Volatility: neutralising Vega

The volatility relevant to options is forward-looking: it reflects the uncertainty between now (the moment the option is traded) and the expiration date.

While the past may provide insights into future behaviour, it is meaningful solely in its ability to inform expectations.

It is critical to understand that “common” market behaviour does not mean “systematically”

A common market phenomenon is the inverse relationship between equity prices and volatility - volatility tends to rise during equity declines and fall during rallies. However, this relationship is asymmetrical as sharp increases in volatility are more common than sudden decreases, except when a decline follows directly after a volatility spike.

It is critical to understand that “common” market behaviour does not mean “systematically”. Volatility is very difficult to model as one needs to integrate properly possible future events. Moreover, volatility is mean-reverting; therefore, to maximise volatility exposure, one needs to focus on short-term volatility.

For a plain vanilla option, the (forward looking) volatility is the only unknown, but the models would typically consider it as known and constant.

The sensitivity of the option price versus a change in one percentage point in volatility is called Vega.

It should be noted that if there are dividends, for example, then there is an additional unknown as future dividends can vary. And contrary to Black-Scholes’ hypothesis, volatility is not typically the same for Puts and Calls and needs to be differentiated with the option Strike.

We build on the example from the first article but now with a focus on Vega: considering again a hypothetical European Long Call option on 8 April with Spot: 4’982.77, Strike: 5’000, 1-month expiration, Risk free: 4.36%, Dividend yield: 1.27% and Volatility: 52.33%: its price would have been 297.64 (Forward: 4’995.62) and its Vega: 5.718405.

The very next day, future volatility closed at: 33.62%, the identified massive drop. Using the risk measure, the loss in value should have been: Vega * (33.62% - 52.33%) = 5.718405 * (-18.71) =-106.9914. Comparing to a full repricing that ignored the change in Spot (but that’s unrealistic) and expiration: 190.55, a loss of 190.55 - 297.64 =-107.09… That’s pretty close to the risk measure.

In order to neutralise Vega, we need to use other options with a negative Vega or go for derivatives that have volatility as their underlying.

There are currently a large number of structured products with S&P 500 implied volatility as the direct underlying, in other words, they are linked to VIX, which is considered "implied" because it is reverse engineered from the market prices of derivatives using a pricing model.

Moreover, published implied volatility measures exist also for other markets such as oil, for example.

Additionally, (implied) volatility is always expressed as an annualised figure, even when derived from short-term data - a standard practice that requires careful interpretation when working with this complex metric.

Now, in the context of structured products, the case of the reverse convertible is of particular interest when it comes to dealing with Vega: for the investment bank, the issuer of the product, Vega (and Delta and Gamma!) can easily be hedged by issuing simultaneously, on the same underlying, with the same expiration and Strike, a reverse convertible and a call warrant! That’s thanks to the Put-Call Parity.

A reverse convertible gives, in option terms, a Short Put exposure for the investor and therefore a Long Put for the issuer. When the investment bank issues a call warrant, it is short that (typically long-term) Call.

Put-Call Parity is the following relationship, true for European style options: +C–P = +S–Present Value of X (using continuous compounding).

The investment bank is therefore [ -C+P = -S + PV(X)] short the underlying.

The table below shows Vega-hedging in action. The European style options have the same underlying and are issued on 16 May, with Spot: 5’958.38; Strike: 6’000, 12-month expiration; Risk free: 4.13%; Dividend yield: 0.00% (for simplification) and Volatility: 21.00% (at the 1-year horizon).

We take the investment bank’s side after the issuance of both, a reverse convertible (the Long Put) and a call warrant (the Short Call):

Instrument	Value	Delta	Gamma	Vega
Long Put	395.7983	-0.394150	0.000308	22.928800
Short Call	596.9310	-0.605850	-0.000308	-22.928800
Greeks combined		-1.000000	0.000000	0.000000

_{Source: Evolids Finance} 

We observe that the combined Vega is 0, that Put-Call parity works with -596.9310 + 395.7983= --201.1327 compared with -5’958.38 + [6’000 * EXP (-4.13%)] = -201.1327.

The investment bank which is short the underlying hedges its exposure by buying the underlying. This straightforward hedging should enable better pricing, resulting in a win-win situation.

In the next episode, we will address the risk aggregation matter and the related structured products Risk Business Intelligence (RBI).

_{Image: Adobe Stock}

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Disclaimer

• This content is not intended as a solicitation or an offer; it is provided solely for informational purposes to professional investors.
• The information presented herein has been prepared with great care; however, errors may still occur.