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36 Cards in this Set
- Front
- Back
- 3rd side (hint)
Assumptions to Derive BSM Differential Equation
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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 |
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Warrants & Employee Stock Options
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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 |
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Hedge Strategies
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1. Naked position
2. Covered position 3. Stop-loss strategy 4. Delta hedging 5. Delta-Gamma-Vega Hedging |
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Pros & Cons of Monte Carlo Simulation
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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 |
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Types of Variance Reduction Procedures
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1. Antithetic Variable technique
2. Control variate technique 3. Importance Sampling 4. Stratified Sampling 5. Moment Matching 6. Using quasi-random sequences |
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Static Option Replication
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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 |
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Advantages of Static Options Replication
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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 |
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Three Types of Levy Process
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1. Diffusion Model
2. Mixed jump-diffusion model 3. Pure jump model |
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Martingale
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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) ) |
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Reasons Why Interest Rate Derivatives are more Difficult to Value than Equity & Foreign Exchange Derivatives
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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 |
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Three Types of Adjustments
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1. Convexity Adjustments
2. Timing Adjustments 3. Quanto Adjustments |
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Problems of Using HJM Model
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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 |
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Problems of Using the Deterministic Actuarial Approach
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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 |
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Pros & Cons of a LogN Model
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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 |
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Pros & Cons of AR(1) Model
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Pros:
1. Capture autocorrelation in the data Cons: 1. Don't capture the extreme values or the volatility bunching |
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Pros & Cons of the ARCH(1) Model
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Pros:
1. Model stoch changes in volatility Cons: 1. Do not allow autocorrelation in the data |
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Pros & Cons of the GARCH(1) Model
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Pros:
1. More flexible and better fit for many econometric applications than the ARCH model Cons: None listed |
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Pros & Cons of the RSLN-2 Model
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Pros:
1. Simple 2. Tractability 3. More accurately capture the more extreme observed behavior 4. Intro stoch vol Cons: None listed |
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Pros & Cons of the Empirical Model
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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 |
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Pros & Cons of the Stable Dist'n Family
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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 |
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Pros & Cons of Wilkie Model
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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 |
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Properties of Maximum Likelihood Estimators
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1. Stationary dist'ns
2. Asymptotic unbiasedness 3. Asymptotic min var 4. Asymptotic N dist'n |
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Limitations of MLE
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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 |
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Problems of Matching Moments
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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 |
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Applications of Interest Rate Models
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1. Value all IR-sensitive securities
2. Manage IR risks 3. Enhance financial innovations 4. Invoke many other applications |
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Callibration Model
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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 |
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Compare Val'n Models to Pricing Models
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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 |
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Arbitrage-Free Modeling
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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 |
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Equilibrium Modeling
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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 |
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Risk Neutral Scenarios
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1. Are not approp for all purposes
2. Assume that all term premia are zero |
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Real World Scenarios
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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 |
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Uses of IR Model
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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 |
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Unrealistic Assumps in BS Formula
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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 |
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Stylized Facts
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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 |
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Stylized Statistical Properties of Asset Returns
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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 |
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Advantages of using a Pareto Dist'n
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1. Tractable
2. Applicable |
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