The Arbitrage Free Valuation Framework Flashcards
What is arbitrage free valuation?
- determining the value of securities by assuming no arbitrage opportunity exists
What is law of one price?
- law of one price: two goods that are perfect substitutes should have the same price. If different prices existed, trader could buy the cheaper one &sell the more expensive one to lock in a risk-free profit based on the price differential.
What is the difference between stripping and reconstitution?
this refers to the principal that fixed-income security can be considered a package of zero-coupon bonds. (eg. dealers can separate the cash flows with a process called stripping)
- stripping: 5 year annual coupon bond stripped into 6 components, 5 coupons and the principal payment
- reconstitution: purchase combination of zero coupon bonds that replicate cash flows of a coupon paying bond
What is the difference between value additivity & dominance for the 2 types of arbitrage opportunities?
- value additivity: value of the whole can differ from the value obtained by adding the values of parts. (eg. the value of a bond could be less than the value of the sum of its individual cash flows)
- dominance: one security consistently offers a better yield than another security, despite both having the same characteristics, including the buying price
What’s the formula to determine a arbitrage-free value for a benchmark bond?
1 = (par rate/ (1+spot rate)) + (par rate/ (1+spot rate^2)) + (par rate +1 / (1+ spot rate^n)
What are lattice models?
- model to value bonds with putable or callable options, takes into account volatility and determine the timing, amount, and discount rates for future cash flows.
Describe what a binomial lattice for 3 time periods looks like and formulas for each time period.
Time 0= i0
Time 1= i1 e^1Sd, i1 e^-1sd
Time 2 = i2 e^2Sd, i2, i2 e^-2Sd
Time 3 = i3 e^3Sd, i3 e^1sd, i3 e^-1Sd, i3 e^-3sd
What are the advantages of using a log normal distribution in a binomial tree?
- lognormal random walk model: negative interest rates are not possible & that interest rates become more volatile as they reach higher levels.
What’s the difference between variance and standard deviation?
- Variance & Standard Deviation: Variance measures the dispersion of a probability distribution, while the standard deviation is its square root.
What is the standard deviation of a one year rate a product of?
- standard deviation of the one-year rate is the product of the current rate and assumed volatility between now and one year.
What’s the difference between historical volatility and implied volatility?
- Historical Volatility: Based on past interest rate fluctuations.
- Implied Volatility: Derived from current market prices of interest rate derivatives.
What is the backward induction valuation method, and formula to find value of bond at each node?
Starting at time 2 when bond matures then using binomial tree backwards to get value at Time 0 (aka present value)
V = 0.5 [(VH + c /1 +i) + VL + c /1+i)] = C +0.5 (VH + VL) / 1+i
VH = interest rate high
VL = interest rate low
C = coupon payment
i = interest rate
goal is to take average price between high and low at time 1, add the coupon payment; then divide by 1 + interest rate
What’s the formula for valuing an option free bond at time 2 that expires at time 3?
- take par price 1,000 + coupon payment, discount at 1+ discount rate
What are the steps in the pathwise model for valuing an option free bond?
- Specify all paths through a tree period without replicating paths.
- Determine the present value of the bond along each path.
- Calculate the average present value from all the paths.
What is the Monte Carlo method best for and how do interest rate paths work for Monte Carlo method?
- Simulates interest rate movements by randomly selecting interest rates within a normal distribution
- Useful for securities with path-dependent cash flows (e.g., mortgage-backed securities).
What is a drift adjusted model?
- model is drift-adjusted if a constant drift term is added to all short-term rates in order to make the benchmark bond’s value match its current market price. (Similar to z spread)
What are term structure models?
- models are used to describe how interest rates evolve over time. can answer specific questions based on their assumptions about interest rates that they incorporate into these models.
- term structure models will not provide a completely accurate depiction of the real world, but they will be sufficient to inform bond valuations and hedging decisions.
What is the general formula for dynamic interest rates that apply to all interest rate models?
- each model uses its own process, all formulas have the same two terms:
1) a drift term that describes the interest rate path assuming zero volatility
2) a dispersion term to add volatility to the process.
dr = 0tdt + SddZ
dr = dynamic short term rate
0tdt = drift term
SddZ = dispersion term incorporates randomness, allowing the model to be used for pricing bonds with embedded options and interest rate derivatives
What are the 2 types of models, describe them.
- Arbitrage-free models: designed to fit current market prices of fixed income securities exactly, ensuring there are no arbitrage opportunities. models primarily used for pricing and hedging.
- Equilibrium models: derived from economic principles and attempt to describe the fundamental forces that drive interest rates over time from a stochastic approach. (stochastic approach: interest rates move in random manner)
What’s the 2 types of drift terms, describe.
- mean reverting: mean-reverting drift means that the interest rate tends to move back toward a long-term average over time.
- time-dependent drift means the drift term is explicitly a function of time, rather than a function of the current interest rate
What are the 2 types of equilibrium models?
- Cox-Ingersoll-Ross Model
- Vasicek Model
What’s the Cox-Ingersoll-Ross Model, and formula?
- CIR model is a equilibrium model based on the assumption that the short-term interest rate will revert to its long-run mean over time
dr = k(0-rt)dt + SD * SQRT rt * dZ
dr = is the expected change (differential) in the short-term rate next period
rt = is the current short-term rate
SD = is the volatility of the short-term rate
0 = is the short-term rate’s long-run average
k = is an adjustment factor that determines how quickly rt is expected to revert to 0.
dt = is the number of periods over which the rate change will occur
dZ = is a normally distributed random variable with a value between 0 and 1 that is used to model stochastic changes
What is Vasicek Model formula?
dr = k(0-rt)dt + SD * dZ
dr = is the expected change (differential) in the short-term rate next period
rt = is the current short-term rate
SD = is the volatility of the short-term rate
0 = is the short-term rate’s long-run average
k = is an adjustment factor that determines how quickly rt is expected to revert to 0.
dt = is the number of periods over which the rate change will occur
dZ = is a normally distributed random variable with a value between 0 and 1 that is used to model stochastic changes
What are the 2 types of arbitrage models?
- Ho-Lee Model
- Kalotay-Williams-Fabozzi (KWF) Model
What’s the difference between Ho-Lee Mode Kalotay-Williams-Fabozzi (KWF) Model?
- Ho Lee Model: simple arbitrage free models, lacks ability to handle embedded options.
- Kalotay-Williams-Fabozzi (KWF): complex and specifically designed for pricing fixed income securities with embedded options like callable and putable bonds.
What’s the difference in formula between ho Lee mode and KWF?
ho Lee model:
dr = 0t*dt + SD *dZ
KWF:
d(in)(rt) = 0t*dt + SD *dZ
d(in)(rt) = log of short term rate
dr = is the expected change (differential) in the short-term rate next period
SD = constant volatility
0 = drift term
dt = is the number of periods over which the rate change will occur
dZ = is a normally distributed random variable with a value between 0 and 1 that is used to model stochastic changes
