Unformatted text preview: Debt Instruments and Markets Professor Carpenter 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 Reading
!!Veronesi, Chapter 9
!!Tuckman, Chapter 9 No Arbitrage Pricing of Derivatives 1 Debt Instruments and Markets Professor Carpenter 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. 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 of Derivatives 2 Debt Instruments and Markets Professor Carpenter No Arbitrage Pricing
The no arbitrage pricing approach for valuing a
derivative proceeds as follows:
1. 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. 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 of Derivatives 3 Debt Instruments and Markets Professor Carpenter 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 in fact
observable.
!!Future price or payoff distributions aren't really
known in practice, although the theory treats them as
known.
!!Modeling future payoffs for no arbitrage pricing in
practice is a problem of forecasting and financial
engineering. Motivation: A OnePeriod Model
for Pricing 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 $1000 par of a zero maturing at time 1.
!!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 0.5 and
the strike price $975. No Arbitrage Pricing of Derivatives 4 Debt Instruments and Markets Professor Carpenter Call Option Payoff
!!Let 0.5d1 represent the price at time 0.5 of $1 par
of the zero maturing at time 1.
!!The payoff of the call is
Max[1000x0.5d1975, 0] .
!!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 1, will be a 6month bond at time 0.5, the
option expiration date. Future Payoffs of the Underlying
!!Suppose that at the option expiration date, the
underlying bond can take on only two possible
prices (per $1 par):
0.5d1=0.972290 or 0.5d1=0.976086 !!This means that the option has only two
possible payoffs:
“up (high interest rate) state”: $0 “down state”: $976.086$975=$1.086
!!Recall that today (time 0) the prices of 6month and 1year zeroes are:
0d0.5=0.973047 No Arbitrage Pricing of Derivatives and 0d1=0.947649. 5 Debt Instruments and Markets Professor Carpenter OnePeriod Binomial Tree
Let’s organize this information about current prices
and future payoffs in a oneperiod binomial tree:
Time 0.5
1
0.972290 Time 0 0
0.5year zero
1year zero
Call option 0.973047
0.947649
C=?
1
0.976086
1.086 General Bond Portfolio in the Tree
Consider a portfolio with N0.5 par in the 0.5year zero and
N1 par in the 1year zero. Here's how its price and possible
payoffs would appear in the tree:
Time 0 0.5year zero
1year zero
Call option General portfolio No Arbitrage Pricing of Derivatives 0.973047
0.947649
C=?
0.973047 N0.5
+0.947649 N1 Time 0.5
1
0.972290
0
1 N0.5+0.97229 N1 1
0.976086
1.086
1 N0.5+0.976086 N1 6 Debt Instruments and Markets Professor Carpenter Replicating the Call Payoff
If there are only two possible future values of the
underlying asset, then the call can be replicated with
a portfolio of the 0.5year and 1year 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 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 N0.5 and N1 denote the number of the 0.5year
bonds and the number of the 1year bonds in the
portfolio.
!!Then the payoffmatching equations are:
Up state portfolio payoff = Up state call payoff
N0.5 x 1 + N1 x 0.97229 = 0
Down state portfolio payoff = Down state call payoff
N0.5 x 1 + N1 x 0.976086 = 1.086 No Arbitrage Pricing of Derivatives 7 Debt Instruments and Markets Professor Carpenter The Replicating Portfolio
!!The replicating portfolio (the solution to
the simultaneous equations) is
N0.5 = 278.163 and N1 = 286.091
!!In other words, a portfolio that is
–!long 286.091 par of the 1year zero and
–!short 278.163 par of the 0.5year zero
!!will have exactly the same payoff as the
call at time 0.5, regardless of which state
is actually realized. Class Problem
What is the noarbitrage call price? No Arbitrage Pricing of Derivatives 8 Debt Instruments and Markets Professor Carpenter General Bond Portfolio in the Tree
Consider a portfolio with N0.5 par in the 0.5year zero and
N1 par in the 1year zero. Here's how its price and possible
payoffs would appear in the tree:
Time 0 0.5year zero
1year zero
Call option General portfolio 0.973047
0.947649
C=?
0.973047 N0.5
+0.947649 N1 Time 0.5
1
0.972290
0
1 N0.5+0.97229 N1 1
0.976086
1.086
1 N0.5+0.976086 N1 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,
!!And then 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. No Arbitrage Pricing of Derivatives 9 Debt Instruments and Markets Professor Carpenter 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 $1000 par of the zero
maturing at time 1,
–!expiration date is time 0.5, and
–!the strike price $975.
!!The put payoff is max[9751000x0.5 d1,0] Put Option in the Tree
Time 0 0.5year zero
1year zero
Call option 0.973047
0.947649
? General portfolio 0.973047N0.5
+ 0.947649N1 Put option ? Time 0.5
1
0.972290
0
1N0.5 + 0.97229N1
2.71 1
0.976086
1.086
1N0.5 + 0.976086N1
0 No Arbitrage Pricing of Derivatives 10 Debt Instruments and Markets Professor Carpenter Class Problem: Replicating and Pricing the Put
!! Again, the put can be replicated with a portfolio of
the 0.5year and 1year zeroes.
!! Let N0.5 and N1 denote the par amounts of the 0.5year bonds and 1year bonds in the portfolio.
1)!What are the payoffmatching equations that
determine the replicating portfolio par amounts? 2)!What are the par amounts?
3)!What is the noarbitrage price of the put? Put Option in the Tree
Time 0 0.5year zero
1year zero
Call option 0.973047
0.947649
? General portfolio 0.973047N0.5
+ 0.947649N1 Put option ? Time 0.5
1
0.972290
0
1N0.5 + 0.97229N1
2.71 1
0.976086
1.086
1N0.5 + 0.976086N1
0 No Arbitrage Pricing of Derivatives 11 Debt Instruments and Markets Professor Carpenter General Bond Derivative
Any security whose time 0.5 payoff is a function of the time
0.5 price of the zero maturing at time 1 can be priced by no
arbitrage. Suppose its payoff is Ku in the up state, and Kd in
the down state.
Time 0 0.5year zero
1year zero 0.973047
0.947649 General portfolio 0.973047 N0.5
+ 0.947649 N1 Time 0.5
1
0.972290 1N0.5 + 0.97229N1 Ku General derivative ?
1
0.976086
1N0.5 + 0.976086N1 Kd Replicating and Pricing the
General Derivative
1) Determine the replicating portfolio by
solving the equations
1N0.5 + 0.97229N1 = Ku
1N0.5 + 0.96086N1 = Kd
for the unknown N's. (The two possible K's
are known.)
2) Price the replicating portfolio as
0.973047N0.5 + 0.947649N1
This is the no arbitrage price of the derivative. No Arbitrage Pricing of Derivatives 12 ...
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 Spring '15
 Jennifer
 Derivative, Strike price, arbitrage pricing, Professor Carpenter

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