Risk neutral

The above stochastic model takes a decision merely based on first-stage and expected second-stage costs leading to an assumption that the decision-maker is risk-neutral. The generic representation of risk can be written as ... 

Pregnancy in bipolar women was found to be a risk-neutral condition, in that it neither protected against nor increased episode risk in a comparison of 42 pregnant with 42 nonpregnant women who stopped lithium either rapidly (over 1-14 days) or gradually (over 15-30 days). Stopping lithium was not risk-neutral, and the risk was especially high in those who stopped rapidly (482) (see... 

Risk neutrality is assumed here. In the case of risk aversion higher investments in environmental technology might be efficient since they can remove disutility of risk fromrisk-averse persons. 

Real option valuation (ROV) Recently, Gupta and Maranas (2004) revisited a real-option-based concept to project evaluation and risk management. This framework provides an entirely different approach to NPV-based models. The method relies on the arbitrage-free pricing principle and risk neutral valuation. Reconciliation between this approach and the above-described risk definitions is warranted. 

When people are presented choices that have uncertain outcomes, they react in different ways. In some situations, people find gambling to be pleasurable. In others, people will pay money to reduce uncertainty for example, when people buy insurance. SEU theory distinguishes between risk neutral, risk averse, risk seeking, and mixed forms of behavior. These different types of behavior are described by the shape of the utility function (Figure 2). 

A risk-neutral decision maker will find the expected utility of a gamble to be the same as the utility of the gamble s expected value. That is, expected M(gamble) = (gamble s expected value). For a risk-averse decision maker, expected M(gamble) < (gamble s expected value) for a risk-... 

Finally, we obtain the risk-neutral bond price dynamics... 

Later on, we need to change the Gog) bond price proeess from the risk-neutral to the appropriate forward measure in order to compute the price of a bond option by summing over the single risk-neutral exercise probabilities. 

Starting from the risk-neutral bond price dynamics (5.4), we derive the well known closed-form solution for the price of a zero-coupon bond option. Thus, as shown in section (2.1) the price of a call option on a discount bond is given by... 

Then, by changing from the risk-neutral measure Q to the Tq—forward we obtain the moments... 

Following chapter (5.2) we obtain the price of a zero-coupon bond option by computing the risk-neutral probabilities... 

Again, we need to transform the process for the (log) bond price dynamics dXit, T) from the risk-neutral measure Q to the forward measure Tq. Thus, following section (5.1) we derive a measure transformation specially adapted to the additional innovation of the stochastic volatility. The bond price can be computed by integrating from t to 7b via... 

Note that the impact of this correlation effect is not in contradiction to the results found by Bakshi, Cao and Chen [5], Nandi [62] and Schobel and Zhu [69] for equity options. They found higher option prices given positive correlations and vice verca. On the other hand, we have a risk-neutral bond price process, where the source of uncertainty is negatively assigned (see e.g. (7.2)). Thus, assuming a USV bond model with negative correlated Brownian motions is the fixed income market analog of a stochastic volatility equity market model, with positive correlated sources of uncertainty. ... 

The assumption of complete capital markets states that, as a result of arbitrage-free pricing, there is a unique probability measure Q, which is identical to the historical probability P, under which the continuously discounted price of any asset is a Q-martingale. This probability level Q then becomes the risk-neutral probability. 

All valuation models must capture a process describing the dynamics of the asset price. This was discussed at the start of the chapter and is a central tenet of derivative valuation models. Under the Black-Scholes model for example, the price dynamics of a risk-bearing asset St under the risk-neutral probability function Q are given by... 

Under these four assumptions, the price of an asset can be described in present value terms relative to the value of the risk-free cash deposit M, and, in fact, the price is described as a Q-martingale. A European-style contingent liability with maturity date t is therefore valued at time 0 under the risk-neutral probability as... 

Equation (3.43) shows that the bond price is equal to the expected value of the bond, discounted at the prevailing one-period rate. Therefore, x is the implied risk-neutral probability. 

Essentially, there are two dimensions to consider risk-neutral versus realistic and equilibrium versus no-arbitrage. There are situatiOTis under which each approach may be applied with validity. 

Therefore, the break-even analysis allows to determine the spread that equals the price of a conventional bond to the one of an inflation-linked bond. This approach assumes a risk-neutral pricing by which an investor treats conventional and inflation-linked bonds the same. Under break-even hypothesis, both bonds have the same nominal yield. Note if the inflation breakeven is greater than expected inflation, for an investor is favorable to buy a conventional bond. Conversely, the inflation-linked bond is more attractive. If inflation breakeven and expectations are equal, the investor bond s choice will be then indifferent. Figure 6.2 shows the trend of UKGGBEIO and UKGGBE20 Index... 

Under risk-neutrality assumption, the most appropriate discount rate is the risk-free rate. The model is more sensitive to the change of recovery rates, while less sensitive to the change in interest rates. If we consider a zero-coupon bond rated R with maturity at time T, the price is given by Equation (8.28) ... 

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