On the implied volatility of Inverse options under stochastic volatility models

Over the last several decades, option pricing models have been developed for conventional assets such as stocks, bonds, interest rates, and foreign currencies. Today, cryptocurrency derivatives, financial contracts whose value depends on an underlying cryptocurrency asset, have emerged as a new class of securities that have garnered significant attention. A key peculiarity of crypto derivatives is how one defines a cryptocurrency: Is it a security, a currency, or a commodity? With regard to options, the answer to this question affects the pricing methodology. A detailed discussion on this topic can be found in Alexander et al. [1]. Unfortunately, there is no clear legal answer to this question; see Bolotaeva et al. [12]. However, a closer examination of the issue leads to the following conclusions.

Cryptocurrency (at least Bitcoin and Ethereum) cannot be considered a security, as it is fully decentralized, and no central authority controls its issuance, whereas securities are issued by a central authority. Furthermore, cryptocurrencies cannot be treated as conventional (fiat) currencies. A key question is whether they preserve the fundamental characteristics of money. Although cryptocurrencies can occasionally be used to buy and sell goods, they are not widely accepted as a means of payment. Furthermore, historical data shows the extreme volatility of cryptocurrencies, leading to the conclusion that their purchasing power is not stable enough over time, and therefore they cannot serve as a store of value. In contrast, Central Bank Digital Currency addresses the volatility issue by controlling the issuance. Finally, although some companies may accept cryptocurrencies as payment, most still use traditional currencies to measure the value of goods and services. For further discussion on the similarities between cryptocurrencies and regular currencies, see Hazlett and Luther [20] and Ammous [10].

On the other hand, if we consider the classical Garman and Kohlhagen [16] foreign exchange (FX) pricing model, the construction of the delta-hedged portfolio for FX options is conceptually different from regular options, as we cannot buy and sell units of the FX spot rate. Instead, hedging is done by buying and selling units of the underlying foreign bond. This difference fundamentally undermines the idea of pricing crypto options using FX models, as cryptocurrencies can be bought and sold in a manner similar to regular tradable assets.

Alternatively, some people consider Bitcoin to be digital gold, but how similar is it to the actual commodity? In Goutteet al. [17], the authors define the following characteristics of hard commodities: they are costly to mine or extract, storable, not controlled by any single government or institution in terms of global supply, demand, or price, and they have intrinsic value, which means that they can be consumed or used as inputs in the production of other goods. The first three properties are naturally satisfied by Bitcoin, but the fourth remains debatable. As a result, we cannot conclusively classify crypto as a commodity. For a more detailed discussion, see Ankenbrand and Bieri [11] and Gronwald [18]. Finally, Dyhrberg [15] provides an empirical analysis using GARCH models for Bitcoin volatility to explore whether Bitcoin is more similar to gold or the US dollar.

The aim of this paper is to solve the pricing problem of crypto options using only the payoff function. A natural way to define the payoff of crypto options is through Inverse options, which are settled in cryptocurrency rather than fiat currency. We will focus on the case of an Inverse European call, whose payoff is given by:

where $(x)_{+}=\max(x,0)$, $S_{T}$ denotes the price of Bitcoin in US dollars at maturity $T$ and $K$ is a fixed strike. In simple terms, if the option becomes in-the-money, the payoff is made in cryptocurrency rather than fiat currency.

Inverse European options are the only type of options traded on the Deribit exchange, which controls over 80 percent of the global crypto options market. For example, on June 10, 2023, the open interest in Bitcoin options on Deribit was 7.5 billion, while the closest competitors, OKX and Binance, had open interest of 0.5 billion and 0.17 billion, respectively. As a result, accurately pricing and hedging Inverse European options is crucial from a practical standpoint. However, this is quite challenging due to the mechanics of the Deribit exchange. Deribit does not allow fiat currency, and all options are margined in cryptocurrency. This is particularly beneficial for professional crypto traders. For instance, consider a crypto hedge fund or a crypto market maker, businesses that deal exclusively with crypto assets. It is natural for them to manage their trading books in cryptocurrency rather than fiat currency. While they are exposed to cryptocurrency depreciation risk, it is easier to manage this risk at the book level rather than on a trade-by-trade basis. This is one of the key reasons for the development of Inverse European options.

The literature on crypto derivatives is still relatively new but has been gaining increasing attention from researchers. For example, Alexander et al. [1] price Inverse European options under the constant volatility Black-Scholes model. Matic et al. [23] study empirical hedging of Inverse options under different stochastic volatility models. Alexander et al. [2] use a delta adjusted for skew to hedge Inverse options under constant and local volatility. Hou et al. [21] price Inverse options under stochastic volatility models with correlated jumps. Finally, Siu and Elliott [27] use the SETAR-GARCH model to model Bitcoin return dynamics.

In this paper, we study the behavior of implied volatility for Inverse European options under general stochastic volatility models. Specifically, we provide general sufficient conditions on a stochastic volatility model to derive short-time maturity asymptotic formulas for the at-the-money level and the skew of the implied volatility of Inverse options. We then apply these results to two well-known models: the SABR and fractional Bergomi models.

The main tool for proving these results is the anticipating Itô’s formula from Malliavin calculus; see Appendix A for an introduction to this topic. The first step is to apply this formula using a similar approach to Alòs [3], in order to derive a decomposition formula for the option price; see Theorem 2. This approach is also used in Alòs et al. [7] for jump-diffusions with stochastic volatility. In that paper, the short-time limit of the implied volatility skew for European calls is also derived, which corresponds to the second step of our paper. This approach has been further developed in other contexts. Recently, Alòs et al. [5] applied this methodology to VIX options, and Alòs et al. [8] extended it to Asian options. These two papers demonstrate how studying Asian or VIX options under stochastic volatility reduces to studying European-type options, where the underlying is represented by a stochastic volatility model with a modified volatility process that depends on maturity.

In the present paper, we extend the methodology outlined above to Inverse European calls. The main novelty, compared to the papers mentioned, is the following.

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  To the best of our knowledge, we are the first to provide a rigorous analytical study of the asymptotic behavior of the implied volatility of Inverse European options under fractional and stochastic volatility models. When dealing with Inverse options, the Black-Scholes formula differs from the classical one, so our results cannot be directly derived from existing literature. Instead, we work with the Black-Scholes formula for Inverse European calls under constant volatility, as obtained in Alexander et al. [1]. This formula turns out to be a non-monotonic function (see Figure 8), meaning its inverse is not uniquely defined, which complicates computations since implied volatility is defined as the inverse of the Black-Scholes formula. We overcome this challenge by noting that for sufficiently small maturities, which is the case studied in this paper, the function is strictly monotonic. However, the inverse is not explicit as in the classical Black-Scholes case, so a careful study of its behavior and derivatives is required, as shown in Appendix B.

• •

  Our formulas can be applied to general stochastic and fractional volatility models. We provide numerical simulations using the SABR and fractional Bergomi models. Additionally, we present an empirical example with Bitcoin options to demonstrate how the results of this paper can be used to model real market data for such options, which have become increasingly popular in financial markets, as discussed at the beginning of this introduction.

Understanding the behavior of the implied volatility of Inverse options is crucial for hedging purposes and can also be used to derive approximation formulas for their price, as mentioned at the end of Section 2. In a world with a flat implied volatility surface, one could simply use the Black-Scholes delta to replicate the derivative payoff. However, in the presence of implied volatility skew, the classical Black-Scholes hedge will not be accurate. The correct approach is to use the Black-Scholes delta with a correction based on the implied volatility skew. In our empirical example, we focus on the fractional Bergomi model.

One of the key parameters in this model is the Hurst exponent $H$. Its value determines the skew that can be reproduced from market data. Since different markets exhibit different behaviors, we expect to see varying values of $H$ that best fit the observed data. For example, in equity markets, the implied volatility skew exhibits a blow-up, which can be reproduced with $H<0.5$, corresponding to rough volatility See, for instance, Cont and Das [14]. In our case, we find that $H>0.5$ best fits our market data and produces zero skew.

Additionally, our analysis allows us to derive the same formulas for Quanto-Inverse European options, whose payoff is given by:

where $R$ is a fixed exchange rate. From a mathematical perspective, Quanto-Inverse options do not differ significantly from Inverse options. In particular, as shown in Section 2, the implied volatility level and skew of both options are equal up to a factor of $R$, so it is sufficient to study Inverse options. However, from a practical perspective, Inverse and Quanto-Inverse options differ, as they are traded on different markets. Unlike Inverse options, which are available to the general public, Quanto-Inverse options are over-the-counter derivatives.

The paper is organized as follows: Section 2 presents the problem statement and the main result, Theorem 1, where the at-the-money implied volatility level is derived in (6) and the skew in (7). The intermediate steps leading to the proof of this theorem are provided in Section 3. Specifically, Theorem 2 enables us to obtain (6), and Propositions 1 and 2 are crucial for deriving (7). Note that Proposition 1 relies on (6). The proof of Theorem 1 is provided in Section 4. Section 5 presents a numerical study for the SABR and fractional Bergomi models, along with an empirical application to the implied volatility of Bitcoin options.

Consider the following model for the asset price $S_{t}$ on the time interval $[0,T]$

where $S_{0}>0$ is fixed and $W_{t}$, $W_{t}^{\prime}$, and $B_{t}$ are three standard Brownian motions on $[0,T]$ defined on the same risk-neutral complete probability space $(\Omega,\mathcal{G},\mathbb{P})$. We denote by $\mathbb{E}$ the expectation with respect to $\mathbb{P}$. For simplicity, we assume that the interest rate is zero, see for instance in Alòs et al. [5]. We assume that $W_{t}^{\prime}$ and $B_{t}$ are independent and $\rho\in[-1,1]$ is the correlation coefficient between $W_{t}$ and $W_{t}^{\prime}$.

We consider the following hypotheses on the volatility process $\sigma_{t}$:

We observe that when the volatility $\sigma_{t}$ is constant, the model (1) is the usual Black-Scholes model. We also observe that although the volatility can be driven by a fractional Brownian motion (see Section 5.2), we are assuming that the volatility is not a traded asset. Thus, there exists a risk-neutral probability $\mathbb{P}$ and there are no arbitrage opportunities (see Comte and Renault [13]). Moreover, if $(V_{t})_{t\in[0,T]}$ and $(V_{t}^{Q})_{t\in[0,T]}$ denote the values of an Inverse European call and a Quanto Inverse European call options with fixed strike $K$, respectively, we have that

and

where $R$ is a fixed exchange rate. Recall that $\mathbb{E}$ denotes the expectation under a risk-neutral probability $\mathbb{P}$. Note that as in [7], we assume that the market selects a unique risk-neutral measure $\mathbb{P}$ under which these derivative contracts are priced.

Notice that the difference between $V_{0}$ and $V_{0}^{Q}$ arises from the currency in which the options are quoted. In our case, $V_{0}$ represents the crypto price of the option, while $V_{0}^{Q}$ represents the dollar value of the option.

We denote by $BS(t,x,k,\sigma)$ the Black-Scholes price of an Inverse European call option with time to maturity $T-t$, log-underlying price $x$, log-strike price $k$ and volatility $\sigma>0$. Then, it is well-known that (see Alexander et al. [1]),

where $N$ is the cumulative distribution function of a standard normal random variable. Moreover, the Black-Scholes price for a Quanto Inverse European call is given by $BS^{Q}(t,x,k,\sigma)=R\times BS(t,x,k,\sigma)$.

One can easily check that the Black-Scholes price satisfies the following PDE

Moreover, one can also check that the classical relationship between the Gamma, the Vega and the Delta holds, that is,

As in Alòs [3], we consider the log price $X_{t}=\log(S_{t})$, which satisfies

Next, we observe that, as $BS(T,x,k,\sigma)=e^{k}\times(e^{-k}-e^{-x})_{+}$, the price of an Inverse call option $V_{0}=e^{k}\times\mathbb{E}(e^{-k}-e^{-X_{T}})_{+}$ can be written as

and by continuity we have $\lim_{t\to T}v_{t}=\sigma_{T}$. In particular, $V_{T}=BS(T,X_{T},k,v_{T})$. Notice that $V^{Q}_{t}=R\times V_{t}$. This implies that the implied volatility level and skew of Inverse and Quanto Inverse European call options are equal up to the factor $R$. Hence, we will only state the main results of this paper for the Inverse options.

We define the at-the-money implied volatility (ATMIV) of an Inverse European call option as the quantity $I(0,k^{\ast})$ satisfying

where $k^{\ast}=X_{0}$. That is, $I(0,k^{\ast})=BS^{-1}(0,X_{0},k^{\ast},V_{0})$. As it is shown in Appendix B, this inverse is only well-defined for $T$ sufficiently small, which is the case we are studying in this paper. Precisely, the aim of this paper is to apply the Malliavin calculus techniques developed in Alòs [3] in order to obtain formulas for

under the general stochastic volatility model (1).

The main results of this paper are given in the following theorem.

We observe that when prices and volatilities are uncorrelated then the short-time skew equals to zero. Observe also that since the term $\mathbb{E}(D_{r}^{W^{\prime}}\sigma_{u})$ is of order $(u-r)^{H-\frac{1}{2}}$ (see Hypothesis 4), the quantity multiplying the term $T^{\max(\frac{1}{2}-H,0)}$ of the right hand side of (7) is bounded by $cT^{H-\frac{1}{2}}$. In particular, its limit is $0$ if $H>1/2$. This suggests that, in the case $H<\frac{1}{2}$, we need to multiply by $T^{\frac{1}{2}-H}$ in order to obtain a finite limit. See the examples in Section 5.

The results of Theorem 1 can be used in order to derive approximation formulas for the price of an Inverse and Quanto Inverse European call options. Notice that, as

by Taylor’s formula we can use the approximations

Of course, this approximation is only linear, and one would expect to obtain better results if one has a short maturity asymptotic formula for the curvature $\partial_{kk}^{2}I(0,k^{*})$. The short-time maturity asymptotics for the at-the-money curvature of the implied volatility for European calls under general stochastic volatility models is computed in Alòs and León [6]. A Taylor expansion for short maturity asymptotics for Asian options when the underlying asset follows a local volatility model is obtained in Pirjol and Zhu [26]. In our setting, computing the curvature is more challenging and we leave it for further work.

In this section, we provide closed-form decomposition formulas for the price and the ATMIV skew of an Inverse call option under the stochastic volatility model (1).

We begin with the following preliminary lemma. For the standard European call option case, see Lemma 6.3.1 in Alòs et al. [5].

The first result of this section is the decomposition of the price of inverse options. The proof follows exactly the same lines as Theorem 4.2 in Alòs et al. [7]. See also Theorem 6 in Alòs [3] and Theorem 26 in Alòs et al. [5]. Note that the derivatives of the Black-Scholes formula for inverse options satisfy the same equations, (4) and (5), which is the key ingredient of the proof, together with the anticipating Itô’s formula. For completeness, we provide a sketch of the proof, as the derivatives of the Black-Scholes formula are different. Thus, Lemma 1 (instead of Lemma 2 in [7]) is needed to verify that all the integrals are well-defined.

We next derive an expression for the ATMIV skew of an Inverse European call option under the stochastic volatility model (1). The proof follows similarly to Theorem 4.2 in Alòs et al. [7], but with the use of properties (12) and (13) of the Black-Scholes function for Inverse options, which differ from the standard Black-Scholes function.

To compute the limit of the skew slope of the ATMIV, we need to identify the leading-order terms of the numerator in equation (11). The following decomposition formula will be crucial in achieving this goal. The proof follows similarly to that in Alòs et al. [7]. For completeness, we provide a sketch of the proof.