Directional Volatility Trading
Derivatives-based Portfolio Solutions
A BRIEF HISTORY OF VOLATILITY TRADINGWhile standardized exchange traded options only started trading in 1973 when the CBOE (Chicago Board Options Exchange) opened, options were first traded in London from 1690. Pricing was made easier by the Black-Scholes-Merton formula (usually shortened to Black- Scholes), which was invented in 1970 by Fischer Black, Myron Scholes and Robert Merton. The derivatives explosion in the 1990s was partly due to the increasing popularity of hedge funds, which led to volatility becoming an asset class in its own right. New volatility products such as volatility swaps and variance swaps were created, and a decade later futures on volatility indices gave investors listed instruments to trade volatility. In this section we shall concentrate on option trading. LONG OR SHORT STRATEGIES ARE POSSIBLE WITH OPTION TRADINGA European call is a contract that gives the investor the right (but not the obligation) to buy a security at a certain strike price on a certain expiry date (American options can be exercised before expiry). A put is identical except it is the right to sell the security. A call option profits when markets rise (as exercising the call means the investor can buy the underlying security cheaper than it is trading, and then sell it at a profit). A put option profits when markets fall (as you can buy the underlying security for less, exercise the put and sell the security for a profit). Options therefore allow investors to put on long (profit when prices rise) or short (profit when prices fall) strategies. Option trading allows investors to take a long or short position on volatility If the volatility of
an underlying is zero, then the price will not move and an option’s payout is
equal to the intrinsic value. Intrinsic value is the greater of zero and the
‘spot – strike price’ for a call and is the greater of zero and ‘strike price –
spot’ for a put. Assuming that stock prices can move, the value of a call and
put will be greater than intrinsic due to the time value (price of option =
intrinsic value + time value). If an option strike is equal to spot (or is the
nearest listed strike to spot) it is called at-the-money (ATM). If volatility
is zero, an ATM option has a price of zero (as intrinsic is zero). However, if
we assume a stock is $50 and has a 50% chance of falling to $40 and 50% chance
of rising to $60, it has a volatility above zero. In this example, an ATM call
option with strike $50 has a 50% chance of making $10 (if the price rises to $60
the call can be exercised to buy the stock at $50, which can be sold for $10
profit). The fair value of the ATM option is therefore $5 (50% × $10); hence,
as volatility rises the value of a call rises (a similar argument can be used
for puts). An ATM option has the greatest time value. This can be seen in the
same example by looking at an out-of-the-money (OTM) call option of strike $60
(an OTM option has strike far away from spot and zero intrinsic value). This
OTM $60 call option would be worth zero, as the stock in this example cannot
rise above $60. CHOOSING THE STRIKE OF AN OPTION STRATEGY IS NOT TRIVIALWhile it is relatively simple to pick the option strategy, choosing the strike and expiry is the most difficult part of an options strategy. Choosing the maturity of the option is easier if there is a specific event (eg, an earnings date) that is anticipated to be a driver for the stock. Choosing the strike of the trade is not trivial either. Investors could choose ATM to benefit from greatest liquidity. Alternatively, they could look at the highest expected return (option payout less the premium paid, as a percentage of the premium paid). While choosing a cheap OTM option might be thought of as giving the highest return, in reality the highest returns come from in-the-money (ITM) options (ITM options have a strike far away from spot and have intrinsic value). This is because an ITM option has a high delta (sensitivity to equity price); hence, if an investor is relatively confident of a specific return, an ITM option has the highest return (as trading an ITM option is similar to trading a forward). Forwards (or futures) are better than options for pure directional playsA forward is a contract that obliges the investor to buy a security on a certain expiry date at a certain strike price. A forward has a delta of 100%. An ITM call option has many similarities with being long a forward, as it has a relatively small time value (compared to ATM) and a delta close to 100%. While the intrinsic value does make the option more expensive, this intrinsic value is returned at expiry. However, for an ATM option, the time value purchased is deducted from the returns. ATM or OTM options are only the best strike (if an investor is very confident of the eventual return) if the anticipated return is very large (as leverage boosts the returns). For pure directional plays, forwards (or futures, their listed equivalent) are more profitable than options. The advantage of options is in offering convexity: if markets move against the investor the only loss is the premium paid, whereas a forward has a virtually unlimited loss. OPTION LIQUIDITY CAN BE A FACTOR IN IMPLEMENTING TRADESIf an underlying is relatively illiquid, or if the size of the trade is large, an investor should take into account the liquidity of the maturity and strike of the option. Typically, OTM options are more liquid than ITM options as ITM options tie up a lot of capital. This means that for strikes less than spot, puts are more liquid than calls and vice versa. We note that as low-strike puts have a higher implied than high-strike calls, their value is greater and, hence, traders are more willing to use them. Low strike put options are therefore usually more liquid than high-strike call options. In addition, demand for protection lifts liquidity for low strikes compared with high strikes. Single stock liquidity is limited for maturities of two years or more. For single stock options, liquidity starts to fade after one year and options rarely trade over two years. For indices, longer maturities are liquid, partly due to the demand for long-dated hedges and their use in structured products. While structured products can have a maturity of five to ten years, investors typically lose interest after a few years and sell the product back. The hedging of a structured product, therefore, tends to be focused on more liquid maturities of around three years. Hedge funds tend to focus around the one-year maturity, with two to three years being the longest maturity they will consider. The two-to-three year maturity is where there is greatest overlap between hedge funds and structured desks. DELTA IS THE DIVIDEND RISK, AS WELL AS THE EQUITY RISKThe delta of the option is the amount of equity market exposure an option has. As a stock price falls by the dividend amount on its ex-date, delta is equal to the exposure to dividends that go ex before expiry. The dividend risk is equal to the negative of the delta. For example, if you have a call of positive delta, if (expected or actual) dividends rise, the call is worth less (as the stock falls by the dividend amount). If a dividend is substantial, it could be in an investor’s interest to exercise early. DIFFERENCE BETWEEN DELTA AND PROBABILITY EXPIRES ITMA digital call option is an option that pays 100% if spot expires above the strike price (a digital put pays 100% if spot is below the strike price). The probability of such an option expiring ITM is equal to its delta, as the payoff only depends on it being ITM or not (the size of the payment does not change with how much ITM the spot is). For a vanilla option this is not the case; hence, there is a difference between the delta and the probability of being ITM. This difference is typically small unless the maturity of the option is very long. Delta takes into account the amount an option can be ITM While a call can have an infinite payoff, a put’s maximum value is the strike (as spot cannot go below zero). The delta hedge for the option has to take this into account, so a call delta must be greater than the probability of being ITM. Similarly, the absolute value (as put deltas are negative) of the put delta must be less than the probability of expiring ITM. A more mathematical explanation (for European options) is given below: Call delta > Probability call ends up ITM Abs(Put delta) < Probability put ends up ITM Mathematical proof option delta is different from probability of being ITM at expiryCall delta = N(d1) Put delta = N(d1) - 1 Call probability ITM = N(d2) Put probability ITM = 1 - N(d2) where: Definition of d1 is the standard Black-Scholes formula for d1. d2 = d1 - σ(T) ^{1/2}σ = implied volatility T = time to expiry N(z) = cumulative normal distribution As d2 is less than d1 (see above) and N(z) is a monotonically increasing function, this means that N(d2) is less than N(d1). Hence, the probability of a call being in the money = N(d2) is less than the delta = N(d1). As the delta of a put = delta of call – 1, and the sum of call and put being ITM = 1, the above results for a put must be true as well. The difference between delta and probability being ITM at expiry is greatest for long-dated options with high volatility (as the difference between d1 and d2 is greatest as well). STOCK REPLACING WITH LONG CALL OR SHORT PUTAs a stock has a delta of 100%, the identical exposure to the equity market can be obtained by purchasing calls (or selling puts) whose total delta is 100%. For example, one stock could be replaced by two 50% delta calls, or by going short two -50% delta puts. Such a strategy can benefit from buying (or selling) expensive implied volatility. There can also be benefits from a tax perspective and, potentially, from any embedded borrow cost in the price of options (price of positive delta option strategies is improved by borrow cost). As the proceeds from selling the stock are typically greater than the cost of the calls (or margin requirement of the short put), the difference can be invested to earn interest. It is important to note that the dividend exposure is not the same, as only the owner of a stock receives dividends. While the option owner does not benefit directly, the expected dividend will be used to price the option fairly (hence investors only suffer/benefit if dividends are different from expectations). Stock replacing via calls benefits from convexity As a call option is convex, this means that the delta increases as spot increases and vice-versa. If a long position in the underlying is sold and replaced with calls of equal delta, then if markets rise the delta increases and the calls make more money than the long position would have. Similarly, if markets fall the delta decreases and the losses are reduced. The downside of using calls is that the position will give a worse profile than the original long position if the underlying does not move much (as call options will fall each day by the theta if spot remains unchanged). Using call options is best when implied volatility is cheap and the investor expects the stock to move by more than currently implied. Put underwriting benefits from selling expensive implied volatilityTypically the implied volatility of options trades slightly above the expected realized volatility of the underlying over the life of the option (due to a mismatch between supply and demand). Stock replacement via put selling therefore benefits from selling (on average) expensive volatility. Selling a naked put is known as put underwriting, as the investor has effectively underwritten the stock (in the same way investment banks underwrite a rights issue). The strike should be chosen at the highest level at which the investor would wish to purchase the stock, which allows an investor to earn a premium from taking this view (whereas normally the work done to establish an attractive entry point would be wasted if the stock did not fall to that level). This strategy has been used significantly recently by asset allocators who are underweight equities and are waiting for a better entry point to re-enter the equity market (earning the premium provides a buffer should equities rally). If an investor does not wish to own the stock and only wants to earn the premium, then an OTM strike should be chosen at a support level that is likely to remain firm. If OTM puts are used, put underwriting benefits from selling skewPut underwriting gives a similar profile to a long stock, short call profile, otherwise known as call overwriting. One difference between call overwriting and put underwriting is that if OTM options are used, then put underwriting benefits from selling skew (which is normally overpriced). |

### Directional Volatility

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