Asymptotic Expansions of the Lognormal Implied Volatility : A Model Free Approach

In a market with no arbitrage, the price of a call option can take two extreme values : its "intrinsic value" which is equal to the payoff of the option (lower boundary value) and the spot price (upper boundary value). For simplicity, we assume a market with no interest rate. Otherwise, we would consider the forward price. We will consider here the case when the call price is close to its boundary value and we will obtain in that case an approximation of the corresponding lognormal implied volatility. This case happens in particular when the maturity of the option is small. To be precise, in the case when T → 0 (resp. T → +∞), we will obtain an asymptotic expansion of the implied lognormal volatility as a sum of terms of the form λ i ln j (λ) with j < i and λ = -1 ln C(T,K)-(S-K) + S (resp. λ = -1 ln S-C(T,K)

)

where C(T, K) denotes the price of a call option with strike K, maturity T and spot price S (Proposition 5). Note that here, the spot price S is present only to insure that the ratio C(T, K) -(S -K) + S (resp.

)is with no-dimension. The important quantity is the “time-value” TV(T, K) := C(T, K) -(S -K) + (resp. “covered call” CC(T, K) := S -C(T, K)) The computations involve no complicated formulas except may be a well known asymptotic expansion for the incomplete Gamma function (Equation ( 16)). The interest of such a formula is twofold. First, it gives quickly an easy approximation of the true implied lognormal volatility. This can serve as a starting point for the calculus of the exact implied lognormal volatility using a Newton method for instance. The formula can also be useful to transform theoretical approximations of a call price into approximations of implied lognormal volatility. Indeed, asymptotics of call prices can be obtained with the help of stochastic differential equations of partial differential equations using perturbation methods. Then, a transformation has to be made to obtain the implied lognormal volatility which is of a fundamental interest for the practitioner. All our work is based on a single inversion formula. Explicitly we invert the following equation for λ ≪ 1 and β > 0 (see Note 2):

The good framework for solving this problem (i.e. obtain v in terms of λ) is the theory of transseries (see [6]). In the expansion of v in terms of λ coming from (1) it is important to go up to order 5 (for us, order 0 is λ, order 1 is λ 2 ln(λ), … order 5 is λ 3 ) to see α 1 (see Lemma 1):

In a Black-Scholes world, the dynamic of a stock (S t ) is given by:

with initial value S at t = 0. The so-called lognormal volatility σ LN is related to the price of a call BS (S, K, T, σ) struck at K with maturity T by the Black-Scholes formula (See [2]):

with

σ LN √ T To simplify matters, we have considered r = 0. Otherwise, we would consider the forward price F t = S t e rt instead of the spot price S t . Following Ropper-Rutkowski ( [4]), we set:

• TV(S, K, K, T ) (or simply TV(T, K) or TV) the time-value of a European call option struck at strike K with maturity T : TV (S, K, T, σ) := BS (S, K, T, σ) -(S -K) +

• x := ln( K S ) (the log-moneyness)

• θ := σ LN √ T (the square root of the time-variance)

The spot price S is assumed to be fixed by the market. We will consider the two following cases: K is fixed and σ √ T is small (case 1) and σ √ T is fixed and K S is large (case 2). In both cases, we will obtain a similar expression for the asymptotic expansion of the implied lognormal volatility.

First let us assume that x = 0.

2.1 Asymptotic expansions of a European call option for x = 0.

We note that the expression giving the time-value of a call-option in the case (θ ≪ 1 and x fixed) is very similar to the case (|x| ≫ 1 and θ fixed).

Proposition 1 (Case 1.) Let N ∈ N. When θ → 0 and x fixed, the asymptotic expansion of the time-value T V = C(T, K) -(S -K) + of a call price is given at order N by:

with

and for j ∈ Z, (2j + 1)!! := (Case 2.) Let N ∈ N. When |x| → +∞ (i.e., K → 0 or K → +∞) and θ fixed, the asymptotic expansion of the time-value of a call price is given at order N by:

.

(Case 3.) Let N ∈ N. When θ → +∞ and x fixed, the asymptotic expansion of the covered call CC = S -C(T, K) of a call price is given at order N by:

Proof. Case 1. For n ∈ N, we denote by ẽn the function defined by

Then, it is classical (properties of alternate series) that

Now, let us fix N ∈ N. We start from:

with x = ln K S as before. This formula can be obtained by deriving the Black-Scholes formula with respect to θ and then integrating the result (See [RR], Lemma 3.1). We have:

So, by (11),

So, with the change of variables u := x 2 2ξ 2 , we get:

We have:

We recall the following asymptotic expansion valid for z → +∞ and m ∈ N (see Formula 6.5.32 in ( [1])):

with

In particular, with a = -N -3 2 , z = x 2 2θ 2 and m = 0, in the limit when θ → 0, we get:

Moreover, for any n < N , we have by ( 16)

Moreover, when θ → 0, we have by (17):

Therefore, by (15), ( 18), ( 19), (20), we obtain:

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