Volatility Trading using Options
Derivatives-based Portfolio Solutions


 

While directional investors typically use options for their equity exposure, volatility investors delta hedge their equity exposure. A delta-hedged option (call or put) is not exposed to equity markets, but only to volatility markets. We demonstrate how volatility investors are exposed to dividend and borrow cost risk and how volatility traders can ‘pin’ a stock approaching expiry.


VOL TRADING VIA CALLS AND PUTS IS IDENTICAL (PUT-CALL PARITY)
A forward is a contract that obliges the investor to buy (or sell if you have sold the forward) a security on a certain expiry date (but not before) at a certain strike price. A portfolio of a long European call and a short European put of identical expiry and strike is the same as a forward of that expiry and strike. This means that if a call, a put or a straddle is delta hedged with a forward contract (not stock), the end profile is identical. We note put-call parity is only true for European options, as American options can be exercised before expiry (although in practice they seldom are).

Delta hedging must be done with forward of identical maturity for put call parity
It is important to note that the delta hedging must be done with a forward of identical maturity to the options. If it is done with a different maturity, or with stock, there will be dividend risk. This is because a forward, like a European call or put, gives the right to a security at maturity but does not give the right to any benefits such as dividends that have an ex date before expiry. A long forward position is therefore equal to long stock and short dividends that go ex before maturity (assuming interest rates and borrow cost are zero or are hedged). This can be seen from the diagram below, as a stock will fall by the value of the dividend (subject to a suitable tax rate) on the ex date. The dividend risk of an option is therefore equal to the delta.


BORROW COST IMPACT ON OPTION PRICING
From a derivative pricing point of view, borrow cost (or repo) can be added to the dividend. This is because it is something that the owner of the shares receives and the owner of a forward does not. While the borrow cost should, in theory, apply to both the bid and offer of calls and puts, in practice an investment bank’s stock borrow desk is usually separate from the volatility trading desk (or potentially not all of the long position can be lent out). If the traders on the volatility trading desk do not get an internal transfer of the borrow cost, then only one side of the trade (the side that has positive delta for the volatility trading desk, or negative delta for the client) usually includes the borrow cost. While the borrow cost is not normally more than 40 bps for General Collateral (GC) names, it can be more substantial for emerging market (EM) names. If borrow cost is only included in one leg of pricing, it creates a bid-offer arbitrage channel.

Zero delta straddles still need to include borrow cost on one leg of the straddle
Like dividends, the exposure to borrow cost is equal to the delta. However, a zero delta straddle still has exposure to borrow cost because it should be priced as the sum of two separate trades, one call and one put. As one of the legs of the trade should include borrow, so does a straddle. This is particularly important for EM or other high borrow cost names

Zero delta straddles have strike above spot
A common misperception is that ATM options have a 50% delta; hence, an ATM straddle has to be zero delta. In fact, a zero delta straddle has to have a strike above spot (an ATM straddle has negative delta). The strike of a zero delta straddle is given below.
Strike (%) of zero delta straddle = e(r + (σ.σ)/2).T


DELTA HEDGING AN OPTION REMOVES EQUITY RISK
If an option is purchased at an implied volatility that is lower than the realized volatility over the life of the option, then the investor, in theory, earns a profit from buying cheap volatility. However, the effect of buying cheap volatility is dwarfed by the profit or loss from the direction of the equity market. For this reason, directional investors are usually more concerned with premium rather than implied volatility. Volatility investors will, however, hedge the equity exposure. This will result in a position whose profitability is solely determined by the volatility (not direction) of the underlying. As delta measures the equity sensitivity of an option, removing equity exposure is called delta hedging (as a portfolio with no equity exposure has delta = 0).

Delta hedging example
As the delta of a portfolio is equal to the sum of the deltas of the securities in the portfolio, a position can be delta hedged by purchasing, or going short, a number of shares (or futures in the case of an index) equal to the delta. For example, if ten call options have been bought with a delta of 40%, then four shares (10 × 40% = 4) have to be shorted to create a portfolio of zero delta. The shares have to be shorted as a call option has positive delta; hence, the delta hedge has to be negative for the sum of the two positions to have zero delta. If we were long a put (which has negative delta), then we would have to buy stock to ensure the overall delta was zero.

Constant delta hedging is called gamma scalping
The rate delta changes as spot moves is called gamma; hence, gamma is the convexity of the payout. As the delta changes, a volatility investor has to delta hedge in order to ensure there is no equity exposure. Constantly delta hedging in this way is called gamma scalping, as it ensures a long volatility position earns a profit as spot moves.

Gamma scalping (delta re-hedging) locks in profit as underlying moves
We shall assume an investor has purchased a zero delta straddle (or strangle), but the argument will hold for long call or put positions as well. If equity markets fall the position will become profitable and the delta will decrease from zero to a negative value. In order to lock in the profit, the investor must buy stock (or futures) for the portfolio to return to zero delta. Now that the portfolio is equity market neutral, it will profit from a movement up or down in the equity market. If equity markets then rise, the initial profit will be kept and a further profit earned. 

Long gamma position can sit on the bid and offer
A long gamma (long volatility) position has to buy shares if they fall, and sell them if they rise. Buying low and selling high earns the investor a profit. Additionally, as a gamma scalper can enter bids and offers away from current spot, there is no need to cross the spread (as a long gamma position can be delta hedged by sitting on the bid and offer). A short gamma position represents the reverse situation, and requires crossing the spread to delta hedge. While this hidden cost is small, it could be substantial over the long term for underlyings with relatively wide bid-offer spreads.

Best to delta hedge on key dates or on turn of market
If markets have a clear direction (i.e. they are trending), it is best to delta hedge less frequently. However, in choppy markets that are range bound it is best to delta hedge very frequently. For more detail on how hedging frequency affects returns and the path dependency of returns, see the section Stretching Black-Scholes Assumptions. If there is a key announcement (either economic or earnings-related to affect the underlying), it is best to delta hedge just before the announcement to ensure that profit is earned from any jump (up or down) that occurs.


GAMMA HEDGING CAN ‘PIN’ A STOCK APPROACHING EXPIRY
As an investor who is long gamma can delta hedge by sitting on the bid and offer, this trade can pin an underlying to the strike. This is a side effect of selling if the stock rises above the strike, and buying if the stock falls below the strike. The amount of buying and selling has to be significant compared with the traded volume of the underlying, which is why pinning normally occurs for relatively illiquid stocks or where the position is particularly sizable  Given the high trading volume of indices, it is difficult to pin a major index. Pinning is more likely to occur in relatively calm markets, where there is no strong trend to drive the stock away from its pin.

Large size of Swisscom convertible pinned underlying for many months
One of the most visible examples of pinning occurred in late 2004/early 2005, due to a large Swiss government debt issue, (Swisscom 0% 2005) convertible into the relatively illiquid Swisscom shares. As the shares traded close to the strike approaching maturity, the upward trend of the stock was broken. Swisscom was pinned for two to three months until the exchangeable expired. After expiration, the stock snapped back to where it would have been if the upward trend had not been paused. A similar event occurred to AXA in the month preceding the June 2005 expiry, when it was pinned close to €20 despite the broader market rising (after expiry AXA rose 4% in four days to make up for its earlier underperformance).


OPTION TRADING RULES OF THUMB
To calculate option premiums and volatility exactly is typically too difficult to do without the aid of a tool. However, there are some useful rules of thumb that can be used to give an estimate. These are a useful sanity check in case an input to a pricing model has been entered incorrectly.
  1. Profit from delta hedging is proportional to square of return. Due to the convexity of an option, if the volatility is doubled the profits from delta hedging are multiplied by a factor of four. For this reason, variance (which looks at squared returns) is a better measure of deviation than volatility.
  2. ATM option premium in percent is roughly (2 × π)-1/2  × σ × t1/2. If one assumes zero interest rates and dividends, then the formula for the premium of an ATM call or put option simplifies to 0.4 × σ × √T. OTM options can be calculated from this estimate using an estimated 50% delta.
  3. Historical annualized volatility roughly equal to 16 × % daily moveHistorical volatility can be estimated by multiplying the typical return over a period by the square root of the number of periods in a year (i.e. 52 weeks or 12 months in a year). Hence, if a security moves 1% a day, it has an annualized volatility of 16% (as 16 ≈ √252 and we assume there are 252 trading days).

While a simple view on both volatility and equity market direction can be implemented via a long or short position in a call or put, a far wider set of payoffs is possible if two or three different options are used. We investigate strategies using option structures (or option combos) that can be used to meet different investor needs.

ATM option premium in percent is roughly (2 × π)-1/2  × σ × t1/2

Call price = S.N(d1) – K.N(d2) e-rT
ATM call price = S N(σ × √T / 2) – S N(-σ × √T / 2) as K=S (as ATM)
ATM call price = S × σ × √T / √(2π)
ATM call price = σ × √T / √(2π) in percent
ATM call price ≈ 0.4 × σ × √T in percent

where:
We assume zero interest rates and dividends (r = 0)
Definition of d1 and d2 is the standard Black-Scholes formula.
σ = implied volatility
S = spot
K = strike
R = interest rate
T = time to expiry
N(z) = cumulative normal distribution

OTM options can be calculated by assuming 50% delta
If an index is 3000 pts and has a 20% implied then the price of a 1y ATM option is approximately 240pts (3000×8% as per the formula above). A 3200 call is therefore approximately 240 - 50% (3200-3000) = 140 pts assuming a 50% delta. Similarly, a 3200 put is approximately 340 pts.

Profit from delta hedging is proportional to percentage move squared
Due to the convexity of an option, if the volatility is doubled, the profits from delta hedging are multiplied by a factor of four. For this reason, variance (which looks at squared returns) is a better measure of deviation than volatility. Assuming constant volatility, zero interest rates and dividend, the daily profit and loss (P&L) from delta hedging an option is given below.

Daily P&L from option = Delta P&L + Gamma P&L + Theta P&L
Daily P&L from option = S.δ + S.2γ/2 + tθ where S is change in Stock and t is time
Daily P&L from option - S.δ = S.2γ/2 + tθ = Delta hedged P&L from option
Delta hedged P&L from option = S.2γ/2 + cost term (tθ does not depend on stock price)

where:
δ = delta
γ = gamma
t = time
θ = theta

If the effect of theta is ignored (as it is a cost that does not depend on the size of the stock price movement), the profit of a delta hedged option position is equal to a scaling factor (γ/2) multiplied by the square of the return. This means that the profit from a 2% move in a stock price is four times the profit from a 1% move in stock price. 


VARIANCE IS THE KEY, NOT VOLATILITY
Partly due to its use in Black-Scholes, historically, volatility has been used as the measure of deviation for financial assets. However, the correct measure of deviation is variance (or volatility squared). Volatility should be considered to be a derivative of variance. The realization that variance should be used instead of volatility led volatility indices, such as the VIX, to move away from ATM volatility (VXO index) towards a variance-based calculation.


VARIANCE, NOT VOLATILITY, IS THE CORRECT MEASURE FOR DEVIATION
There are three reasons why variance, not volatility, should be used as the correct measure for volatility. However, despite these reasons, even variance swaps are normally quoted as the square root of variance for an easier comparison with the implied volatility of options (but we note that skew and convexity mean the fair price of variance should always trade above ATM options).
  1. Variance takes into account implied volatility at all stock prices. Variance takes into account the implied volatility of all strikes with the same expiry (while ATM implied volatility will change with spot, even if volatility surface does not change).When looking at how rich or cheap options with the same maturity are, rather than looking at the implied volatility for a certain strike (ie, ATM or another suitable strike) it is better to look at the implied variance as it takes into account the implied volatility of all strikes. For example, if an option with a fixed strike that is initially ATM is bought, then as soon as spot moves it is no longer ATM. However, if a variance swap is bought, then its traded level is applicable no matter what the level of spot. The fact that a variance swap (or log contract) payout depends only on the realized variance and is not path dependent makes it the ideal measure for deviation.
  2. Deviations need to be squared to avoid cancelling. Mathematically, if deviations were simply summed then positive and negative deviations would cancel. This is why the sum of squared deviations is taken (variance) to prevent the deviations from cancelling. Taking the square root of this sum (volatility) should be considered a derivative of this pure measure of deviation (variance).
  3. Profit from a delta-hedged option depends on the square of the return. Due to the convexity of an option, if the volatility is doubled, the profits from delta hedging are multiplied by a factor of four. For this reason, variance (which looks at squared returns) is a better measure of deviation than volatility.

VOLATILITY SHOULD BE CONSIDERED A DERIVATIVE OF VARIANCE
The three points above show why variance is the natural measure for deviation. Volatility, the square root of variance, should be considered a derivative of variance rather than a pure measure of deviation. It is variance, not volatility, that is the second moment of a distribution. 

VIX and VDAX moved from old ATM calculation to variance
Due to the realization that variance, not volatility, was the correct measure of deviation, on Monday, September 22, 2003, the VIX index moved away from using ATM implied towards a variance-based calculation. Variance-based calculations have also been used for by other volatility index providers. The old VIX, renamed VXO, took the implied volatility for strikes above and below spot for both calls and puts. As the first two-month expiries were used, the old index was an average of eight implied volatility measures as 8 = 2 (strikes) × 2 (put/call) × 2 (expiry). We note that the use of the first two expiries (excluding the front month if it was less than eight calendar days) meant the maturity was on average 1.5 months, not one month as for the new VIX.

Similarly, the VDAX index, which was based on 45-day ATM-implied volatility, has been superseded by the V1X index, which, like the new VIX, uses a variance swap calculation. All recent volatility indices, such as the vStoxx (V2X), VSMI (V3X), VFTSE, VNKY and VHSI, use a variance swap calculation, although we note the recent VIMEX index uses a similar methodology to the old VIX (potentially due to illiquidity of OTM options on the Mexbol index).

The variance term structure is not always flat
While average variance term structure should be flat in theory, in practice supply and demand imbalances can impact variance term structure. The buying of protection at the long end should mean that the variance term structure is on average upward sloping, but in turbulent markets it is usually inverted.
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Surévaluation de la volatilité implicite l’anomalie de marché à exploiter  165k v. 1 Jun 25, 2014, 1:08 PM QMS Advisors