Replicating portfolio

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 porlfolio. 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. 

Since the six-month zero-coupon bond in the replicating portfolio matures at period 1, it is worth 100 percent of face value at point 1, no matter where interest rates stand. The value of the one-year at period 1, when it is a six-month bond, depends on the interest rate at the time. At the higher rate, it is 97.3236 percent of face at the lower rate, it is 97.5562 percent of face. The two equations state that the total value of the portfolio must equal that of the option, which at the higher interest rate is zero and at the lower 0.1562. 

Solving the two equations gives Cj = —65.3566 and C2 = 67.1539. This means that to construct the replicating portfolio, you must purchase 67.15 of one-year zero-coupon bonds and sell short 65-36 of the six-month zero-coupon bond. The reason for constructing the portfolio, however, was to price the option. The portfolio and the option have equal values. The portfolio value is known it is the price of the six-month bond at period 0 multiplied by Cj, plus the price of the one-year bond multiplied by Cj, or... 

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. 

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... 

The discussion so far has focused on asset allocation as generally applicable to a broad cross-section of investors. In reality, the vast majority of individual investors face special restrictions that limit their flexibility to implement optimal portfolios. For example, many investors hold real estate, concentrated stock holdings, VC or LBO partnerships, restricted stock, or incentive stock options that for various reasons cannot be sold. For these investors, standard MV optimization is stiU appropriate and they are weU advised to target the prescribed optimum portfolio. They should then utilize derivative products such as swaps to achieve a synthetic replication. 

FIGURE 6.3 shows that the payoff profile illustrated in figure 6.1 can be replicated by a portfolio composed of one unit of the underlying asset... 

The general relationship between the forward price and the spot prices is demonstrated in FIGURE 6.4. As in Figure 6.3, the first step is to replicate a forward s payoff profile with a portfolio composed of R units of the underlying asset, whose creation is funded by borrowing a sum equal to the present value of the forward price. Again, as in F ure 6.3, the loan is settled on the forward expiry date by paying an amount equal to (Fr xr = F. 

The management of a portfolio of bonds may be undertaken either passively or actively. Passive fund management does not involve any actual analysis or portfolio selection, because the manager merely constructs the bond portfolio to mirror the benchmark or index whose performance he wishes to replicate. As such, passive fund management is more of an administrative function than an analytical or strategic one. 

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