First, we show that implied normal volatility is intimately linked with the incomplete Gamma function. Then, we deduce an expansion on implied normal volatility in terms of the time-value of a European call option. Then, we formulate an equivalence between the implied normal volatility and the lognormal implied volatility with any strike and any model. This generalizes a known result for the SABR model. Finally, we adress the issue of the "breakeven move" of a delta-hedged portfolio.
This article comes within the scope of the study of the asymptotics of implied volatility which has been considered extensively ( [19], [11], [7]). Asymptotics of implied volatility are important for different reasons. First, they give information on the behaviour of the underlying through the moment formula ( [17]) or the tail-wing formula ( [4]). Second, they allow a full correspondence between vanilla prices and implied volatilities. With such a correspondence, asymptotics in call prices can be easily transformed into asymptotics in implied volatilities. When applied to a specific model, asymptotics are widely used as smile generators ( [13]). In practice, other models are then used for pricing options using tools like Monte-Carlo simulations.
So far, all the asymptotics studied by authors concern asymptotics for implied lognormal volatility. In this article, we consider implied normal volatility which refers to the Bachelier model. Why is it interesting to consider normal implied volatility? First, for short maturities, the Bachelier process makes more sense than the Black-Scholes model. Indeed, the behaviour of the underlying from one day to another is generally well approximated by a Gaussian random variable (see [20]). That’s the reason why the Bachelier model is very popular in high frequency trading ([3]). Second, the “breakeven move” of a delta-hedged portfolio option is easily interpreted as normal volatility. Generally, the P & L of a book of delta-hedged options is positive if the (historical) volatility of the underlying is greater than a breakeven volatility which has to be expressed in normal volatility. Moreover, it makes more sense to compare implied normal volatilities with historical moves of the underlying as can be done by a market risk department. Likewise, some markets such as fixed-income markets with products like spread-options are quoted in terms of implied normal volatility ( [15]). Finally, the skewness of swaption prices is much reduced if priced in terms of normal volatility instead of lognormal volatility. Therefore, it is important to have a robust and quick way to compute implied normal volatilities from market prices and also to be able to switch between lognormal volatilities and normal volatilities.
What kind of asymptotics should we consider? Most of the approximations in option pricing theory are made under the assumption that the maturity is either small (see the Hagan et al formula [13] for instance) or large ( [10]); it is actually assumed that a certain time-variance σ 2 T is either small or large. A possible way to derive such approximations is to replace the factor of volatility σ by εσ and then set ε = 1. This can be done at the partial differential equation level (see all the techniques coming from physics [13]) as well as directly at the stochastic differential equation level with the help of the Wiener chaos theory for instance [21]. Other types of asymptotics are obtained by considering large strikes. In our approach, we unify all these types of asymptotics (see [11] and [12] for the lognormal case). Indeed, we obtain an approximation of the implied normal volatility as an asymptotic expansion in a parameter λ for λ ≪ 1 and it turns out that λ → 0 when T → 0 or K → +∞.
This study is organized as follows. We first give another expression for the pricing of a European call option which involves an incomplete Gamma function (Proposition 1). Then, we inverse this function asymptotically and obtain an expansion of normal implied volatility. This is particularly important if we want to quickly obtain the implied normal volatilities from call prices as is the case in high frequency trading ( [3]). The formula is also potentially useful theoretically if, given an approximation for the price of a European call option or a spread option (for instance in the framework of the Heston or the SABR model), we want to obtain an approximation of the normal implied volatility. Finally, we restrict our formula to the order 0 and we compare it to a similar formula for the lognormal case. Then, we obtain an equivalence between normal volatility and lognormal volatility. We use it also to compare the Black-Scholes greeks to the Bachelier greeks. Finally, we consider a delta-hedged portfolio and we compute the “breakeven move” in the normal case as well as in the lognormal case.
In a Bachelier model, the dynamic of a stock (S t ) is given by:
with initial value S at t = 0. The so-called normal volatility σ N is related to the price of a call C(T, K) struck at K with maturity T by the formula [20]:
with
Following Ropper-Rutkowski ( [19]), we can isolate the volatility σ N in the pricing formula.
Definition 1 Let us denote by TV(K, T ) (or simply TV) the time-value of a European call option struck at strike K with maturity T : TV(T,
Proposition 1 In the Bachelier model,
where Γ (a, z) is the incomplete Gamma function:
The proof is given in the Appendix. It is clear from P
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