Arbitrage-Free Pricing

Assume that the current six-month and one-year rates are 5.00 and 5-15 percent, respectively. Assume further that six months from now the six-month rate will be either 5-01 or 5.50 percent, and that each rate has a 50 percent probability of occurring. Bonds in this hypothetical market pay semiannual coupons, as they do in the U.S. and U.K. domestic markets. This situation is illustrated in FIGURE 11.2. 

For the six-month zero-coupon bond, all the factors necessary for pricing—the cash flow and the discount rate—are known. In other words, only one world state has to be considered. The situation is different for the one-year zero coupon, whose binomial price lattice is shown in FIGURE 11.4. 

Deriving the one-year bonds price at period 0 is straightforward. Once again, there is only one future cash flow— the period 2 redemption payment at face value, or 100—and one possible discount rate the one-year interest rate at period 0, or 5.15 percent. Accordingly, the price of the one-year zero-coupon bond at point 0 is 100/(1 + [0.0515/2] ), or 95 0423-At period 1, when the same bond is a six-month piece of paper, it has two possible prices, as shown in figure 11.4, which correspond to the two possible sbc-month rates at the time 5.50 and 5.01 percent. Since each interest rate, and so each price, has a 50 percent probability of occurring, the avert e, or expected value, of the one-year bond at period 1 is [(0.5 x 97.3236) + (0.5 x 97.5562)], or 97.4399. 

Using this expected price at period 1 and a discount rate of 5 percent (the six-month rate at point 0), the bonds present value at period 0 is 97.4399/(1 + 0.05/2), or 95.06332. As shown above, however, the market price is 95.0423. This demonstrates a very important principle in financial economics markets do not price derivative instruments based on their expected future value. At period 0, the one-year zero-coupon bond is a 

Assume now that the one-year zero-coupon bond in the example has a call option written on it that matures in six months (at period 1) and has a strike price of 97.40. FIGURE 11.5 is the binomial tree for this option, based on the binomial lattice for the one-year bond in figure 11.4. The figure shows that at period 1, if the six-month rate is 5.50 percent, the call option has no value, because the bond s price is below the strike price. If, on the other hand, the six-month rate is at the lower level, the option has a value of 97.5562 - 94.40, or 0.1562. 

Using this expected price at period 1 and a discount rate of 5 percent (the six-month rate at point 0), the bond s present value at period 0 is 97.4399/(1 -F 0.05/2), or 95-06332. As shown previously, however, the market price is 95.0423. This demonstrates a very important principle in financial economics markets do not price derivative instruments based on their expected future value. At period 0, the one-year zero-coupon bond is a riskier investment than the shorter-dated six-month zero-coupon bond. The reason it is risky is the uncertainty about the bond s value in the last six months of its life, which will be either 97-32 or 97.55, depending on the direction of six-month rates between periods 0 and 1. Investors prefer certainty. That is why the period 0 present value associated with the single estimated period 1 price of 97-4399 is higher than the one-year bond s actual price at point 0. The difference between the two figures is the risk premium that the market places on the bond. 

What is the value of this option at point 0 Option pricing theory states that to calculate this, you must compute the value of a replicating portfolio. In this case, the replicating portfolio would consist of six-month and one-year zero-coupon bonds whose combined value at period 1 will be zero if the six-month rate rises to 5.50 percent and 0.1562 if the rate at that time is 5.01 percent. It is the return that is being replicated. These conditions are stated formally in equations (11.4) and (11.5), respectively. 

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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. 

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. 

In the HIM model, the processes for the bond price and the spot rate are not independent of each other. As an arbitrage-free pricing model, it differs in crucial respects from the equilibrium models presented in the previous chapter. The core of the HIM model is that given a current forward rate curve, and a function capturing the dynamics of the forward rate process, it models the entire term structure. 

The principle of arbitrage-free pricing requires that the hedged portfolio s return equal the risk-free interest rate. This equivalence plus an expansion of dC(F,t) produces partial differential equation (8.33). 

The result of this calculation, 0.06, is the arbitrage-free price of the option if the option were priced below this, a market participant could earn a guaranteed profit by buying it and simultaneously selling short the replicating portfolio if it were priced above this, a trader could profit by writing the option and buying the portfolio. Note that... 

In this section, we start from a simple multi-factor HJM term structure model and derive the drift term of the forward rate dynamics required to obtain an arbitrage-free model framework (see HJM [35]). Furthermore, we derive the equivalence between the HJM-firamework and a corresponding extended short rate model. Then, by applying our option pricing technique (see chapter (2)) we are able derive the well known closed-form solution for the price of an option on a discount bond (e.g. caplet or floorlet). 

Thus, going forward we only have to adapt our pricing framework to the new multi-factor dynamics of the arbitrage-free bond price given by... 

Along the lines of HIM it can be shown that the existence of an arbitrage-free setup implies that the drift ji t,T) is fully determined by the volatility function cr (f, T) and the stochastic state variable V (f). Applying Ito s lemma to the bond price... 

The Ho-Lee (1986) model was one of the first arbitrage-free models and was presented using a binomial lattice approach, with two parameters the standard deviation of the short-rate and the riskpremium of the short-rate. We summarise it here. Following Ho and Lee, let ( ) be the equilibrium price of a zero-coupon bond maturing at time T under state i. That is F( ) is a discount... 

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

In an arbitrage-free model, the initial term structure described by spot rates today is an input to the model. In fact such models could be described not as models per se, but essentially a description of an arbitrary process that governs changes in the yield curve, and projects a forward curve that results from the mean and volatility of the current short-term rate. An equilibrium term structure model is rather more a true model of the term structure process in an equilibrium model the current term structure is an output from the model. An equilibrium model employs a statistical approach, assuming that market prices are observed with some statistical error, so that the term structure must be estimated, rather than taken as given. 

The bond and the money market are both risk-free and have identical payouts at time T, and neither will generate any cash flow between now and time T. Since the interest rates involved are constant, the bond must have a value equal to the initial investment in the money market account g -r(r q other words, equation (3.13) must hold. This is a restriction placed on the zero-coupon bond price by the requirement for markets to be arbitrage-free. 

The Vasicek model was the first term-structure model described in the academic literature, in Vasicek (1977). It is a yield-based, one-factor equilibrium model that assumes the short-rate process follows a normal distribution and incorporates mean reversion. The model is popular with many practitioners as well as academics because it is analytically tractable—that is, it is easily implemented to compute yield curves. Although it has a constant volatility element, the mean reversion feature removes the certainty of a negative interest rate over the long term. Nevertheless, some practitioners do not favor the model because it is not necessarily arbitrage-free with respect to the prices of actual bonds in the market. 

Selecting the appropriate term-structure model is more of an art than a science, depending on the particular application involved and the users individual requirements. The Ho-Lee and BDT versions, for example, are arbitrage, or arbitrage-free, models, which means that they are designed to match the current term structure. With such models—assuming, of course, that they specify the evolution of the short rate correctly—the law of noarbitrage can be used to determine the price of interest rate derivatives. 

Both portfolios have the same value at maturity. Since prices are assumed to be arbitrage free, the two sets of holdings must also have the same initial value at start time t. The put-call relationship expressed in (8.26) must therefore hold. 

Hence, the absence of arbitrage opportunities implies tbat the drift term of the bond price dynamics equals the risk-free interest rate. This implies... 

What is the significance of this Here we take it as given that because price processes can be described as equivalent martingale measures (which we do not go into here) they enable the practitioner to construct a risk-free hedge of a market instmment. By enabling a no-arbitrage portfolio to be described, a mathematical model can be set up and solved, including risk-free valuation models. 

A short-rate model can be used to derive a complete term structure. We can illustrate this by showing how the model can be used to price discount bonds of any maturity. The derivation is not shown here. Let P t, T) be the price of a risk-free zero-coupon bond at time t maturing at time T that has a maturity value of 1. This price is a random process, although we know that the price at time T will be 1. Assume that an investor holds this bond, which has been financed by borrowing funds of value C,. Therefore, at any time t the value of the short cash position must be C,= —P(t, T) otherwise, there would be an arbitrage position. The value of the short cash position is growing at a rate dictated by the short-term risk-free rate r, and this rate is given by... 

A default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

The price of the forward contract is thus a function of the current underlying spot price, the risk-free or money market interest rate, the payoff, and the maturity of the contract. It can be shown that neither F > P r/l(f nor F

futures price is at fair value. 

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