Risk-neutral probabilities

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

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. 

These models are two more general families of models incorporating Vasicek model and CIR model, respectively. The first one is used more often as it can be calibrated to the observable term structure of interest rates and the volatility term structure of spot or forward rates. However, its implied volatility structures may be unrealistic. Hence, it may be wise to use a constant coefficient P(t) = P and a constant volatility parameter a(t) = a and then calibrate the model using only the term structure of market interest rates. It is still theoretically possible that the short rate r may go negative. The risk-neutral probability for the occurrence of such an event is... 

Use of credit spread data to estimate the risk-neutral probabilities. 

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. 

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). 

---