Risk Neutral Valuation

3 Key excerpts on "Risk Neutral Valuation"

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  eBook - ePub

  Pricing, Risk, and Performance Measurement in Practice

  The Building Block Approach to Modeling Instruments and Portfolios

  • Wolfgang Schwerdt, Marcelle von Wendland(Authors)
  • 2009(Publication Date)
  • Academic Press

    (Publisher)

  . Using this and expanding it we find the definition of the risk-neutral probability in terms of discount factors

  are positive, less than or equal to one, and sum to one, so they are a legitimate set of probabilities. Using and , assuming a risk-free asset, we can rewrite the asset pricing formula for risk-neutral investors as

  which is the formulation we need to justify the equivalent martingale measure in the previous section; that is, . denotes the conditional expectation with respect to the modified probability distribution.

  The assumption of risk neutrality of the investor allows eliminating the utility function from the stochastic discount factor. As a consequence, the probabilities differ from the subjective probabilities π 0 (ω ) in that they are not dependent on a utility function, which is equivalent to assuming that the investor(s) is risk-neutral, hence the name risk-neutral probabilities . They are not, in general, the real probabilities associated with states by individual investors.

  The use of risk-neutral probabilities has a very fundamental interpretation: risk aversion is equivalent to paying more attention to unpleasant states, relative to their actual probability of occurrence. People who report high subjective probabilities of unpleasant events may not have irrational expectations, but simply may be reporting the risk-neutral probabilities or the product q 0 (ω ) = π 0 (ω )m 0, t (ω

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  eBook - ePub

  Paul Wilmott on Quantitative Finance

  • Paul Wilmott(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  risk-neutral expectation.

  Figure 15.19

  Pricing the option.

  Damn and blast! They have found the correct answer for the wrong reasons! To put it in a nutshell, they have twice used their basic assumption of pricing via simple expectations to get to the correct answer. Two wrongs in this case do make a right.

  And this technique will always work. In the risk-neutral world they have exactly the same price for the option (but for different reasons).

  15.10.10 Non-zero Interest Rates

  When interest rates are non-zero we must perform exactly the same operations, but whenever we equate values at different times we must allow for present valuing.

  With r = 0.1 we calculate the risk-neutral probabilities from

  So The expected option payoff is now And the present value of this is And this must be the option value. (It is the same as we derived the ‘other’ way.)

  Risk-neutral pricing is a very powerful technique, and we will be seeing a lot more of it. Just remember one thing for the moment, that the risk-neutral probability p′ that we have just calculated (the 0.5 in the first example) is not real, it does not exist, it is a mathematical construct. The real probability of the stock price was always in our example 0.6, it’s just that this never was used in our calculations.

  15.11 AND NOW USING SYMBOLS

  In the binomial model we assume that the asset, which initially has the value S, can, during a time step δt, either

  • rise to a value u × S or
  • fall to a value v × S,

  with 0 < v < 1 < u (see Figure 15.20 ).

  Figure 15.20

  The model, using symbols.

  • The probability of a rise is p and so the probability of a fall is 1 – p.

  Note: By multiplying the asset price by constants rather than adding

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  eBook - ePub

  Property Investment Appraisal

  • Andrew E. Baum, Neil Crosby, Steven Devaney(Authors)
  • 2021(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  Does this mean the risk premium for this property should be lower? Not necessarily. What is the level of risk aversion? If we assume that an 84%/16% probability around the risk-free rate meets our risk aversion parameters, then the appropriate risk premium is the standard deviation and we would modify our target rate from 9% to 8.7% accordingly. Is 84%/16% good enough odds to proceed or should it be, say, 95%/5%? In that case the risk premium would have to be 4.2%. If the risk-free rate were 6%, the target rate ought to have been 10.2%.

  There are two problems with this kind of analysis. First, although it seems preferable to picking a risk premium out of the ether, regardless of whether it is related to first principles as set out earlier in this text, it is still based on some qualitative judgement as to the variation in the inputs and the investor's level of risk aversion (and measure for it). Second, there is the issue of IRR versus NPV. Discounting at varying IRRs introduces different weightings within the appraisals for cash flows at the beginning and end of the timeframe, and NPV is supposed to solve that issue as the target rate is set.

  Using NPV means that the first step is to identify a target rate, but we still have the same problem as before, i.e. the appropriate risk premium is not known. The analysis assumes a target rate of 9%.

  At the individual asset level, we need to determine the risk of the asset to qualify the risk premium. The standard deviation is approximately £225,000 around a valuation of approximately £1.225 million, a SD/valuation ratio of approximately 20%. In terms of pricing individual assets and determining the adjustments to risk premium, this NPV analysis does not really give us much more help. How can we use the NPV approach to help determine what the risk premium should be?

  Adapting another approach to discounting cash flows could provide some additional insights to help to solve this problem. Certainty-equivalence valuation is set out in some basic finance texts and is applied to real estate in, amongst others, Geltner et al. (2014 ).

  8.4.2 Certainty-Equivalent Cash Flows

  Avoiding the subjective determination of the discount rate within a RADR approach implies the adjustment of the income stream for risk instead. Risk is held within the cash flow inputs so it gives the opportunity to adjust differentially for risk at the point at which that risk exists. The projected income flow is a function of a complex relationship of gross rental, operating expenses, financing arrangements, taxation and capital return, all of which are subject to potential variance and hence risk.

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