Risk-Neutral Pricing

Although, as noted, the market does not price instruments using expected values, it is possible to derive risk-neutral probabilities that generate expected values whose discounted present values correspond to actual prices at period 0. The risk-neutral probabilities for the example above are derived in (11.7). 

Solving equation (11.6) gives p = 0.5926 and - p - 0.4074. These are the two probabilities for which the probability-weighted average, or expected, value of the bond discounts to the true market price. These risk-neutral probabilities can be used to derive a probability-weighted expected value for the option in figure 11.5 at point 1, which can be discounted at the six-month rate to give the option s price at point 0. The process is shown in (11.8). 

The option price derived in (11.8) is virtually identical to the 0.062 price calculated in (11.6). Put very simply, risk-neutral pricing works by first finding the probabilities that result in an expected value for the underlying security or replicating portfolio that discounts to the actual present value, then using those probabilities to generate an expected value for the option and discounting this to its present value. 

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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... 

In the academic literature, the risk-neutral price of a zero-coupon bond is expressed in terms of the evolution of the short-term interest rate, r t)—the rate earned on a money market account or on a short-dated risk-free security such as the T-bill—which is assumed to be continuously compounded. These assumptions make the mathematical treatment simpler. Consider a zero-coupon bond that makes one payment, of 1, on its maturity date T. Its value at time ris given by equation (3.14), which is the redemption value of 1 divided by the value of the money market account, given by (3.12). 

The next step is to use this tree to describe the bond s price evolution, ignoring its call feature. The tree is constructed from the final date backwards, using the bond s ex-coupon values. At each node, the ex-coupon bond price is equal to the sum of the expected value plus the coupon six months forward, discounted at the appropriate six-month yield. At year 3, the bond s price at all the nodes is 100.00, its ex-coupon par value. At year 2.5, the bond s price at the highest yield, 7-782 percent, is calculated by using this rate to discount the bond s expected price six months forward. The price in six months in both the up and the down state is 103.00—the ex-coupon value plus the final coupon payment. The bond s price at this node, therefore, is derived using the risk-neutral pricing formula as follows ... 

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. 

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... 

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. 

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) ... 

In practice, assuming the discrete time case, the transition matrix includes the transition probabilities between the possible states. Therefore, in this model, market prices are used to find the credit spread and convert the matrix of transition probabilities to the time-dependent risk-neutral matrices Qt t+i- The credit spread is given by Equation (8.32) ... 

In the first step, we determine the interest rate path in which we create a risk-neutral recombining lattice with the evolution of the 6-month interest rate. Therefore, the nodes of the binomial tree are for each 6-month interval, and the probability of an upward and downward movement is equal. The analysis of the interest rate evolution has a great relevance in callable bond pricing. We assume that the interest rate follows the path shown in Figure 11.4. In this example, we assume for simplicity a 2-year interest rate. We suppose that the interest rate starts at time tg and can go up and down following the geometric random walk for each period. The interest rate rg at time tg changes due to two main variables ... 

It is obvious that = 0 and therefore the volatility of the forward rate determines the drift as well. In other words all that is needed for the HJM methodology is the volatility of the bond prices. The short rates are easily calculated from the forward rates. Once a model for short rates is determined under the risk-neutral measure Q the bond prices are calculated from... 

In this section we consider the pricing of a European option on the money fund. (This is the same as a bank account when the initial value B(0) = 1.) Thus, the payoff of a European call option with exercise price K is max[B(T) - K,0], The continuous version of the Ho-Lee model is assumed for the short interest rate process. The risk-neutral valuation methodology provides the solution as ... 

The statistical transition matrix is adjusted by calibrating the expected risky bond values to the market values for risky bonds. The adjusted matrix is referred to as the risk-neutral transition matrix. The risk-neutral transition matrix is key to the pricing of several credit derivatives. 

The JLT model allows the pricing of default swaps, as the risk neutral transition matrix can be used to determine the probability of... 

Various credit derivatives may be priced using this model for example, credit default swaps, total return swaps, and credit spread options. The pricing of these products requires the generation of the appropriate credit dependent cash flows at each node on a lattice of possible outcomes. The fair value may be determined by discounting the probability-weighted cash flows. The probability of the outcomes would be determined by reference to the risk neutral transition matrix. 

The default swap market is not unlike the lottery ticket. What if the shipping and handling fee for the winning ticket was unknown or turned out to be zero In that case, if an investor observed these lottery tickets trading at a price of 4, it may appear that the probability of winning was simply 4.00%. In the case of a default swap this is what is referred to as the risk-neutral probability of default. The risk-neutral probability of our default swap is approximately equal to the premium of 4.00%. By applying the lottery ticket example to our default swap, it is easy to see how the hazard rate is dependent on both the risk-neutral probability as well as the recovery value assumption, and thus can be approximated hy X = P/(l - R). 

In the academic literature, the bond price given by equation (3.15) evolves as a martingale process under the risk-neutral probability measure P. This process is the province of advanced fixed-income mathematics and lies outside the scope of this book. An introduction, however, is presented in chapter 4, which can be supplemented by the readings listed in the References section. 

The price of coupon bonds can also be derived in terms of a risk-neutral probability measure of the evolution of interest rates. The formula for this derivation is (3.22). 

Most option pricing models use one of two methodologies, both of which are based on essentially identical assumptions. The first method, used in the Black-Scholes model, resolves the asset-price model s partial differential equation corresponding to the expected payoff of the option. The second is the martingale method, first introduced in Harrison and Kreps (1979) and Harrison and Pliska (1981). This derives the price of an asset at time 0 from its discounted expected future payoffs assuming risk-neutral probability. A third methodology assumes lognormal distribution of asset returns but follows the two-step binomial process described in chapter 11. 

Muchofthe re search has assumedthateach agent in a supply chain is risk-neutral and his objective is to maximize (minimize) the expected profit (cost). Under this assumption, a supply chain is said to be coordinated when the summation of individual expectedprofits (costs) are maximized (minimized). A number of contractual forms have been studied recently under the risk-neutral assumption. These include the three popular contracts wholesale price (WP) contracts (Lariviere Porteus, 2001 Corbett et al., 2004), buy back (BB) contracts (Pasternack, 1985), and quantity flexibility (QF) contracts (Tsay, 1999 Cachon, 2003). 

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