CAIA L2 - 9.2 - Volatility, Correlation, and Dispersion Products and Strategies Flashcards
Formula
Theta
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Partial derivative of the price of the option, with respect to time
measures time decay
(while holding other factors such as asset price and anticipated volatility constant),
which quantifies how quickly the value of an option decreases as the option approaches expiration
θ’r=0’ = −(S N’(d) σ) / (2√T)
θ’r=0’ = theta, assuming the risk-free interest rate is zero (a simplifying assumption)
S = price of the underlying asset
N’(d) = (non-cumulative) probability density function of the normal distribution
σ = return volatility of the underlying asset
T = option’s time to expiration (i.e., its tenor) (annual basis => 3 months = T= 0.25)
Obs:
* Theta < 0 => because all of the components in this formula are positive
* Theta increases as the option nears expiration (i.e., as T→0).
* Gráfico de theta x stock => formato de uma normal invertida (maior theta = ATM)
The option theta will result in an ATM option’s price changing proportionally with the square root of time.
As a result, a six-month option will have
√0.5 = 71% of a one-year option’s premium,
and a three-month option will have
√0.25 = 50% of a one-year option’s premium.
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Describe
writing option
straddles and strangles
butterflies and condors
as short volatility strategies
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Straddle => long: \/ short: /\ - long call, long put, same underlying asset, same sexpiration date and strike price
Strangle => long: \/ short: /¯\ - short call and short put. differing strike prices for the call and put
Iron butterflies => long: ¯\/¯ short: /\. Short iron butterfly involves shorting a bull spread and shorting a bear spread with the middle strike price identical
Iron condor => long: ¯\/¯ short: /¯\. short iron condor involves shorting an OTM bull spread and an OTM bear spread
Short straddle / strangle = short vol (betting on lower realized volatility); it has almost zero beta and zero delta
Short Butterflies and Condors = short vol with limited downside
‘–
Obs:
Long iron butterflies and long iron condors are convex to the underlying asset price, meaning they curve upwards.
Those strategies have positive gamma and are long volatility, and therefore, result in small losses and larger profits when the asset moves in either direction.
.
Short iron butterflies and short iron condors are concave to the underlying asset price and curve downwards. Short strategies are short volatility and result in larger losses and smaller profits when the asset moves in either direction.
However, when an asset value moves significantly, the short volatility in a short butterfly or condor becomes long volatility and convex in the outer strike prices.
Neither iron butterfly or iron condor is exposed to sudden large price changes in the underlying assets
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
List
2 primary performance drivers
of a delta-neutral hedged portfolio
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
-
Price variation.
The driver refers to the amount and timing of price variation in the assets underlying the options. Here it is necessary to consider not just the amount of asset return variation but also the shape of that return distribution, when the variation happens, and how near to the money the variation occurs. -
Rehedging strategy.
The frequency of rehedging must also be considered, which will be based either on passage of time between rebalancing, or on a price band to allow price movements before rebalancing. Rebalancing may have a net bullish or net bearish position, although in a delta-neutral strategy the directional price risk is removed.
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Identify
4 key points
that surround
delta-neutral option portfolios
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
-
Effect of realized variation
Performance = “realized vol - implied vol” and “specifics of return variation”
The performance of a delta-neutral portfolio will primarily be determined by the difference between the realized volatility of the underlying asset versus the initial implied volatility of the options. However, a portfolio with long positions in options will also have long vega and long gamma. Performance will depend not only on the realized volatility of the underlying asset’s returns but also on the specifics of that variation including timing, variation relative to the strike price, and higher moments of the frequency distribution of price changes such as kurtosis and skewness. -
Rebalancing frequency
Rebalancing refers to the trades required to rehedge back to a delta-neutral position after a change in the underlying asset price. If the underlying asset’s price changes form a random walk (i.e., neither trending nor mean reverting), the time interval chosen will not (aside from trading costs) impact expected profitability and should not generate an expected risk-adjusted profit. -
Impact of rebalancing on returns
The optimal rebalancing frequency will differ depending on whether the underlying asset’s price change is trending or mean reverting. If the returns of the asset that underlies the options are mean reverting, more frequent rebalancing will (ignoring transaction costs) increase expected returns. The opposite is true if the returns of the underlying asset trend. However, transaction costs will increase with increasing frequency of rebalancing. -
Directional risk and recognition of profits
return not predictable = no rebal return
As mentioned previously, in the absence of a predictable pattern (trending, mean reverting) in the returns of the assets underlying the options, hedging itself is not expected to produce a return. However, the rebalancing process will tend to accelerate profit recognition, and also lower directional risk.Delta-neutral portfolios with positive gamma perform well when volatility is high, perform poorly when volatility is low and have no directional risk wrt changes in value of the underlying asset.
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Define
vega normalization
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
A way to “correct” real life variation of volatilities differences:
Options with higher implied volatility are likely to have larger absolute changes in volatility than options with lower implied volatility.
Ex:
volatility change from 5% to 5.5% for one option
would be equivalent to
a change from 10% to 11% for another option
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Define
Vertical intra-asset
option spreads
x
ratio intra-asset
option spread
x
Horizontal intra-asset
option spreads
x
Inter-Asset
option spreads
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
vertical spread (which is a type of skew spread)
involves long and short calls or short and long puts
with the same expiration dates,
but with different strike prices
Ex:
long 50 April 125 strike calls +
short 50 April 135 strike calls
‘–
ratio spread (which is a type of skew spread)
similar to a vertical spread but
does not have the same number of options on both sides
Ex:
long 50 April 125 strike calls +
short 25 April 135 strike calls
‘–
horizontal spread (which is a type of skew spread)
involves either long and short calls or short and long puts
with same strike prices
but with different expiration dates
Ex:
long 50 April 125 strike calls +
short 50 May 125 strike calls
‘–
Taking a long position in an asset whose implied volatility is anticipated to increase more than that of the implied volatility of the short position asset.
Ex:
long position in the NASDAQ Index option and a
short position in the S&P 500 option
is a long volatility position that will
perform well when there is financial stress (assuming that delta hedging is in place)
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
List and Describe
2 derivative strategies
that create payoffs driven by
realized variance
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
-
Variance swaps
are over-the-counter (OTC) instruments that generate returns that are directly linked to the realized volatility of an underlying asset. They can be used as an alternative to avoid the high transactions costs of hedging as well as the path dependence of options. Variance swaps involve the receipt of the realized annualized variance of a given asset and the payment of a fixed variance (or strike rate). For example, the strike rate could be set as 5.25 at the outset of the swap. Subsequently, suppose the realized variance of the underlying asset is 6. Therefore, the swap buyer will have a gain equal to 0.75 times the notional swap amount (paid by the swap seller). On the other hand, if the realized variance on the underlying asset is 4.5, then the swap buyer will have a loss (the swap seller will have a gain) equal to 0.75 times the notional swap amount (paid by the swap buyer). The computation of the strike rate is dependent on the option-implied volatilities. -
Variance futures
Less liquid than swaps
are relatively lightly traded compared to OTC variance swaps. Variance futures are available on such exchanges as the Eurex, the Chicago Mercantile Exchange (CME), or the EURO STOXX 50 Index. Variance futures contracts differ from other futures contracts in that the settlement amounts of variance futures are based on the observed (realized) variances in the returns of the associated assets or on observed variance swap rates.
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Describe
CBOE Volatility Index (VIX)
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
- market-based estimation of the 30-day implied volatility of the S&P 500 Index
- VIX products comprise most of the total volatility trading in the marketplace
- Variance swap rate is computed for two expiration dates: immediately before 30 days, and immediately after 30 days
- calculated as the square root of the interpolated values of the two variance rates
VIX Future:
* implied vol of D+1 to D+30 (D = settlement date)
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Formula
Pricing an
S&P VIX Short-Term Futures
Contract
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
30-day hypothetical contract price =
P’s’ (T1-30)/(T1-Ts) + P1 (30-Ts)/(T1-Ts)
Ps = price of the shorter-term contract
Ts = number of days to settlement of the shorter-term contract
P1 = price of the longer-term contract
T1 = number of days to settlement of the longer-term contract
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
List
VIX-Related Financial Derivatives
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
- VXX => ETN - long S&P 500 Short-Term VIX Futures Index.
- XIV , SVXY => ETN - short S&P 500 Short-Term VIX Futures Index.
- S&P 500 VIX Mid-Term Index => long rollover strategy 4-7m VIX term structure
- Other volatility-based indices exist that cover equities, commodities, and currencies.
- There are now more than 40 ETPs based on those indices, including the VIX of the VIX (VVIX), which has exposure to the volatility of VIX options.
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Relate
VIX term structure
to
portfolio insurance
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
- term structure curve is usually upward sloping (i.e., contango—futures prices above spot prices).
- long VIX futures positions usually incur losses
=> Such losses can be thought of as insurance payments to hedge long positions in risky assets
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Define
Correlation swap
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
is used to transfer risk between counterparties.
The value is based on the difference between the average correlation of a stated group of stocks versus the swap’s fixed strike correlation.
The payments are netted. The swap seller pays (and the swap buyer receives) the notional amount times the average actual correlation, which is a market-weighted average of the returns of each pair of stocks within the stated group of stocks. The fixed strike correlation rate is negotiated between the counterparties at the beginning of the transaction. The average correlation relates to the implied volatility smile.
Ex:
Consider a correlation swap of five equally weighted stocks in a portfolio. The swap has a notional value of $1 million and a fixed strike correlation rate of 0.48. One pair of stocks has an actual correlation of 0.25 with each other, two pairs of stocks have an actual correlation of 0.3 with each other, three pairs of stocks have an actual correlation of 0.5 with each other, and four pairs of stocks have an actual correlation of 0.55 with each other.
Determine which side needs to make a net payment, and calculate the amount of net payment.
Answer:
Five stocks can be labeled into 10 pair combinations. The average actual correlation is calculated as follows:
(0.25 + 0.3 + 0.3 + 0.5 + 0.5 + 0.5 + 0.55 + 0.55 + 0.55 + 0.55) / 10 = 0.455.
The payment is made by the buyer to the seller and is calculated as follows:
(0.455 – 0.48) × $1,000,000 = -$25,000.
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Assumed (average)
correlation coefficient
of the portfolio assets
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
ρ’average’ ≈ (σ’p’^2) / (σ’i’ ^2)
σ’p’^2 = variance of the portfolio return
σ’i’^2 = variance of asset i in the portfolio
Obs:
* portfolio is equally weighted
* return variance of each asset is equal
* return correlation (ρaverage) of each asset pair is equal (and positive)
* portfolio has a very large number of assets
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Recognize
Motivations to
correlation trading
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Profit in crisis (for correlation swap buyers)
In real life,
correlations usually increase during periods of financial stress especially when there is a spike in volatility, so correlation swap buyers, who are long volatility and short beta, should profit during such times.
The negative beta is a result of the negative correlations between equities and equity returns.
Compared to realized correlation,
correlation swaps (based on implied correlation) are frequently overpriced,
and profits can often be earned by selling correlation swaps.
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
Define
Dispersion trades
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
- position (long or short) in an index option +
- the opposite positions in the options of the index’s underlying stocks.
long index option + short options of the index underlying stocks = profit in crisis (dispersion is low)
short index option + long options of the index underlying stocks = profit in normal times (dispersion is high)
The value of a portfolio of options on individual assets tends to be higher than an option on a portfolio (index) of the same assets
Therefore, investors who are long the option on the portfolio and short the portfolio of options tend to generate profits.
The option positions are generally delta hedged
9.2 - Volatility, Correlation, and Dispersion Products and Strategies
