable price, you are safe. So you have to "outsmart" other people. You need to use "what other people think" to price your CDS. This is so-called arbitrage free model.

If you hold the CDS to maturaty, your risk is whether the company really default or not. There is not so much to do with what other people think. You are dealing with "real world" risk - the company really defaults or not.

Of course, how to link this two approaches? Efficient market hypothesis. Only if this comes true, the two will converge, ie, all the information are fully available for investors to make a reasonable judgement, and people are really making a reasonable judgement. So what people think and what is goning to happen in the market are the same, then the two may reach the same results. But for how many times this is the case?
 
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quantmeh
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Tue Mar 17, 09 03:06 PM
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Quote

Originally posted by: coolwindsg


Of course, how to link this two approaches? Efficient market hypothesis. Only if this comes true, the two will converge, ie, all the information are fully available for investors to make a reasonable judgement, and people are really making a reasonable judgement. So what people think and what is goning to happen in the market are the same, then the two may reach the same results. But for how many times this is the case?


RN and real-world probabilities dont have to converge in efficient markets.
 
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crmorcom
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Tue Mar 17, 09 03:12 PM
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Quite so. In fact they are almost guaranteed not to so long as the marginal trader is not risk-neutral - except perhaps in 2006-2007, when risk premia all seemed to be zero!

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"Where so many hours have been spent in convincing myself that I am right, is there not some reason to fear I may be wrong?" - Jane Austen
 
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list
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Tue Mar 17, 09 04:30 PM
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It might be I am not right but when we talk about neutral trading the neutral relates to the strategy or idea that we could neglect the seller and buyer differences in their returns otherwise one could state about arbitrage opportunity. If we combine the trade with borrowing then if we relate borrowing with say repo or similar fixed income borrowing then there should be tri-party relationship whether or not total deal implies negative, neutral or positive income. This type of observation calls for macro regulation. In particular if we observe increasing volatility or it might be close to it increasing frequency in positive-negative fluctuations around market indexes with respect to government securities then one could expect cutting risk-free rates. The problem is what to do if risk free close to 0. Chairman of FR a month or so told that they have some ideas or instruments that could work when government rate close to 0. From neutral point of view it looks that in that situation the best way is to find a legal way to swindle all together especially primary debt holders with moral: money should move otherwise its just a papers.

 
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quantmeh
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Tue Mar 17, 09 05:36 PM
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Originally posted by: list
The problem is what to do if risk free close to 0.


this has nothing to do with RN rate. risk free rate = 0 means there's no time value of money. it doesnt say anything about the price of risk. it's not zero

 
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list
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Tue Mar 17, 09 06:11 PM
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of cause you right " nothing to do with risk free rate = 0 " but if overnight is 0 then bank rate could be say 0.2% and secondary market can borrow for 0.5%.
But general idea was risk neutral trading is when buyer and seller rates actually close. But its corresponding return is not risk free and therefore “neutral “ from risk neutral world and neutral from neutral trading have different meaning.
 
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