Everything Flashcards
What is an event-driven hedgefund
Attempts to profit from situations such as M&A, restructuring, bankruptcy, etc.
Describe long-short equity hedge
Equity-oriented posittions on either side, depending on outlook, not meant to be market neutral.
Describe portable alpha
Market neutral pure play, buy a stock you think will increase, while for example neutralizing market risk by selling index futures to achieve zero beta. So a market neutral position on the stock.
What is statistical arbitrage
Quantitative and often automated trading systems that seek out temporary and modest misalignments in prices. Different from convential as its not risk-free based on unambiguous mispricing.
Describe price smoothing
Hedge funds with illquid assets may smoothe reported prices, leading to serial correlation between returns
What is the bakcfill bias
Hedge funds only report when they choose to, and thus probably only when they perform well.
What is survivorship bias
Ill-performing funds cease operations and leave the database, leaving only the successful ones behind
What are changing factor-loadings?
Changing exposure to different risk-factors due to hedge fund flexibility, making performance hard to judge
Why has alpha decreased for many HFs?
-Decrease of transaction costs on financial markets
-increaes of capital (more HFs, more competition)
-Information more readily available -> less anomalies
-Tightening of regulations and effort of compliance after financial crisis
-Decrease of financial incentives for HF managers because they have so much invested that the small returns are enough to make them rich
Can hedge funds hedge market risk?
-You would expect so, but correlation is higher than you would think due to high number of HFS
-Diversification benefits by FoFs are thus negligible
-Most HF strategies are thus not a good hedge
Potential reasons for HF outperformance
-Skill
-Investment flexibility
-Fraud
-Data issues
-Risk
-Problems measuring returns for HFs with CAPM
Model classifications
-Empirical: from data to model
-Theoretical: from theory to model -> black scholes
-Deterministic: random term
Probabilistic: no random term
Discrete: variable can only take discrete values
Continuous: variable can take any value within range
Cross-sectional: comparing multiple units at a point in time
Time series: track one unit over time
Panel: track multiple units over multiple time periods
What is a high water mark?
If a fund experiences losses, incentive fee is only paid when it makes up for those losses -> creates incentive to shut down fund after poor performance and start over
Why do we care about both alphas and betas for HFs?
We can achieve beta by simply buying (levered) ETFs -> we only care about alpha, but need beta to determine it
Investors could short market index to remove market risk from HF investment
What is alpha transfer, and how can it be achieved?
Portable alpha strategy earns beta in one asset and alpha in another
-Often implemented using futures contracts
Steps:
1. invest where you can find alpha, e.g., Jap. small caps
2. hedge systematic beta away, e.g., short Nikkei futures
3. Establish exposure to desired asset class by using futures or ETF
Why can hedge fund risk not be described with Stdev and sharpe?
Returns do not follow a normal distribution
What can break an alpha transfer strategy?
-CAPM needs to hold
-Alpha forecast needs to be correct
-Beta forecast needs to be correct
-Epsilon should be negatiev or positive, as it cannot be hedged away
Imagine you have a stock and you write a put option. Does this increase or decrease down-market beta?
Increase
Describe the sharpe ratio and advantages and disadvantages
measures return compensation per unit of total risk, with risk, measured by std. dev. -> slope of CAL
Advantages
Simple to use
Allows comparison across asset classes
Intuitive conceptual interpretation (reward per unit of total risk)
Not affected by leverage
Disadvantages
Assumes that returns follow normal distribution
Does not distinguish between good and bad volatility
Does not distinguish between systematic and idiosyncratic risk
Comparison needed
Describe the sortino ratio and lists advantages and disadvantages
Measures return compensation per unit of bad riks, which is measured by downside deviation (stdev of all returns that are lower than RF)
Advantages
Distinguishes between good and bad volatility
Does not assume that returns follow normal distribution
Allows comparison across asset classes
Intuitive conceptual interpretation
Disadvantages
Noisier than Sharpe ratio when return distribution is symmetric
Does not distinguish between systematic and idiosyncratic risk
Comparison needed
Describe the traynor ratio, and what are advantages and disadvantages
Return compensation per unit of systematic risk, measured by market beta
Advantages
Distinguishes between systematic and idiosyncratic risk
Intuitive conceptual interpretation
Disadvantages
Assumes that beta is positive and constant
Does not distinguish between good and bad beta
Requires beta estimation -> sensitive to chosen benchmark
Comparison needed
Describe jensen’s alpha, and list advantages and disadvantages
Excess return of the portfolio above CAPM
Alpha is the difference between actual return and predicted return
Appropriate when portfolio is part of broader, fully diversified portfolio
Advantages
Simply to compute
Statistical significance easy to test
Level has intuitive interpretation
Direct measure of return added by manager
Disadvantages
Noise in beta estimation leads to noise in alpha estimation
Beta may vary over time due to timing or style switching
Does not distinguish between god and bad beta
Alpha depends on benchmark (factors) in models
Alpha can be simply boosted by using
What are problems for performance measurement?
-Survivorship bias
-Luck
-‘‘Beta in disguise’’
-Lack of data/data manipulation
-> smoothing of returns, non-linear strategies, deviating from proclaimed investment style (style drift)
Why do HF managers increase risk?
Flow-performance relationship: outflows from bad performance are smaller than inflows from good performance -> may give poorly performing managers incentive to gamble
Why do we want to measure performance?
-identifies talent
-Criteria for compensation (only for abnormal returns that cannot be achieved through passive investing)
-Incentivizes HF managers to decide well
What is the information ratio, and its advantages and disadvtages?
Alpha divided by idiosyncratic risk
Pros:
accounts for idiosyncratic risk (deviation from benchmark)
Intuitive conceptual interpretation (active return/active risk)
Cons:
Same as for any measure that uses alpha as input, the same drawbacks as jensen’s alpha
It needs to be compared, it doesn’t say much by itself. IN practice IR of 0.5 is good, IR of >1 is excetional
Describe style analysis, and the steps
We choose style analysis to evaluate funds comapred to passive similar styles. So to evaluate fund strategies, in some cases to replicate them, to build liquid alternatives and to manage risk exposure.
The process
1. regress fund returns on passive style portfolios/benchmarks
2.Regression coefficients constrainted to be nonnegative (MFs cannot short sell) and sum to 1 -> solver required
a. benchmark coefficients reveal implicit weight of each asset class
b. intercept measures average return due to security selection and timing
c r2 measures % of returrn variation due to style choice rather than selection -> style choice matters more than security selection specifics
Describe style analysis issues
-Ambiguity on definitions: what is a value style?
-Style drift
-Depends on appropriate selection of benchmarks
-Not immune to manipulation (return smoothing)
-Statistical testing is difficult, bnechmarks are strongly correlateid
-Yields average style but does not capture rapid style change
Style analysis pros
-Can handle any strategy for which passive indices exist
-Uses only return information (no holding needed)
-Powerful analysis which detects style drift
-Gives insights in how to replicate the funds returns
If alpha is 2%, what is the expected return of a market neutral position?
Alpha + RF
Describe problems with mean-variance optimization
-Highly concentrated portfolios
-Unstable portfolio weights that change drastically with input changes
-Not implementable -> extreme short selling
-Not possible to incorporate own view
-Variance is not a good risk measure if return distributino is non-normal
-Thus often poor out-of sample performance
How can we solve MPT problems?
Constrain the output
-No short selling constraint
-Restrict maximum weights
Improve the inputs
-Covariance matrix (by increasing data frequency, factor models and shrinkage methods)
-Expected return (by extending data sample and black litterman approach)
Describe Black Litterman model and the steps
-Approach to combine investors own subjective view with markets opinion
Steps
1. retrieve equilibrium market weights and compute covariance matrix of assets return using historical data
2. calculate benchmark - implied expected assets return through reverse optimization
3. express investor view Q and confidence level of views omega
4. compute investors adjusted ER
Describe BL pro’s and cons
Pros
-Avoid extreme portfolios
-More stable weights
-Allows investors to integrate own views
-Ensure that forecasts are internally consistent
-Manager does not need to have a view on all assets
Cons
-Requires assumption about market risk aversion
-Unclera how to determine own view and its uncertainty
-Assumes that returns follow normal distribution
What is Auto-correlation (function)?
Correlation of current value with previus value
Problem’s with MVO
- concentration
- unstable weights
- unimplementable portfolios
- can’t incorporate own views
- sometimes returns are nonnormally distributed
What is delta T for a 0.5 month period?
Its 1 /24th of a year so 1/42
Garch volatility estimator
σ_n^2 = γ * VL + αu_{n-1}^2 + βσ_{n-1}^2
With 7 being gamma, VL being long-term volatility, the second term being alpha * return^2 and the third term being beta * var
EWMA volatility estimator
σ_n^2 = λσ_{n-1}^2 + (1 - λ)u_{n-1}^2
Lambda * var+ (1-lamda) * return^2
T-day X% value at risk with
VaR = Pt√TσpN^(-1)(X)
With PT = portfolio value
N^(-1)(X) equalling Z-value of the Xth percentage
Stock price path in the risk-neutral world
S_t+δt = S_t * e^((rf - (σ^2)/2)δt + σ√δtZ)
So stock price * exp (return-var/2)*delta t + stdev * sqrt (delta T) * Z
Z being random number of deviations from the mean
Sharpe ratio
Sp = (rp - rf) / σp
Information ratio
αp / σ(εp)
So alpha portfolio / idiosyncratic stdev
Treynor ratio
Tp = (rp - rf) / βp
Gamma and vega hedging
[ option1* ] = [ r1 g1 ]^-1 [ -Γp ]
[ option2* ] [ v0 g2 ] [ -νp ]
So Mmult matrix inverse of option greeks * portfolio values
Then recalculate delta
Hedge delta with -delta stocks
State prices and risk-neutral state prices
State-prices:
qu = (R - D) / (R(U - D))
qd = (U - R) / (R(U - D))
Risk-neutral state prices
πU = (R - D) / (U - D)
πD = (U - R) / (U - D)
With pay-off for risk-neutral needing to be discounted
Opinion-adjusted weights (so for black litterman/envelope portfolio/market portfolio with RF rate)
z = S^(-1)(E(r)^adj - rf)
With S being the var/covar matrix and ^-1 being its inverse
Black-Litterman steps with formulas
- E(R) market= E(r)_market = Sm* + rf -> so not weights * av. return!
- Normalization factor: (m^opinion - rf) / (m^T Sm)
(M^opinon / E(r)) also seems to work!
- Normalized expected returns
E(r)_normalized = Sm^opinion - rf + rf / (m^T Sm* + rf)
Although old return * normalization factor also seems to work
- Adjust returns for N individual stock opinions
μ^adj_i = μ_i + σ_i,1^2 * τ_1 + … + σ_i,j^2 * τ_j + … + σ_i,N^2 * τ_N
Which is basically the old stock price + (varstock/varotherstock) * delta for otherstock
Of course a delta for the stock itself will have varstock/varstock = 1 and thus be fully incorporated
- calculate opinion adjusted weights
z = S^(-1)(E(r)^adj - rf)
With S being var/covar matrix
M* being weights
M^opinion opinion on market returns
Portfolio var
σp^2 = w^T Σw
σp^2 is the variance of the portfolio.
w is the vector of weights of the assets in the portfolio.
w^T is the transpose of the vector of weights.
Σ (Sigma) is the covariance matrix of the asset returns.
Delta
N(D1) for call options
N(-D1) for put option
Which is norm.s.dist function (or perhaps normsdist also works)
Calculation of expected shortfall
Average loss within VaR
So if 5% VaR would be a loss of 6% which equals 60k
The average loss if the loss of 6% is met or exceeded could be 7.5%
Then expected shortfall would be 75k
Stock movement in up state
U = e^(stdevsqrt(delta t)
or exp(stdevsqrt(delta t)
D = exp(stdev*-sqrt(delta t)
Optimization through shrinkage
Create var-covar matrix, and another version where only the variance is included
Then weigh them, so for example 0.3 * S + 0.7 * S-shrunk
Makes extreme covariances smaller and thus outcomes more realistic
Put-call parity
C + Ke^(-rT) = P + S
where:
C = call premium
Ke^(-rT) = present value of the strike
P = put premium
S = the current price of the underline
Option-price change formula
First order approximation df = Δ x δS
Second order approximation df = Δ x δS + 1/2 Γ x (δS)^2
So you can approximate the delta in option price with delta * delta option price
However to be accurate, take this, and 0.5 * gamma * delta stock^2
So for example a delta is 0.6, and gamma is 0.03, stock price increases with 1.
Estimated new option price is 0.6 higher (0.6 * 1)
Accurate estimate includes gamma: 0.6 * 1 + 0.03 * 0.05 * 1^2 = 0.0615
New delta after price increase is 0.6 + 0.03 = 0.63
An option has a delta of 0.5. Is it a put or a call?
Call, puts always have 0 or lower delta.
What is a volatility smile?
-Distribution of rates tends to exhibit kurtosis (fat tails)
-Lognormal distribution assumed by BS model underestimates the probability of extreme price movements, especially for FX rates
-OTM calls and puts have higher probability of ending ITM than under lognormal distribution -> obeserved market prices reflect this probability and are thus higher than implied by BS
-So higher prices due to higher implied volatility
-Called a smile because the implied volatility for both extremely low and high strike prices is higher, thus creating a smiley without eyes.
Describe volatility skew
Relating to options
-Typically seen for equity options
-Stock prices have negative skewness (heavy left tail) -> equity options with lower prices usually have higher volatility
-Extreme negative movements are underestimated by BS
-OTM call has lower probability of ending up ITM than under lognormal distribution -> lower expected payoff -> lower price and IV
-OTM put has higher probability of ending up ITM than underl ognormal distribution -> higher expected payoff -> higher price and IV
-Called a skew because volatility decreases as strike price increases
How do calls and puts respond to dividends
Call prices will be lower, as cash dividends will decrease stock price increases.
Vice versa puts will be higher, as price increases are lower.
Stock-dividends are more complex, and accounted for in the contract.
Why may options be preferred over stocks?
They create a levered payout and can thus be useful if you want to achieve something with limited upfront payment. Conversely, risk is higher.
Through leverage, we can achieve insurance value. A relatively cheap options can insure against a disaster.
Name three main goals of options
hedging
Speculation
Arbitrage
Name some advantages of options
-Redundant assets (synthetically) but make risk transfer and speculation cheaper and more effective due to
-Reduced TX fees
-Lump many transactions together
-Provide ways to make leveraged bets
-Regulatory arbitrage: sometimes options can circumvent regulatory restrictions
Describe the covered call and its use
Long stock + short call
Reduces volatility by capping upside profits and reducing downside losses. To make money, you should correctly anticipate future stock volatility to be lower than market’s expectation.
Describe a protective put
Long stock, l;ong put
Strongest performane during bear markets. Protects the portfolio from decline, but you can capitalize on portfolio returns if the market performs very well.
Describe a collar
Long stock, long put, short call (different strikes)
Brackets the portfolio betweeen two bounds and reduces the cost of the protective put at the expense of giving up some profit potential.
Describe straddles and strangles
Bets on volatility. Pay off is a V-shape for a straddle, or a V shape with space in between for a strangle.
Straddle is better, but more expensive.
Payoff is achieved for both upward and downard volatility.
What is the put-call parity formula and intution
c + Ke^r-t = p + s0
Use it it to calculate prices, or find out arbitrage opportunities if violated
For European options, describe the impact of these factors on call & put option prices, so negative or positive relation
- Price of underlying
- Strike price
- Time to expiration
- Volatility of underlying
- Interest rate
- Dividends
Price of underlying: positive for call, negative for put
Strike price: negative for call, positive for put
Time to expiration: unknown
Volatility: both positive
Interest rate: positive for call, ngative for put
Dividends: negative for call, positive for put
so meaning: higher strike price is lower value for calls, since the relationship is negative.
For interest rates:
European Call Options:
Increase in Interest Rates: When interest rates increase, the present value of the strike price (which is paid at expiration) decreases, making it less costly to carry the position until expiration. Therefore, it’s cheaper to buy the stock at a later date for a fixed price, which increases the value of the call option.
Decrease in Interest Rates: Conversely, if interest rates decrease, the present value of the strike price increases, making the option less attractive since it becomes more expensive to buy the stock at the strike price in the future. Hence, the value of the call option decreases.
European Put Options:
Increase in Interest Rates: An increase in interest rates makes it more attractive to sell the stock at the strike price in the future since the proceeds can be reinvested at the higher rate. Additionally, the cost of holding cash (the proceeds you’d get if you sold the underlying asset now and held the cash until the option expiration) is higher. This makes the put option less valuable.
Decrease in Interest Rates: A decrease in interest rates reduces the future value of the cash you would get from selling the underlying asset at the strike price, making the put option more valuable because the alternative (holding cash) is less attractive.
What is the hedge ratio
The number of options that need to be held to hedge a stock..
(Payoff up - Payoffdown)/(Return up - Return down)
Probably not a formula that needs to be known. The intuition is that you would need 3 options to hedge 5 shares for example. Then the hedge ratio would be 3/5th (or -3/5th?)
What is the intuition behind the risk-neutral world for option pricing
Risk-neutral world is a hypothetical world in which investors do not require premium for risk, the expected return on all assets is the risk-free rate, but the option price computed in the risk-neutral world equals the option price in the real world.
Name the assumptions underlying Black-scholes
- No TX costs /taxes
- Riskless borrowing and lending and short selling are possible
- No arbitrage
(These assumptions imply perfect markets) - Underlying asset prices follows brownian motion -> lognormal distribution and continuous asset price ( no jumps)
- No dividends during life of derivative
- security trading is continuous
- Rf rate and vol are constant
If vol is time varying, asset price no longer has a lognormal distribution so returns are no longer normal.
intuition:
If we cut δt small enough and add enough time steps, binomial tree converges to distribution behavior of geometric Brownian motion.
If stock price follows geometric Brownian motion, option price and stock price depend on the same underlying source of risk.
So also, in continuous time we can set up a riskless portfolio consisting of stock and option
Hedging argument: choose delta such that the delta of the stock - f is risk-free rate → Continuous rebalancing needed to keep portfolio risk-free
Portfolio is riskless (in this small-time interval) and must earn risk-free rate → Derive BS formula by solving PDE
Magic → Only volatility matters, we do not need to worry about risk and risk premium if we can hedge away the risk completely (in all states).
Under black scholes, does the option value depend on the expected rate of return on a stock?
No, this information is already built into the formula with the inclusion of the stock price, which already reflects the stock’s return and risk characteristics.
Given the BS formula: c = e^(-rT)[S_0 N(d1)e^(rT) - KN(d2)]
Describe N(D1) for calls or n(-d1)
and N(d2)
N(d1) is the delta for call and N(-d1) is the delta for a put. So appreciation per appreciation of 1 of the underlying.
N(d2) is risk-neutral probability that a call will be exercised. K*N(d2) is the expected cost of exercise in the risk neutral world.
What is the intuition behind delta for a stock/option
Stock delta is always 1, as it always increases by 1 relative to a 1 increase of the underlying (the stock itself)
Delta of an option varies. Deep ITM should be close to 1, deep OTM should be close to 0.
What kinds of volatility are there? Describe ways to calculate
- Investors own estimate
Calculate with historical volatility, EWMA, Garch
Used as input in BS - Implied volatility (forward looking)
Solve BS for volatility to determine what volatility is implied
Forward looking
In simple words, describe Delta, Gamma and Vega
Delta: impact of small change in asset price (similar to duration)
Gamma:P impact of large change in asset price - 2nd order effect (similar to convexity)
Vega: impact of change in volatility
Greeks are partial derivatives of the BS formula
Why use greeks? And what are their drawbacks?
Benefits:
-Simple to compute in real time (analytical expressions)
-Allow us to approximate change in position value when underlying risk factors change
-Express exposures of different positions to underlying risk factors in comparable terms (same scale)
-They can be aggregated across assets in portfolio
-Easy to hedge by trading in derivatives and underlying
Drawbacks:
-Greeks change when underlying risk factor changes → Should be recomputed frequently and hedge position should be adjusted
-Greeks depend on validity of pricing model used to compute them
-Greeks for American options need to be computed based on a tree
What is the delta for a call? And for a put? express in terms of a formula.
What are normal values?
Delta call = (Cu-Cd)/(Su-Sd)
Delta put = (Pu-Pd)/(Su-Sd)
Between 0 and 1 for a call and -1 and 0 for a put. Can be higher or lower when positions are aggregated.
What is an absolute value?
The numerical part of a value:
So absolute value of -1 equals 1, and absolute value of 1 equals 1.
Simply put, you take the value rather than the negative or positive value.
Will delta increase or decrease as maturity increases?
Delta will decrease for an in the money option, and increase for an OTM option.
Intuiton is that the longer the maturity, the higher the chance the option may not be ITM/OTM in the end.
Describe Gamma in relation to Delta
Gamma is the amount Delta will increase or decrease if the stock price changes with 1
So for a delta of 0.7 and a gamma of 0.2, if price increases with 1, delta will become 0.68 afterwards
So delta is a quick approximation, but you need gamma for the exact figures.
Explain option ‘‘convexity’’
Call options have positive convexity, stock price increase has larger effect on call price than stock price decrease (which is nice for option holder)!
This goes for both call & put, it’s the same for both
Option gamma measures rate of change of delta with respect to price of underlyinhg asset
So Delta can be seen as driving speed, gamma can be seen as acceleration
What does high gamma imply?
Frequent changes to delta, so frequent need for rebalancing to be delta-neutral, which makes hedging more expensive
Describe VaR
is the loss level V during time period of length T that we are X% certain will not be exceeded → with probability X, we will not lose more than V dollars on the portfolio over the next T
days.
What determines VaR
Portfolio composition (exposure to risk)
Investment horizon (size of risk)
confidence level (type of risk)
Can actual los frequency exceed VaR?
Yes, due to chance.
Advantages and disadvantages of VaR
Pros:
-Single number that summarizes total risk of FI → Aggregates all Greek letter for all risk factors underlying portfolio into 1 number
-Intuitive: easy to understand for top management
Cons:
-VaR does not tell how large loss can be if VaR is exceeded
-VaR is not an adequate measure of tail risk
-VaR is inherently backward-looking → Sensitive to data input → Perform stress test
-VaR is not a coherent risk measure for non-normal distributions, VaR of portfolio can exceed sum of VaRs of individual positions because it does not take into account that diversification lowers risk.
-The main weakness of VaR is tail risk, because VaR does not take into account the shape of the tail
Advantages and disadvantages of historical simulation method
Pros:
-Conceptually simple and historical data usually available
-Historical correlation between risk factors automatically -incorporated
-VaR can be computed for portfolio with non-linear assets
-Non-parametric: no distribution assumptions needed for risk factors
Cons:
-Depends fully on one historical price path → historical sample may not include any extreme events would produce a too low VaR
-Accuracy of VaR depends on number of observations on risk factors → trade-off between timelines (recent observations) and precision (more observations)
-Computationally intensive (slow) → portfolio revalued many times
Advantages and disadvantages of delta normal method
Pros:
-Easy to implement and computationally fast (for simple portfolio)
Theoretically appealing (based on Markowitz portfolio theory)
-Variance-Covariance matrix can be updated using DCC-GARCH models → Can allow for time-varying volatilities and correlations
-In theory, VaR estimate more precise than with historical simulations
Cons:
–If key assumptions fail, VaR estimate can be severely biased (we would prefer less precision over high bias, such as with the historical simulations)
-Cannot handle assets that are non-linear in risk factors (options).
-Linear models fail to capture skewness in probability distribution of option portfolio, even if stock distribution is normal. When assuming normal distribution, VaR underestimated for negative gamma portfolio and overestimated for positive gamma portfolio.
-Cannot handle risk factors that have a non-normal distribution
-Estimating variance-covariance matrix of a large portfolio is hard
Monte carlo simulation approach advantages and disadvantages
Pros:
-Flexible: can assume various distributions to account for non-normality
-Can handle non-linear products, such as options
-Does not require a lot of historical data (only for choosing parameters)
-Large number of simulations can be generated → Increases precision
-VaR can be computed for high confidence levels
Cons:
-If the assumed return distribution (model risk) is wrong, VaR can be severely biased
-Computationally inefficient (slow) because the complete portfolio has to be revalued many times
-> may use delta/gamma approximation to speed up the
calculation of change in value of some portfolio components
Describe expected shortfall
ES is the expected loss during time T conditional on the loss exceeding the Xt percentile of the loss distribution. Meaning it is conditional on exceeding VaR and ES measures expected tail loss, so it takes tail-risk into account.
Qu SPSU formula
State price up formula
Qd SPSD formula
State price down formula
Risk neutral state price formula (up scenario)
Risk neutral state price down scenario
Put call parity formula
If up/down state then stock moves with formula
Bond price movement regardless of state
Formula
Formula of a sub period
Delta T = always in terms of years
0.5 month = 24 periods
VaR (delta normal) formula
Variance formula in excel (matrix)
Portfolio variance: weight transposed * variance covariance matrix * weight vector.
What is the formula of Generalized Autoregressive Conditional Heteroskedasticity (GARCH)
Multiply gamma with long term variance + alpha * pervios return^2 + beta * Previous variance
What is the formula of Computing exponentially weighted moving average volatility (EWMNA)
How do you calculate the Z’s and asset returns for monti carlo simulation
=$correlation$Z1+SQRT(1-$correlation$^2)Z2
The stronger the correlation, the more closly the movements of the second asset will follow the first asset. And this is + second assets will be scaled by the square root of 1 - correlation^2
How do you hedge gamma?
-negative of portfolio gamma / the options gamma
How do you hedge delta?
and gamma (not considering vega)
How to hedge delta, gamma and vega?
- matrix: Greeks go in rows, and calls in columns
- portfolio greeks go in table
- Mmult(mininverse(matrix,vector)
Dont forget control shift enter - what delta position do we need? (org delta + delta option 1 + delta option 2
- opposite delta
table: Greeks go in rows, and calls in columns
B) Suppose that the stock price of Coca Cola increases by $2 and its volatility goes down by 2%.
Compute the change in the value of option 1 using a delta-gamma-vega approximation.
What are the steps in risk neutral options simulation?
How can We can simulate stock price paths from the ”real world” by drawing random numbers Z from a standard normal distribution:
We can simulate stock price paths from the ”risk-neutral world” by drawing random numbers Z from a standard normal distribution:
Mu is replaced by RF
What can you expect from the greeks in the covered call?
Long call: long delta
Short call: reduced delta, reduced vega
total: downside of covered call is the reduced ability to get gains from stock price (delta)
Vega: hoping for low vol, so negative vega
What can you expect from the greeks in a straddle?
investors would like volatility -> a high vega
At the money -> delta are slightly higher than 0.5, thats why not 0.5
What is the setup of a binomial pricing model
Formula to calculate a sub period (delta t)
Formula to calculate a up/down state stock move
= Exp(stock vol*sqrt delta t)
Continously Compounded Interest Rate
formula
State price up formula
State price down formula
What is the put-call parity equation
When is it helpful?
This equation essentially states that the difference between the price of a call option and the price of a put option with the same strike price and expiration date is equal to the difference between the current stock price and the present value of the strike price.
Usually when it is asked “what must be the price”. you can use a no arbitrage argument.
what does the payoff of a covered call look like and when is it helpful
BXM provides long equity exposure (positive delta) and short volatility exposure (negative vega)
what does the payoff of a protective put look like and when is it helpful
PPUT provides long equity exposure (positive delta) and long volatility exposure (positive vega)
what does the payoff of a collar look like and when is it helpful
Its often used when you already own the stock. combo of covered call and a protective put. It has a floor and a cap on the price. youre protecting from vol moves, but u cant gain or lose much. You might have a very specific stock.
You own stock
You hedge against downside; so put
When ppl hedge by buying a put option, but with as little upfront capital as possible –> shorting call options. so you cheaply limited downside potential but at the cost of upside potential.
What is a bull spread
Bull spread occurs when traders use strike prices between the high and low prices at which traders want to trade a security = position gets better when stock price increases. Similar to a collor, but where a collar is a non directional move, a spread is directional. We are betting also on low vol move, but also to go up.
Bull with puts advantage; initial cash inflow
short 22 dollar strike, in order to make it a bull spread, it needed to be higher than the strike 18
right top purple; short call that we wrote with a lower strike price must be more valuable (faster itm), so the cost of the long puts is more than offset
What is a bear spread
While bear spread occurs when a trader sells a call option at a strike price, and then buys it at a higher strike price later, = position gets worse when stock price increases. Similar to a collor, but where a collar is a non directional move, a spread is directional. We are betting also on low vol move, but also to go up.
bear with calls advantage; initial cash inflow
Left top purple; short call that we wrote with a lower strike price must be more valuable (faster itm), so the cost of the long call is more than offset
What positions do you take in a straddle and strangle
Value at risk (var) formula (delta normal method)
- As its measured in euros and you need value of portfolio –> pt
- Its a risk measure so sdev; Sdev doesn’t make assets move
- you need sth else to make sigma move, Z
- its scaled with square root T, as u need horizon, (10 day value at risk, 100 day)
portfolio variance in excel formula
Steps in doing historical simulation to calculate Var and expected shortfall
4 steps
What are the steps of monti carlo VAR and expected shortfall simulation?
5 steps
How do you calculate the Z’s and asset returns for monti carlo simulation
Z3 its not just your normal.s. inverse
Z3 is a combo of z1 and z2. z1 force that drives asset 1. and z2 is the internal force of asset2 except asset 2 is not just driven by its own randomness but also of asset 2.
The stronger the correlation, the more closly the movements of the second asset will follow the first asset. And this is + second assets will be scaled by the square root of 1 - correlation^2
What is the formula of exponentially weighted moving average (EWNA)
variance not volatility
What is the formula of Generalized Autoregressive Conditional Heteroskedasticity (GARCH)
Multiply gamma with long term variance + alpha * previous return^2 + beta * Previous variance
portfolio volatility not with matrix algebra
What are the 4 types of risk
What are the steps of computing exponentially weighted moving average volatility (EWMNA)
What are the main differences between hedge funds and index funds?
consider: Sharpe ratio, transparency , liquidity, risk exposures, constraints, capacity, trading, fees
How do hedge fund fees work?
Management fee plus incentive/performance fee
- management: usually 2% of Asset Under Management + incentive 20% of gains (2/20 scheme)
- Incentive fee can be modeled as call option -> may encourage
excessive risk taking by HF manager (asymmetric payoff). for every dollar above hurdle, manager get 20 cents
Why do we care about hedge Fund Alphas and Betas
Why do we care about alphas and betas?
1. No point to pay fee for beta exposure that you can get yourself
through index fund or ETF only pay for alpha!
2. Allows to check if HF manager follows proclaimed strategy
3. Investor can short market index (futures/ETF) to remove market risk
Why is the performence of market following strategies hard to measure?
The strategy is often dynamic and non linear
Non-linear HF returns due to market timing
* Intercept is alpha, slope is beta biased!
Solution: allow for separate up- and down-market betas
How do you denote weights in matrix algebra?
How do you denote returns in matrix algebra?
How do you denote the variance covariance in matrix algebra?
How do you calculate portfolio returns using matrix algebra
How do you calculate portfolio variance using matrix algebra
How to go from variance to volatility (standard deviation) using matrix algebra?
How do you calculate the variance of the CML portfolio?
Market weight^2*variance
How do we calculate the minimum variance portfolio returns
step 2: standardise
Sign next to s is a matrix 1, so multiply var cov with 1 vector
What are the steps to calculate the efficient portfolio with short sale constraints
- choose some random values for weights
- make sure to have a total of 100%
- Calc portfolio statistics including sharpe
- open solver “max” sharpe by including constraints
What are the steps in the black-litterman model
Step 2 BL: Calculate the expected returns that are implied by the mcap-weighted portfolio
Step 3 BL: Incorporate our opinion of the expected market return
output is the factor
Our opinions are always in excess of rf / variance of market.
Step 4 BL: Re-calculate expected returns with normalization
*also give the check you do here
Var cov * marketweights * normalisation + rf
Check: mmult weights * new implied returns = your opinion
Step 5 BL: incorporate our stock opinions
We use the covariance matrix which makes elegant. (opinion on one stock impacts other stocks)
formula:
Market implied return + (small component that depends on the covariance of the stock we’re adjusting with stock we have opinion about / stock we have opinion about) * opinion
always fix opinion and denominator (variances)
Step 6: Calculate opinion-adjusted optimal portfolio
Look a lot like market portfolio but with adjusted er
RBSA step by step
- make a returns row with portfolio returns, replication and error row next to the building block row. Add a weights row above the building blocks
- Replication formula: sumproduct(buildingblocks, weight row), error formula= portfolio-replication)
- make an error variance, total weights and r2 cell
- use solver objective; min error variance with the weights row. Add constraints
What is the result of the black jensen scholes research?
The SML is too flat. On average, portfolios with a low beta have historically had positive alpha’s and for alpha’s its the other way around.
A) would you say portfolio X outperformed or underperformed compared to the S&P 500
b) Briefly explain the conflict of results
What is the sharpe ratio and when is it appropiate to use?
Sharpe ratio measures return compensation per unit of total risk,
with risk measured by std. dev. -> slope of capital allocation line
Appropriate when portfolio represents entire investment for an
individual -> total volatility matters
What is the information ratio and what is the formula
The information ratio divides the alpha of the portfolio by nonsystematic risk,
called “tracking error” in the industry. It measures abnormal return per unit of risk that in principle could be diversified away by holding a market
index portfolio
If a stock is part of the index market fund, you can use the information ratio to measure the “contribution to the overal sharpe” . a higher information ratio adds more value to the portfolio
How do you calculate the treynor measure
When employing a number of managers in a fund, the nonsystematic risk of each will be largely diversified away, so only systematic risk is relevant. The appropriate
performance metric when evaluating components of the full risky portfolio is now the Treynor measure. This reward-to-risk ratio divides expected excess return by systematic risk (i.e., by beta).
The index regression model formula
What is the sharpe ratio and when is it appropiate to use?
Sharpe ratio measures return compensation per unit of total risk,
with risk measured by std. dev. -> slope of capital allocation line
Appropriate when portfolio represents entire investment for an
individual -> total volatility matters
What is the Sortino ratio and when is it appropiate to use?
Sortino ratio measures return compensation per unit of bad
(downside) risk, with risk measured by downside deviation
* Downside deviation is std. dev. of all returns < Rf
* Idea: upside volatility is beneficial to investors not a risk
* Std. dev. overstates risk for positively skewed return distributions
-> Sharpe ratio may increase when removing largest + returns!
* Std. dev. understates risk for negatively skewed distributions
What do you see here
Left: positive skewness; trend following hedgefunds
right: negative skewness; hedgefund that sell out of the money
same sdev, negative skew has way more downside risk
right; sdev overstated risk
left; understates true risk
Covered Call
Long stock, short call
BXM provides long equity exposure (positive delta) and short volatility exposure (negative vega)
protective put
long stock, long put
Limits downside, unlimited upside but profits will always be lower than the stock
PPUT has lowest downside deviation -> Sortino ratios of various strategies very similar
PPUT provides long equity exposure (positive delta) and long volatility exposure (positive vega)
Collar
long stock, long put, short call different strike prices,
brackets portfolio between two bounds and reduces costs of only the protective put at the expense of giving up profit potential
Bull spread
Buying a call with X1 and selling a call with 2x > X1
Buying a put with with X1 and selling a put with X2 > X1
pays off if the underlying appreciates
Bull put spread advantage: initial cash inflow
Bear spread
Buying a call with X2 and selling a call with X1 < X2
Buying a put with with X2 and selling a put with X1 < X2
pays off if the underlying depreciates
Long in the higher strike price
Call bear spread advantage: initial cash inflow
Straddle
long call and put with same strike price
