Fete Section 3

Front 1

Assumptions to Derive BSM Differential Equation

Back 1

1. The stock price follows the GB motion w m & s constant
2. Permit short-selling securities w full use of proceeds
3. No transaction costs or taxes
4. No div during the life of the derivative
5. No riskless arbitrage opportunities
6. Security trading is continuous
7. The rfr is constant & the same for all maturities

Front 2

Warrants & Employee Stock Options

Back 2

Exercising warrants & employee stock options leads to the company:
1. Issuing more shares
2. Selling them at a strike price
3. If the strike price is less than the mkt price, dilute the interest of the existing shareholders

Front 3

Hedge Strategies

Back 3

1. Naked position
2. Covered position
3. Stop-loss strategy
4. Delta hedging
5. Delta-Gamma-Vega Hedging

Front 4

Pros & Cons of Monte Carlo Simulation

Back 4

Pros:
1. Numerically efficient
2. Give a std error for the estimates
3. Can accomodate complex payoffs
4. Can be used in path-dependent payoffs

Cons:
1. Computationally time consuming
2. Cannot easily handle early exercise situations

Front 5

Types of Variance Reduction Procedures

Back 5

1. Antithetic Variable technique
2. Control variate technique
3. Importance Sampling
4. Stratified Sampling
5. Moment Matching
6. Using quasi-random sequences

Front 6

Static Option Replication

Back 6

1. Use a portfolio of actively traded options to replicate the exotic option
2. If 2 portfolios are worth the same as a certain boundary, they are also worth the same at all interior points of the boundary

Front 7

Advantages of Static Options Replication

Back 7

1. Does not require frequent rebal
2. Can be used for a wide range of derivatives
3. Has flexibility in choosing the boundary & the options

Front 8

Three Types of Levy Process

Back 8

1. Diffusion Model
2. Mixed jump-diffusion model
3. Pure jump model

Front 9

Martingale

Back 9

1. A zero-drift stoch process
2. A variable follows a martingale if its process follows d(theta) = sigma d(z)
3. Has a property that its exp'd value at any future time is its value today ( E(theta(t)) = theta(0) )

Front 10

Reasons Why Interest Rate Derivatives are more Difficult to Value than Equity & Foreign Exchange Derivatives

Back 10

1. The behavior of an individual IR is more complicated
2. Req a model to describe the behavior of the entire yield curve
3. The vols of diff points on the yield curve are diff
4. IRs are used for discounting the derivative and defining its payoff

Front 11

Three Types of Adjustments

Back 11

1. Convexity Adjustments
2. Timing Adjustments
3. Quanto Adjustments

Front 12

Problems of Using HJM Model

Back 12

1. The process for the short rate is non-Markov
2. Monte Carlo sim has to be used
3. Difficult to use a tree to rep term structure movements
4. Some forward rates are Markov and can be rep'd by recombining trees
5. The model is expressed in terms of instantaneous forward rates
6. Its difficult to calibrate the model to prices of actively traded instruments

Front 13

Problems of Using the Deterministic Actuarial Approach

Back 13

1. The single path lacks credibility
2. Its difficult to interpret the results
3. A single path may not capture the risk appropriately for all contracts

Front 14

Pros & Cons of a LogN Model

Back 14

Pros:
1. Simple & tractable
2. Provide a reasonable approx over short term intervals

Cons:
1. Dont provide a reasonable approx for longer term intervals
2. Fail to capture more extreme price movements
3. Don't allow for autocorrelation in the data
4. Fail to capture volatility bunching

Front 15

Pros & Cons of AR(1) Model

Back 15

Pros:
1. Capture autocorrelation in the data

Cons:
1. Don't capture the extreme values or the volatility bunching

Front 16

Pros & Cons of the ARCH(1) Model

Back 16

Pros:
1. Model stoch changes in volatility

Cons:
1. Do not allow autocorrelation in the data

Front 17

Pros & Cons of the GARCH(1) Model

Back 17

Pros:
1. More flexible and better fit for many econometric applications than the ARCH model


Cons:
None listed

Front 18

Pros & Cons of the RSLN-2 Model

Back 18

Pros:
1. Simple
2. Tractability
3. More accurately capture the more extreme observed behavior
4. Intro stoch vol


Cons:
None listed

Front 19

Pros & Cons of the Empirical Model

Back 19

Pros:
1. A simple & quick method
2. Easy to construct multivariate dist'ns

Cons:
1. Analytical development isnt possible
2. Lose the autocorrelation in the data
3. Vol bunching is not modeled

Front 20

Pros & Cons of the Stable Dist'n Family

Back 20

Pros:
1. Can be very fat-tailed
2. Can convolute the dist'n

Cons:
1. Not easy to use
2. Est'n req's advanced techniques
3. Not easy to sim a stable process
4. Do not incorp autocorrelations arising from vol bunching

Front 21

Pros & Cons of Wilkie Model

Back 21

Pros:
1. A collection of models

Cons:
1. Do not allow for change in the nature of economic time series
2. Inconsistent with some economic theories
3. "Data mining" problem

Front 22

Properties of Maximum Likelihood Estimators

Back 22

1. Stationary dist'ns
2. Asymptotic unbiasedness
3. Asymptotic min var
4. Asymptotic N dist'n

Front 23

Limitations of MLE

Back 23

1. Do not apply for not-strictly-stationary models
2. Can't be relied on if the est'd parameter is near the boundaries
3. Useful only if a large sample is used
4. Do not tell how close the fit is

Front 24

Problems of Matching Moments

Back 24

1. An unreliable method of fitting parameters
2. The overall fit may not be very satisfactory
3. The std errors can be large
4. For a satisfactory overall fit it is better to emplow more of the dist'n than the first two moments
5. A common used is as starting values for an interative optimization procedure
6. Both MLE & moment matching emphasize the fit in the center of the dist'n

Front 25

Applications of Interest Rate Models

Back 25

1. Value all IR-sensitive securities
2. Manage IR risks
3. Enhance financial innovations
4. Invoke many other applications

Front 26

Callibration Model

Back 26

1. Est'd from the observed prices in the mkt at the present time
2. Not tested by its empirical robustness
3. Continually adj'd the implied vol surface to fit the mkt prices
4. Not internally consistent
5. Practical approach to value securities
6. RN prob is used

Front 27

Compare Val'n Models to Pricing Models

Back 27

Val'n Models
1. Derive the value of an option so that it is consistent with the price of the underlying stock

Pricing Models
1. Option priced are det'd by supply & demand
2. Influenced by mkt environment

Front 28

Arbitrage-Free Modeling

Back 28

1. Adjust model time dependent parameters to fit mkt prices exactly
2. Do not model the dynamics of the term structure
3. It may look like the term structure today but will not act like the term structure tomorrow
4. The model is an interpolation system

Front 29

Equilibrium Modeling

Back 29

1. Model the behaviors of the term structure over time
2. Employ a statistic approach
3. Do not exactly match mkt prices today
4. Do not contain time dependent parameters

Front 30

Risk Neutral Scenarios

Back 30

1. Are not approp for all purposes
2. Assume that all term premia are zero

Front 31

Real World Scenarios

Back 31

1. In the RW, term premia are not zero
2. Reflect the changes in the RW
3. Good for stress testing or rx adequacy testing

Front 32

Uses of IR Model

Back 32

1. Arbitrage-free & RN: Current Pricing where input data is reliable
2. Equilibrium & RN: Current pricing were inputs are unreliable or unavailable, horizon pricing
3. Arbitrage-free & realistic: unusable since term premia can't be reliably est'd
4. Equilibrium & realistic: stress testing, rx and asset adequacy testing

Front 33

Unrealistic Assumps in BS Formula

Back 33

1. Known & unchanged vol
2. Smooth changes in the price
3. No changes in the short-term IR
4. Freely borrow or lend
5. Full use of the short-sell proceeds
6. No trading costs
7. No change in taxes
8. No dividends
9. Exercise only at expiration
10. No early end of the option's life

Front 34

Stylized Facts

Back 34

1. Properties that are common across a wide range of instruments, mkts and time periods
2. Formulated in terms of qualitative properties of asset returns
3. May not be precise enought to distinguish among diff parametric models

Front 35

Stylized Statistical Properties of Asset Returns

Back 35

1. Absence of autocorrelations
2. Heavy tails
3. Gain/loss asymmetry
4. Aggregational Gaussianity
5. Intermittency
6. Vol clustering
7. Conditional heavy tails
8. Slow decay of autocorrelation in absolute returns
9. Leverage effect
10. Volume/vol correlation
11. Asymmetry in time scales

Front 36

Advantages of using a Pareto Dist'n

Back 36

1. Tractable
2. Applicable