No Arbitrage Pricing of Derivatives


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1 No Arbitrage Pricing of Derivatives Concepts and Buzzwords Replicating Payoffs No Arbitrage Pricing Derivative, contingent claim, redundant asset, underlying asset, riskless asset, call, put, expiration date, strike price, binomial tree No Arbitrage Pricing of Derivatives
2 Readings Tuckman, chapter 9. Derivative Debt Instruments So far, the course has focused mainly on assets with fixed future cash flows. Now we will begin to study assets with random future cash flows that depend on future bond prices or interest rates. These are sometimes called derivatives or contingent claims. A derivative is an asset whose payoff depends on the future prices of some other underlying assets. No Arbitrage Pricing of Derivatives 2
3 Redundant Securities In some cases, the payoff of the derivative can be replicated by the payoff of a portfolio containing underlying assets whose prices are already known. In that case the derivative is called a redundant security. It is possible to price a redundant security by no arbitrage. No Arbitrage Pricing The no arbitrage pricing or contingent claims pricing approach for valuing a derivative proceeds as follows:. Start with a description (model) of the future payoff or price of the underlying assets across different possible states of the world. 2. Construct a portfolio of underlying assets that has the same random payoff as the derivative. 3. Set the price of the derivative equal to the value of the replicating portfolio. No Arbitrage Pricing of Derivatives 3
4 Prices and Payoffs of Underlying Assets? Nonredundant assets (that can't be synthesized with other assets) cannot be priced by arbitrage. What determines their prices? We can think of their current and future prices as the outcome of market clearing in a general equilibrium. Bond prices (interest rates) are influenced by the size of the deficit, international trade balances, the productivity of real investment (real rates) monetary policy (inflation rates) people's investment horizons, risk preferences, and endowments, correlations with other assets (term premia) No Arbitrage Pricing Approach The no arbitrage pricing approach picks up where equilibrium theory leaves off. It takes the prices and payoffs of the underlying (nonredundant) assets as given. Current prices of underlying assets are observable. Future price or payoff distributions aren't known exactly. Modeling future payoffs for no arbitrage pricing is a problem of forecasting and financial engineering. No Arbitrage Pricing of Derivatives 4
5 No Arbitrage Pricing in a One Period Model: A Call Option Before constructing an elaborate interest rate model, let's see how noarbitrage pricing works in a oneperiod model. To motivate the model, consider a call option on a $ par of a zero maturing at time. The call gives the owner the right but not the obligation to buy the underlying asset for the strike price at the expiration date. Suppose the expiration date is time.5 and the strike price $975. Call Option Payoff Let.5 d represent the price at time.5 of $ par of the zero maturing at time. The payoff of the call is Max[x.5 d 975, ]. To value the option, we need to form an idea about what the possible values of the underlying asset will be on the option expiration date. Note that the underlying asset, a bond maturing at time, will be a 6month bond at time.5, the option expiration date. No Arbitrage Pricing of Derivatives 5
6 Future Payoffs of the Underlying Suppose that at the option expiration date, the underlying bond can take on only two possible prices (per $ par):.5 d = or.5 d = This means that the option has only two possible payoffs: up state : $ down state : $ $975=$.86 Recall that today (time ) the prices of 6 month and year zeroes are: d.5 = and d = OnePeriod Binomial Tree We can organize information about current prices and future payoffs in a binomial tree:.5year zero year zero Call option Time C =? Time No Arbitrage Pricing of Derivatives 6
7 General Bond Portfolio in the Tree Consider a portfolio with N.5 par in the.5year zero and N par in the year zero. Here's how its price and 2 possible payoffs would fit in the tree:.5year zero year zero Call option Time C =? Time N N General portfolio N N N N Replicating the Call Payoff If there are only 2 possible future values of the underlying asset, then the call can be replicated with a portfolio of the.5year and year zeroes (the riskless asset and the underlying asset). A portfolio of the underlying and riskless asset replicates the call if it satisfies two equations: Up state portfolio payoff = Up state call payoff Down state portfolio payoff = Down state call payoff No Arbitrage Pricing of Derivatives 7
8 Replicating the Call Payoff... The portfolio and its payoff are described by specifying the number of securities, or in other words, the face value of the bonds, in the portfolio. Let N.5 and N denote the number of the.5 year bonds and the number of the year bonds in the portfolio. Then the payoffmatching equations are: Up state portfolio payoff = Up state call payoff N.5 x + N x = Down state portfolio payoff = Down state call payoff N.5 x + N x =.86 The Replicating Portfolio The replicating portfolio (the solution to the simultaneous equations) is N.5 = and N = In other words, a portfolio that is long par of the year zero and short par of the.5year zero will have exactly the same payoff as the call at time.5, regardless of which state is actually realized. No Arbitrage Pricing of Derivatives 8
9 The No Arbitrage Call Price Since the call and its replicating portfolio have the same payoff, they must have the same price. Otherwise, there would be an arbitrage opportunity. The no arbitrage price of the call equals the price of its replicating portfolio: (N.5 x.973) + (N x.9476) = ( x.973) + (286.5 x.9476) =.448 No Arbitrage Price of the Call in the Tree.5year zero year zero Call option Time Time No Arbitrage Pricing of Derivatives 9
10 Summary To price the call, we assumed two possible future values of the underlying at the option expiration date, determined the two possible future payoffs of the call, constructed a portfolio that replicates the call using two assets that are already priced: the underlying asset and the zero maturing on the expiration date, set the price of the call equal to the cost of the replicating portfolio. The call price is only as accurate as the future underlying payoffs we assumed. Pricing a Put Option Let's price another derivative  say, a put option. A put gives the owner the right but not the obligation to sell the underlying asset for the strike price at the expiration date. Suppose that, again, the underlying is $ par of the zero maturing at time, expiration date is time.5, and the strike price $975. The put payoff is max[975x.5 d,] No Arbitrage Pricing of Derivatives
11 Put Option in the Tree.5year zero year zero Call option Put option Time General portfolio.97347n N? Time N N N N Replicating the Put Payoff Again, the put can be replicated with a portfolio of the.5year and year zeroes. Let N denote the par amounts of.5 and N the.5year bonds and year bonds in the portfolio. The replicating portfolio par amounts are determined by the payoffmatching equations: up: (N.5 x ) + (N x.97229) = 2.7 down: (N.5 x ) + (N x.97686) = => N.5 = and N = No Arbitrage Pricing of Derivatives
12 Pricing the Put A portfolio that is long $ par of.5year bonds and short $73.95 par of year bonds gives the same payoff as the put, regardless of which state is realized. Therefore, in the absence of arbitrage, the put must have the same value as the replicating portfolio: (696.88x.973)(73.95x.9476)=.52 Put Option in the Tree.5year zero year zero Call option General portfolio Put option Time N N.52 Time N N N N No Arbitrage Pricing of Derivatives 2
13 General Bond Derivative Any security whose time.5 payoff is a function of the time.5 price of the zero maturing at time can be priced by no arbitrage. Suppose its payoff is K u in the up state, and K d in the down state. Time Time.5.5year zero year zero N N General portfolio General derivative? N N K u N N K d Replicating and Pricing the General Derivative ) Determine the replicating portfolio by solving the equations N N = K u N N = K d for the unknown N's. (The two possible K's are known.) 2) Price the replicating portfolio as.97347n N This is the no arbitrage price of the derivative. No Arbitrage Pricing of Derivatives 3
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