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1. INTRODUCTION
● Market prices will adjust until there are no opportunities for arbitrage underpins the
valuation of fixed-income securities, derivatives, and other financial assets
● If net proceeds are 0, risk is 0, and return should be 0
2. THE MEANING OF ARBITRAGE-FREE VALUATION
● Arbitrage-free valuation→ an approach to security valuation that determines
security values that are consistent with the absence of an arbitrage
opportunity, which is an opportunity for trades that earn riskless profits
without any net investment of money
● Principal of no arbitrage= prices adjust until there are no arbitrage opportunities
○ Implication of the idea that identical assets should sell at the same price
● Valuation of a financial asset:
○ 1. Estimate future cash flows
○ 2. Determine appropriate discount rate to discount rates that should be used to
discount the cash flows
○ 3. Calculate the present value of the expected future cash flows found in Step 1
by applying the appropriate discount rate or rates determined in Step 2
● Bond values derived by summing the present value of individual eros (cash flows)
determined by such a procedure can be shown to be arbitrage free
● If the bond’s value was must less than the sum of the values of its cash flows
individually, a trader would perceive an arbitrage opportunity and by the bond while
selling claims to the individual cash flows and pocketing the excess value
● Example of Arbitrage-free valuation: The valuation of a bond as a portfolio of erios based
on using the spot curve, each component must have an arbitrage-free value
●
2.1. THE LAW OF ONE PRICE
● Central idea of financial economics: MARKET PRICES WILL ADJUST UNTIL THERE
ARE NO OPPORTUNITIES FOR ARBITRAGE
● Think of it as “free money”, prices will adjust until there is no free money to be acquired
● Arbitrage opportunities arise as a result of violations of the law of one price
● The law of one price states that 2 goods that are perfect substitutes must sell for the
same current price in the absence of translation costs
● If it were costless to trade, one would simultaneously buy at the lower price and sell at
the higher price. Riskless profit would be the difference in the prices ( repeat until prices
converge)
●
2.2. ARBITRAGE OPPORTUNITY
● Arbitrage Opportunity= a transaction that involves no cash outlay that results in a
riskless profit
● 2 types of arbitrage opportunity:
○ 1. Value Additivity
, ■ The value of the whole equals the sum of the values of the parts
■ Example:
●
Asset Price Today Payoff in One Year
A 0.952381 1
B 95 105
● Asset A: risk-free zero-coupon bond pays off $1 and prices
$0.952381 (1/1.05)
● Asset B: Portfolio of 1-5 units of Asset A that pays off $105 one
year from today and is priced today at $95
● The portfolio does not equal the sum of the parts
● Asset B is cheaper than buying 105 units of Asset A
● This position generates a certain $5 now and generates net ) one
year from today
● An investor would engage in this trade over and over again until
prices adjust
○ 2. Dominance
■ A financial assets with a risk-free payoff in the future must have a positive
price today
■ Example:
●
Asset Price Today Payoff in One Year
C 100 105
D 200 220
● Assets C & D are risk-free zero-coupon bonds
● Asset D is cheap relative to Asset C
● If both risk-free, should have the same discount rate
● To make money: sell 2 units of Asset C at $200 and use proceeds
to purchase 1 unit of Asset D at $200
● No net cash outlay today, generates $10 one year from today
Example 1: Arbitrage Opportunities
Which of the following investment alternatives includes an arbitrage opportunity?
Bond A: The yield for a 3% coupon 10-year annual-pay bond is 2.5% in New York City. The
same bond sells for $104.376 per $100 face value in Chicago
Bond B: The yield for a 3% coupon 10-year annual-pay bond is 3.2% in Hong Kong SAR. The
same bond sells for RMB97.220 per RMB100 face value in Shanghai
Solution: Bond B is correct. Bond B’s arbitrage-free price is RMB98.311 which is higher than
the price in Shanghai, therefore an arbitrage opportunity exists. Buy bonds in Shanghai and
,sell them in Hongkong and a profit of RMB1.091 per bond traded
2.3. IMPLICATIONS OF ARBITRAGE-FREE VALUATION FOR FIXED-INCOME
SECURITIES
● Any fixed-income security should be thought of as a package or portfolio of zero-coupon
bonds.
● Example: 5-year 2% coupon Treasury issue should be viewed as a package of 11 zero-
coupon instruments (10 semiannual coupon payments one of which is made at maturity,
and one principal value payment at maturity)
● Stripping= The dealers ability to separate the bond’s individual cash flows and trade
them as zero-coupon securities.
● Reconstitution= dealers can recombine the appropriate individual zero-coupon
securities and reproduce the underlying coupon Treasury.
● Arbitrage profits are possible when value additivity doesn't hold
● The arbitrage-free valuation approach does not allow a market participant to realise an
arbitrage profit through stripping and reconstitution
● Moreover, 2 cash flows that have identical risks delivered at the same time will be valued
using the same discount rate even though they are attached to two different bonds.
3. INTEREST RATE TREES AND ARBITRAGE-FREE VALUATION
● Develop a method to produce an arbitrage-free value for an option-free bond
● Bonds that are option-free, determining the arbitrage-free value as the sum of the
present values of expected future values using the benchmark spot rates
● Benchmark bonds are assumed to be correctly priced by the market.
● The valuation model we develop will be constructed so as to reproduce exactly the
prices of the benchmarked bonds
Example 2: The Arbitrage-Free Value of an Option-Free Bond
The YTM (“par rate”) for a benchmark one-year annual-pay bond is 2%, for a benchmark 2-
year annual-apy bond is 3%, and for a benchmark 3-year annual-pay bond is 4%. A three
year, 5% coupon, annual-apy bond with the same risk and liquidity as the benchmarks is
selling for $102.7751 today (time zero) to yield 4%. Is this value correct for the bond given the
current term structure?
Solution: First find the correct spot rate (zero-coupon rates) for each year’s cash flows. The
spot rates may be determined using bootstrapping (iterative process).
Par Rate Par Rate Par Rate +1
1= 1
+ 2
+...+ N
[1+r (1)] [1+r (2)] [ 1+r (N )]
0.03 0.03+1 0.03 0.03+1
⇒ 1= + = +
1
(1+ r (1)) (1+ r (2))
2
(1+0.02) (1+r (2))
2
The spot rates are 2%, 3.015%, and 4.055%. The correct arbitrage-free price for the bond,
then, is
P0=5 /1.02+5 /1.03015 2+5 /1.04055 3=102.8102(The bond is mis-priced by 0.0351)
● For option-free bonds, performing valuation discounting with spot rates produces an
arbitrage-free valuation
● For bonds that have embedded options, need a different approach
○ Expected future cash flows are interest rate dependent
○ For bonds with options attached, changes in future interest rates impact the
likelihood the option will be exercised and in so doing impact the cash flows
○ We must allow interest rates to take on different potential values in the future
based on some assumed level of volatility
● Interest rate “tree”= representing possible future interest rates consistent with the
assumed volatility
● Often called “lattice models”, the interest rate tree performs 2 functions:
○ 1. Generate the cash flows that are interest rate dependent
○ 2. Supply the interest rates used to determine the present value of the cash flows
● An interest rate model seeks to identify the elements or factors that are believed to
explain the dynamics of interest rates
● These factors are random or stochastic in nature (we can't predict the path of any
particular factor)
● The interest rate model must specify a statistical process that describes the stochastic
property of these factors in order to arrive at a reasonably accurate representation of the
behaviour of interest rates
● These interest rate models are referred to as one-factor models because only one
interest rate is being modeled over time
● Binomial Lattice Model= short interest rate can take on one of 2 possible values
consistent with the volatility assumption and an interest rate model
● The two possible interest rates next period will be consistent with the following 3
conditions:
○ 1. An interest rate model that governs the random process of interest rates
○ 2. The assumed level of interest rate volatility
○ 3. The current benchmark yield curve
3.1. THE BINOMIAL INTEREST RATE TREE
● 1st step for demonstrating the binomial valuation method is to present the benchmark
par curve by using bonds of a particular country or currency
● We assume that all bonds have annual coupon payments.
● Benchmarks bonds are conveniently priced at par so the yields to maturity and the
coupon rates on the bonds are the same
● We use bootstrapping methodology to uncover the underlying spot rates
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