Chapter 3 Discrete Time and State Models

Discount Functions

Discount Functions Illustration

Discount Functions Illustration (Continued)

The Term Structure Of Interest Rates

The Term Structure Of Interest Rates Illustration

Term Structure Estimation Based Upon A System Of Bond Prices Suppose we are given the cash flows and prices of T bonds over a T year period. As long as the cash flows of the T bonds form a linearly independent set of vectors, then we can determine the discount functions for each time period from time 0 to times 1 through T. Matrix CF will represents cash flows anticipated in each of T years from the T bonds. Bond i cash flows appear in Row i of matrix CF; time t cash flows appear in column t. The prices of these bonds are given by vector b 0. We will calculate from these two matrices vector d, which represents discount functions (the value of $1 to be paid) for each of these T years.

Term Structure Estimation Based Upon A System Of Bond Prices (continued)

Once we have found d, we can then obtain interest rates, yield curves, and price any other bond that may enter the market in the T-year time frame. In order to obtain a unique solution for d, we need to have each row vector of the cash flow matrix be independent of one another. This is required because the inverse CF -1 of the matrix CF exists if and only if its row vectors are linearly independent. In that case, the solution for the vector of discount functions is given by: d = CF -1 × b 0

Term Structure Estimation Based Upon A System Of Bond Prices (continued) Any additional bonds entering the market can be priced based on the values of d. The financial interpretation of there existing T linearly independent cash flow vectors for T time periods is that the market satisfies the important requirement that bond markets be complete. If we were unable to find bonds that produced a set of T linearly independent cash flows, there would be insufficient information in the bond market in order to uniquely determine the discount functions, and hence the interest rates. Such markets are said to be incomplete.

Illustration

Illustration (Continued)

Arbitrage The key components of arbitrage are that: 1.) Arbitrage is riskless. All cash flows, including transactions prices in the market are known. 2.) Arbitrage will never produce a negative cash flow in any time period or under any outcome. An arbitrage opportunity fulfills the above conditions and produces at least one positive cash flow in at least one period and/or outcome. In a perfect market, arbitrage opportunities do not exist since rational and greedy investors will never price securities such that they produce an arbitrage opportunity for a competitor at their own expense. No- arbitrage conditions are used to price securities relative to one another such that they do not produce such an arbitrage opportunity.

No-Arbitrage Bond Markets

No-Arbitrage Bond Markets (Continued)

Pricing Bonds In The Arbitrage-Free Market

Pricing Bonds In terms Already Priced Bonds Typically, in a bond market, new bonds are priced based upon the prices of bonds that have already been priced in the market. In a no arbitrage market with T time periods, as long as there are at least T bonds with linearly independent payoff vectors that have already been priced, then there is a consistent way to price any new bond entering the market.

Pricing Bonds In terms Already Priced Bonds (Continued)

Illustration Of Pricing Bonds In terms Already Priced Bonds

Illustration Of Pricing Bonds In terms Already Priced Bonds (Continued)

Pure Securities This process can involve determining the vector space ℝ n, valuing n “control” securities with linearly independent payoff vectors, and pricing the payoff vectors of the previously unpriced securities based on linear combinations of prices from the “control” securities. We will initially assume the following for our valuations: 1.) There exist n potential states of nature (prices) in a one-time period framework. 2.) Each security will have exactly one payoff resulting from each potential state of nature.

Pure Securities (Continued) 3.) Only one state of nature will occur at the end of the period (states are mutually exclusive) and which state occurs is ex-ante unknown. 4.) Each investor's utility or satisfaction is a function only of his level of wealth; the state of nature that is realized is important only to the extent that the investor's wealth is affected (this assumption can often be relaxed). 5.) Capital markets are in equilibrium (supply equals demand) for all securities. Often, the states of nature are the possible outcomes in the future. For simplicity, we are assuming that there are only a finite number of possible future states. We will consider in later chapters the case when the number of possible future outcomes are unlimited (infinite).

Pure Securities (Continued)

Illustration Of A Market Described In Terms Of Pure Securities

Illustration Of A Market Described In Terms Of Pure Securities (Continued)

Pricing Pure Securities Suppose we are in a one time period n state economy with n already priced securities at time zero such that their payoff vectors at time one are linearly independent. We also assume that there are no opportunities for arbitrage. Then we will be able to obtain the time zero price for each of the n pure securities.

Pricing Pure Securities (Continued)

Synthetic Probabilities If our analysis includes a riskless asset in the type of no arbitrage pricing model that we have been using in this section, every security in this no arbitrage market will have will have the same expected rate of return as the riskless bond under certain circumstances. An important feature of this type of pricing model in such a market is that it can be used to define “synthetic,” “hedging” or “risk-neutral” probabilities q i. These risk- neutral probabilities do not exist in any sort of realistic sense, nor are they assumed at the start of the modeling process. Instead, they are inferred from market prices of securities and interest rates.

Synthetic Probabilities (Continued) These risk-neutral probabilities are essential in that they can be used to calculate the price of any security in the market so that the no arbitrage nature of the market is maintained. Risk neutral probabilities have the useful feature that they lead to expected values that are consistent with pricing by investors that are risk neutral, leading to the term risk neutral pricing. This is important because it means that we do not need to work with unobservable risk premiums when we value securities; in fact, we do not even need to know anything about any investors' risk preferences.

Synthetic Probabilities (Continued)

where ψ i is the price of pure security i and ψ j is the price of each of n pure securities j. In our illustration, synthetic probabilities are q 1 =.7/.8 =.875 and q 2 =.1/.8 =.125. These probabilities are referred to as synthetic probabilities because they are constructed from security prices rather than directly from investor assessments of physical probability.

Complete Markets And No Arbitrage Pricing A complete market is one in which the payoffs for any security can be replicated by a portfolio of existing securities that have already been priced. We also assume that the market is frictionless, which means that there are no transactions costs associated with the purchase or sale of securities. If, in addition, there are no arbitrage opportunities, then any security can be priced with the replicating portfolio of the already priced securities. Suppose there are n possible outcomes (or states) in the market, which implies there are n possible payoffs for each security. Each security’s payoffs are then defined by a n dimensional payoff vector.

Complete Markets And No Arbitrage Pricing (Continued)

The pricing is invariant with respect to investor risk preferences. Pure security prices and synthetic probabilities implicitly reflect risk preferences so that such preferences need not be explicitly input into pricing of other securities. Pure security prices and relative pricing relations are enforced by arbitrage. This is the basis of the risk-neutral valuation models. Risk- neutral valuation means that we are able to price securities such that in the risk neutral probability space (synthetic probability space), risky securities such as stocks will have the same expected return as the return on a riskless asset such as a T-bill. This will be illustrated in the next section.

Derivative Securities And Options Derivative securities are assets whose values are derived from the performance of other securities, indices or rates. Stock options are examples of derivative securities. One type of stock option is a call, which grants its owner the right (but not the obligation) to purchase shares of an underlying stock at a specified “exercise” price within a given time period (before the expiration date of the call). The expiration payoff of a call is the maximum of either zero or the difference between the stock price S T at expiration (at time T) and the exercise price X of the call: c T = MAX[S T - X,0]

Illustration Of pricing A Call In A One Time Period Binomial Setting

Illustration Of pricing A Call In A One Time Period Binomial Setting (Continued)

Put-Call Parity

Put-Call Parity (Continued)