When the Greeks get it wrong: the hidden risk in structured products (part 1)

More complex structured products combine derivatives, including exotic options, priced with models that can have many parameters, each a potential source of error in the price and the issuer’s hedge. That is the raison d’être of this article, illustrated with a classic Reverse Convertible: the issue exists even with plain vanilla options.

Model risk

A model simplifies reality for a given purpose, so every model carries model risk, a form of operational risk. It stems from inputs in production (a manual faux pas, an IT error, or parameters themselves modelled), from misconception (during development) or from inappropriate use (beyond its scope, limitations or assumptions). A new model must therefore be validated and its use controlled.

For structured products with rather simple options, the key input is volatility, unknown and forward-looking, which must be modelled appropriately.

Comparing constant volatility with stochastic volatility

What is stochastic volatility, exactly? Not merely a varying volatility: in a local volatility model (Dupire), it varies with price and time but is fully determined by them. In a stochastic volatility model, the instantaneous volatility is itself a random process with its own source of randomness, possibly correlated with the price but not determined by it. It becomes a second state variable, unknown in advance even given the future price, whose risk cannot be hedged with the underlying alone. It also differs from implied volatility, which is inferred from a market price using a model.

The example par excellence is the Heston model (Steven Heston, The Review of Financial Studies, 1993), probably the best known, although others are more used in practice, notably SABR in interest-rate markets. It describes how the price S and its variance v move together (√v is the volatility):

Price: dS = r S dt + √v S dW1 where dW1 is the random part of the price move over a very short time dt, with mean zero and variance dt

Variance: dv = κ (θ - v) dt + ξ √v dW2 where dW2 is the random part of the variance move, of the same kind

Correlation between the two random moves dW1 and dW2: ρ

The price moves as in Black-Scholes, but with the current volatility √v instead of a constant σ and the variance follows its own random path. ρ links the two: negative for equities, whose volatility typically rises when markets fall, hence the skew.

^{Table: the five Heston parameters in plain words (values used below)}

Parameter	Value	What it means	Illustration
Starting volatility √v₀	30%	Today’s volatility: the starting point	Same as Black-Scholes, for a fair comparison
Long-run volatility √θ	25%	The “average” level to which volatility returns	Above it, volatility tends to fall and below it, to rise
Speed of mean reversion κ	2	How fast volatility returns to √θ	Half the gap from 30% to 25% closes in ln2 / κ ≈ 4 months
Volatility of volatility ξ	0.45	How much volatility moves at random	About ±1.4 vol points on a typical day, ±6.5 in a month
Correlation ρ	-0.7	Link between price and volatility moves	Volatility up by about 0.5 point for each 1% fall

_{Source: Evolids Finance}

Speed κ = 2: an elastic band ties volatility to its long-run level of 25%, κ measuring its stiffness. Without new random moves, volatility would fall from 30% to 26.95% after 6 months and 25.73% after one year, an average of 27.27% (used below). Markets behave alike, though calm can return faster after a crisis spike.

Volatility of volatility ξ = 0.45: how variable volatility is. κ and θ draw the expected path while ξ sets how far volatility wanders around it: one month ahead, between about 19% and 39% in 90% of scenarios, against 29.29% with ξ = 0. By making large moves more likely, ξ bends the flat Black-Scholes line into a smile.

Correlation ρ = -0.70: volatility rises when markets fall, that’s déjà vu in every sell-off. Large falls become more likely than large rises, making out-of-the-money puts dearer and calls cheaper: ρ tilts the smile into a skew.

In short, κ and θ set the average level, ξ bends the line into a smile and ρ tilts it into a skew: the 1-year implied volatility falls from 32.03% at a 70% strike to 22.29% at 130%, against a flat 27.27% without vol of vol.

Illustration with the Reverse Convertible: pricing and the Greeks

Take a 1-year Reverse Convertible (RC) on an underlying at S₀ = 100: strike 100%, coupon 12% p.a. paid semi-annually, rate 5%, no dividends, notional 100. It is a zero-coupon bond plus coupons, minus a European put, the only part depending on the volatility model.

Black-Scholes uses a constant 30% while Heston starts from the same current volatility (√v₀ = 30%), other parameters as in the table above. The difference thus measures what constant volatility leaves out.

^{Table: pricing (per 100 notional)}

	Black-Scholes	Heston	Difference
Zero-coupon bond + Present Value of coupon	106.68	106.68	none
Short put	-9.35	-8.02	1.33
Reverse Convertible price	97.33	98.66	1.33
Volatility implied at the strike	30.00%	26.48%	-3.52 pts
Fair coupon (price = 100)	14.77%	13.39%	-1.38 pts

_{Source: Evolids Finance}

Heston values the Reverse Convertible 1.33 higher, as the put is 14% cheaper (fair coupon 1.38 points lower). Black-Scholes ignores two features: mean reversion, an average volatility of 27.27% over the year (+1.03) and random volatility (+0.30), almost entirely from ξ. At the money, an option’s value is roughly proportional to volatility, so uncertain variance slightly lowers it.

The gap is not a constant bias. With ρ < 0, the skew makes low-strike puts more expensive, offsetting mean reversion: +0.57 at a 90% strike, zero at 80%, -0.28 at 70%. A single flat volatility thus overprices the at-the-money put here but underprices low strikes.

^{Table: Greeks of the RC (per note of 100)}

Greek	Black-Scholes	Heston	Difference
Delta	0.376	0.312	-17%
Gamma	-0.0126	-0.0133	+5% short gamma
Vega (per vol point)	-0.379	-0.371	similar total
		(√v₀: -0.174, √θ: -0.197)
Theta (per day)	0.0238	0.0206	-13%
Rho (per rate point)	-0.568	-0.646	14%
Vol of vol ξ (per 0.10)	n/a	0.142	not in Black-Scholes
Correlation ρ (per 0.10)	n/a	-0.007	not in Black-Scholes

_{Source: Evolids Finance}

Delta is 17% lower under Heston, almost entirely due to the skew (Black-Scholes at 26.48% still gives 0.374): when the spot rises, the strike moves down the skew, where implied volatility is higher, so the put loses less value. On 100 million notional, the issuer holds 6.4 million less underlying. At an 80% strike, prices coincide (104.12 versus 104.11), yet deltas differ by 24%.

Vega totals are close but differ in nature: Heston splits the exposure between current and long-run volatility, the latter neither observable nor tradable, and adds exposures to ξ and ρ.

Gamma, Theta and Rho. Gamma is slightly more negative and Theta lower: lower expected volatility concentrates convexity around the strike and leaves the cheaper put less time value to lose. Rho differs only through the put, via the risk-neutral probability of ending below the strike: 41% under Heston against 49% (with ρ < 0, losses are rarer but slightly deeper).

This article is based on data and analysis provided by the SRP Greeks product. Find out more about SRP Greeks here

_{Image: Who is Danny/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