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). 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.Delta hedging must be done with forward of identical maturity for put call parity BORROW COST IMPACT ON OPTION PRICINGFrom 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. 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 namesZero delta straddles still need to include borrow cost on one leg of the straddle 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.Zero delta straddles have strike above spot
DELTA HEDGING AN OPTION REMOVES EQUITY RISKIf 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.
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.
OPTION TRADING RULES OF THUMB
**Pr****ofit 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.**ATM option premium in percent is roughly (2 × π)**^{-}^{1/2}**× σ × t**^{1/2}. 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.**Historical annualized volatility roughly equal to 16 × % daily move**. Historical 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).
Call price = S.N(d1) – K.N(d2) e-rTATM 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 percentATM call price ≈ 0.4 × σ × √T in percent
We assume zero interest rates and dividends (r = 0)
Daily P&L from option = Delta P&L + Gamma P&L + Theta P&LDaily P&L from option = S.δ + S.2γ/2 + tθ where S is change in Stock and t is timeDaily P&L from option - S.δ = S.2γ/2 + tθ = Delta hedged P&L from optionDelta hedged P&L from option = S.2γ/2 + cost term (tθ does not depend on stock price)
VARIANCE IS THE KEY, NOT VOLATILITYPartly 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 DEVIATIONThere 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).
VOLATILITY SHOULD BE CONSIDERED A DERIVATIVE OF VARIANCEThe 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. |