Appendix - Modeling Volatility Surfaces
Derivatives-based Portfolio Solutions


There are a variety of constraints on the edges of a volatility surface, and this section details some of the most important constraints from both a practical and theoretical point of view. We examine the considerations for very short-dated options (a few days or weeks), options at the wings of a volatility surface and very long-dated options.

Options that only have a few days or a few weeks to expiry have a very small premium. For these low-value options, a relatively small change in price will equate to a relatively large change in implied volatility. This means the implied volatility bid-offer arbitrage channel is wider, and hence less useful. The bid-offer spread is more stable in cash terms for options of different maturity, so shorter-dated options should be priced more by premium rather than implied volatility.

Need to price short-dated options with a premium after a large collapse in the market
If there has been a recent dip in the market, there is a higher than average probability that the markets could bounce back to their earlier levels. The offer of short-dated ATM options should not be priced at a lower level than the size of the decline. For example, if markets have dropped 5%, then a one-week ATM call option should not be offered for less than approx. 5% due to the risk of a bounce-back.

The payout of a put spread (and call spread) is always positive; hence, it should always have a positive cost. If it was possible to enter into a long put (or call) spread position for no cost (or potentially earning a small premium), any rational investor would go long as large a position as possible and earn risk-free profits (as the position cannot suffer a loss). A put spread will have a negative cost if the premium earned by selling the lower strike put is more than the premium of the higher strike put bought. This condition puts a cap on how negative skew can be: forhigh (negative) skew, the implied of the low strike put could be so large the premium is too high (ie, more than the premium of higher strike puts). The same logic applies for call spreads, except this puts a cap on positive skew (ie, floor on negative skew). As skew is normally negative, the condition on put spreads (see figure below on the left) is usually the most important. As time increases, it can be shown that the cap and floor for skew (defined as the gradient of first derivative of volatility with respect to strike, which is proportional to 90%- 100% skew) decays by roughly the square root of time. This gives a mathematical basis for the ‘square root of time rule’ used by traders.

Far-dated skew should decay by time for long maturities (approx. 5 years)
It is possible to arrive at a stronger limit to the decay of skew by considering leveraged ratio put spreads (see chart above on the right). For any two strikes A and B (assume A<B), then the payout of going long A× puts with strike B, and going short B× puts with strike A creates a ratio put spread whose value cannot be less than zero. This is because the maximum payouts of both the long and short legs (puts have maximum payout with spot at zero) is A×B. This can be seen in the figure above on the right (showing a 99-101 101x99 ratio put spread). Looking at such leveraged ratio put spreads enforces skew decaying by time, not by the square root of time. However, for reasonable values of skew this condition only applies for long maturities (approx. 5 years).

Enforcing positive values for put and call spreads is the same as the below two conditions:
  • Change in price of a call when strike increases has to be negative (intuitively makes sense, as you have to pay more to exercise the higher strike call).
  • Change in price of a put when strike increases has to be positive (intuitively makes sense, as you receive more value if the put is exercised against you).
These conditions are the same as saying the gradient of x (= Strike / Forward) is bound by:
Lower bound  = - √ ( 2π ) e ( d12 / 2) [ 1 - N(d2) ] x ( 2π ) e ( d12 / 2) [ 1 - N(d2) ]  = upper bound

It be can shown that these bounds decay by (roughly) the square root of time.

Proof of theoretical cap for skew works in practice
In the above example, for a volatility of 25% the mathematical lower bound for one-year skew (gradient of volatility with respect to strike) is -1.39. This is the same as saying that the maximum difference between 99% and 100% strike implied is 1.39% (ie, 90%-100% or 95%- 105% skew is capped at 13.9%). This theoretical result can be checked by pricing one-year put options with Black-Scholes.
  • Price 100% put with 25% implied = 9.95%
  • Price 99% put with 26.39% implied = 9.95% (difference of implied of 1.39%)

In practice, skew is likely to be bounded well before mathematical limits
While a 90%-100% one-year skew of 13.9% is very high for skew, we note buying cheap put spreads will appear to be attractive long before the price is negative. Hence, in practice, traders are likely to sell skew long before it hits the mathematical bounds for arbitrage (as a put spread’s price tends to zero as skew approaches the mathematical bound). However, as the mathematical bound decays by the square root of time, so too should the ‘market bound’.

While it is popular to plot implieds vs delta, it can be shown for many models that implied volatility must be linear in log strike (ie, Ln[K/F]) as log strike goes to infinity. Hence a parameterisation of a volatility surface should, in theory, be parameterised in terms of log strike, not delta. In practice, however, as the time value of options for a very high strike is very small, modelling implieds against delta can be used as the bid-offer should eliminate any potential arbitrage.