# Journal of Computational Finance

**ISSN:**

1460-1559 (print)

1755-2850 (online)

**Editor-in-chief:** Christoph Reisinger

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Need to know

- We propose a new method for volatility surface construction for FX-options. The result is a sufficiently smooth explicit parameterization of option prices.
- Our results suggest that the model produces volatility surfaces that fit market quotes with an error of few volatility basis points and with calibration time less than 1 ms per expiry.
- We extend the model to allow for interpolation between expiries and present sufficient conditions for absence of arbitrage.
- We apply the calibrated modes to construct local volatility surfaces and for pricing variance swaps.

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Abstract

In this paper, we propose a new method of constructing volatility surfaces for foreign exchange options. This methodology is based on the local variance gamma model developed by P. Carr in 2008. Our model generates smooth volatility surfaces, fits market quotes with an error of a few volatility basis points and allows very fast calibration. Using the Levenberg–Marquardt algorithm, we measure the average calibration time to less than one millisecond per expiry. We suggest a simple and fast yet market-consistent technique for arbitrage-free interpolation of volatility in the maturity dimension, and we derive sufficient conditions for the absence of calendar spread arbitrage within our model. We also apply the methodology to pricing variance swaps.

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Introduction

## 1 Introduction

In this paper, we address a classical long-standing problem in option pricing: “given market-quoted implied volatilities for a sparse grid of vanilla options with different strikes and maturities, generate implied volatility (and thus the vanilla option price) at any given strike and maturity.” We focus on applications in the foreign exchange (FX) options market and introduce a new type of price parameterization, together with a set of no-arbitrage conditions and a methodology for constructing volatility surfaces. Using our proposed method, it is possible to generate extrapolated volatility surfaces that are arbitrage-free, with a calibration time around one millisecond per quoted market expiry.

### 1.1 Implied volatility surface

Constructing implied volatility surfaces is a fundamental part of quantitative finance. Such surfaces allow the possibility to compare option prices of different strikes and maturities using the only nonobservable parameter in the Black–Scholes model: the average volatility during the lifetime of the option. Some methods for constructing the implied volatility surface are derived by introducing stochastic volatility into the model of the underlying asset, such as the stochastic alpha–beta–rho (SABR) model (Hagan et al 2002) and the Heston (1993) model. Such methods are often capable of calibrating to market smiles fairly well for many asset classes. However, there is no guarantee of the quality of the fit, and the models are often unsuitable for extrapolation. In other approaches, notably the stochastic volatility inspired (SVI) model by Gatheral (2004), the volatility or variance smile is parameterized without the calibration of the process representing the underlying asset. In the original formulation of the SVI model, the absence of arbitrage cannot be guaranteed. Suggested conditions for the absence of arbitrage in the SVI model (Gatheral and Jacquirer 2014) add nonlinear constraints to the model parameters, potentially making the calibration process cumbersome and time consuming.

Important factors to consider when constructing volatility surfaces include the smoothness, the (good) fit to the market-quoted instruments and the calibration performance. The number of different models available tells us that these qualities are difficult to combine.

The local volatility model proposed by Dupire (1994) assumes that an implied volatility is already provided, from which a local volatility surface is derived. Achieving smoothness in local volatility while guaranteeing the absence of arbitrage and the possibility of pricing market-quoted vanilla options correctly has proven to be a difficult and computationally challenging task. The problem has been covered in a wide variety of ways by numerous authors. Obtaining regular local volatilities is essential for FX exotic options pricing, as it is an integral part of industry-standard local-stochastic volatility models.

Carr (2008) showed that one finite-difference time step in the Dupire equation can be interpreted as option prices coming from a model called “local variance gamma” (LVG). This model is further examined by Carr and Nadtochiy (2017), who present a calibration technique that assumes the diffusion function is a deterministic piecewise continuous function. We will call the strike dependent diffusion function in the LVG model the LVG volatility function.

Andreasen and Huge (2011) use a numerical technique for generating arbitrage-free call price surfaces from the simplified LVG version of the Dupire equation. Their proposed calibration method assumes piecewise constant LVG volatility and is arbitrage-free. Their methodology allows local volatility surfaces without any singularities, but which inherit the discontinuities of the assumed LVG volatility, to be produced.

## 2 Theory

We will repeat some earlier results from the LVG model and show how it is possible to produce well-behaving price parameterizations suitable for the FX options market. In this section, we only consider mutually independent option price functions with fixed time to maturity.

### 2.1 Local variance gamma

Carr (2008) shows that a gamma subordinator can be introduced into the local volatility model by Dupire (1994). The idea of using such a subordinator process comes from the variance gamma model introduced by Madan and Seneta (1990) and further developed by Milne and Madan (1991) and Madan et al (1998). In the standard LVG model, the characteristic time parameter ${t}^{\star}$ from the gamma process is assumed to be equal to the time to maturity of the option; we also make this assumption. Thus, the gamma process is unbiased, ie, its expected value is the normal time axis. Subordinating the Brownian motion in the local volatility model with such a gamma process introduces additional kurtosis to the return distribution without enhancing the degrees of freedom. It turns out (Carr 2008) that this also allows a simplification of the Dupire equation:

$\frac{{\alpha}^{2}(K)}{2}}{\displaystyle \frac{{\partial}^{2}C}{\partial {K}^{2}}}(K,{t}^{\star})={\displaystyle \frac{C(K,{t}^{\star})-{(x-K)}^{+}}{{t}^{\star}}}\mathit{\hspace{1em}}\text{for all}K\in [L,U],$ | (2.1) |

where $C$ are the European call option prices, $K$ are the corresponding strikes, ${t}^{\star}\ge 0$ is the time to maturity, $\alpha $ is the positive LVG volatility function and $x\in [L,U]$ is the spot price. The parameters $L,U\ge 0$ define the strike region in which the option prices from the above equation are defined. Below, we will assume, like Carr (2008), that all interest rates are zero.

The following theorem serves as the justification for the LVG model.

###### Theorem 2.1.

Let $C\mathit{}\mathrm{(}K\mathrm{,}{t}^{\mathrm{\star}}\mathrm{)}\mathrm{,}K\mathrm{\in}\mathrm{[}L\mathrm{,}U\mathrm{]}\mathrm{,}{t}^{\mathrm{\star}}\mathrm{\ge}\mathrm{0}$, be a solution to (2.1). Then the set of call options $C\mathit{}\mathrm{(}K\mathrm{,}{t}^{\mathrm{\star}}\mathrm{)}$ does not contain butterfly or vertical spread arbitrage.

A proof of Theorem 2.1 can be found in Carr and Nadtochiy (2017). Note that a direct consequence of this theorem is that option prices from the LVG model must be positive and greater than or equal to the intrinsic option values in the region $K\in [L,U]$. Replacing ${(x-K)}^{+}$ with ${(K-x)}^{+}$ yields a similar formula for put options.

### 2.2 LVG volatility function

In addition to testing the LVG model on the FX options market, our approach differs from earlier publications by considering a new assumption on the LVG volatility function $\alpha $. To the best of the authors’ knowledge, all earlier implementations of the LVG model assume $\alpha $ to be piecewise continuous. Option prices derived from such approaches are arbitrage-free, and implied volatility smiles are ${C}^{1}$. However, local volatility surfaces and probability densities derived from models with a discontinuous $\alpha $ will also be discontinuous. Rebonato (2004) argues that this is problematic for calculations of Greeks and digital/barrier option pricing. In this paper, we assume $\alpha $ to be a continuous function restricted to the set of piecewise linear four-interval functions that are constant in the outer subintervals (see the examples in Figure 3). Let us start off by partitioning the space $K\in [L,U]$ into four separate subintervals:

$$ | (2.2) |

Recall that $x$ is the spot value of the underlying asset. The constants ${\nu}_{1}$, ${\nu}_{2}$ are assumed to satisfy $$ and $$.

###### Assumption 2.2.

Let the LVG volatility function $\alpha \mathit{}\mathrm{(}K\mathrm{)}$ be a continuous function defined as

$$\alpha (K)=\sum _{i=1}^{4}{\alpha}_{i}(K){\mathrm{?}}_{\{K\in {I}_{i}\}},$$ | (2.3) |

where the functions ${\alpha}_{i}\mathit{}\mathrm{(}K\mathrm{)}$ are defined as

${\alpha}_{1}(K)$ | $={\gamma}_{1}$ | $\mathit{\text{for all}}K$ | $\in {I}_{1},$ | ||

${\alpha}_{2}(K)$ | $={\gamma}_{2}K+{b}_{2}$ | $\mathit{\text{for all}}K$ | $\in {I}_{2},$ | ||

${\alpha}_{3}(K)$ | $={\gamma}_{3}K+{b}_{3}$ | $\mathit{\text{for all}}K$ | $\in {I}_{3},$ | ||

${\alpha}_{4}(K)$ | $={\gamma}_{4}$ | $\mathit{\text{for all}}K$ | $\in {I}_{4}.$ |

Further, let $\omega \mathrm{\in}{\mathbb{R}}_{\mathrm{+}}^{\mathrm{5}}$ be

$\omega =\{{\sigma}_{1},{\sigma}_{x},{\sigma}_{2},{\nu}_{1},{\nu}_{2}\}.$ | (2.4) |

All elements of the set $\omega $ are positive and bounded from above. The parameters in the function $\alpha $ are related to the underlying set $\omega $ by the following conditions:

${\gamma}_{1}$ | $={\sigma}_{1},$ | $\mathrm{\hspace{1em}\hspace{1em}}{\gamma}_{2}$ | $={\displaystyle \frac{{\sigma}_{x}-{\sigma}_{1}}{x-{\nu}_{1}}},$ | $\mathrm{\hspace{1em}\hspace{1em}}{b}_{2}$ | $={\sigma}_{1}-{\nu}_{1}{\displaystyle \frac{{\sigma}_{x}-{\sigma}_{1}}{x-{\nu}_{1}}},$ | ||

${\gamma}_{3}$ | $={\displaystyle \frac{{\sigma}_{2}-{\sigma}_{x}}{{\nu}_{2}-x}},$ | ${\gamma}_{4}$ | $={\sigma}_{2},$ | ${b}_{3}$ | $={\sigma}_{x}-x{\displaystyle \frac{{\sigma}_{2}-{\sigma}_{x}}{{\nu}_{2}-x}}.$ |

This family of continuous functions is constant on ${I}_{1}$ and ${I}_{4}$ and linear on ${I}_{2}$ and ${I}_{3}$. Under this assumption $\alpha $ is positive, integrable, locally differentiable and bounded from above by $\mathrm{max}({\sigma}_{1},{\sigma}_{x},{\sigma}_{2})$. In this section, $x$ and all the elements in $\omega $ are assumed to be known.

###### Remark 2.3.

We could argue for other, more complex, continuous parameterizations of the LVG volatility function. The standard choice (Carr and Nadtochiy 2017; Andreasen and Huge 2011) would be to take six intervals, with their boundary points being at-the-money (ATM), $25\mathrm{\Delta}$ and $10\mathrm{\Delta}$ strikes, to achieve better calibration for quotes that are further away from ATM. However, in the FX options market, the $10\mathrm{\Delta}$ quotes, meaning the ones furthest away from ATM, are often of quite poor quality. In practical applications, perfectly calibrating a model to the mid of these quotes might introduce a significant risk of overfitting without adding much value.^{1}^{1}The risk of overfitting is especially pronounced for illiquid currency pairs, such as emerging markets currencies. A possible enhancement to the model would be to relax the condition that the LVG volatility function $\alpha $ is constant on the intervals ${I}_{1}$ and ${I}_{4}$. In particular, to generate implied volatility smiles with more appealing asymptotic behavior around $K=0$, we could take the function $\alpha $ to be linear on ${I}_{1}$, such that $\alpha (0)=0$. We have chosen the proposed form of $\alpha $ to avoid introducing too much complexity into the model. As we show below (see Sections 2.5, 3.4 and 5.2), our assumptions are acceptable from a practical point of view.

In equity markets, option quotes are generally given for a large number of strikes. When working with quotes of such high density, the interpolation technique is apparently a minor problem. Carr and Nadtochiy (2017) end up approximating the piecewise constant diffusion function in order to simplify the calculations due to the high density of the quotes. We focus on applications for the FX options market, where quotes generally contain only three or five liquid points for every expiry. In such a situation, the interpolation technique will naturally be more important, as we will typically end up interpolating quite far from market quotes. Hence, the complexity introduced by using a continuous piecewise linear diffusion function is a more attractive trade-off on the FX options market.

### 2.3 Explicit solution

In this section, we present an explicit solution to (2.1) given the LVG volatility function proposed in Assumption 2.2. Our approach will be to solve the homogeneous part of (2.1) locally on each subinterval from (2.2). Boundary conditions will be imposed on each subinterval, such that the resulting option price function is unique and ${C}^{2}$. Note that the ${C}^{2}$ smoothness of the price on $[L,U]$ under our model is the standard result of the existence and uniqueness theorem for ordinary differential equations.

###### Definition 2.4.

Let $\mathrm{\Psi}(K)$ be the time value of European call options $C(K)$, $K\in [L,U]$, with fixed time to maturity ${t}^{\star}$:

$$\mathrm{\Psi}(K)=C(K)-{(x-K)}^{+}\mathit{\hspace{1em}}\text{for all}K\in [L,U].$$ |

The function $\mathrm{\Psi}(K)$ is the solution to the homogeneous part of (2.1), ie, $\mathrm{\Psi}(K)$ solves

$$\frac{{t}^{\star}{\alpha}^{2}(K)}{2}\frac{{\partial}^{2}\mathrm{\Psi}(K)}{\partial {K}^{2}}-\mathrm{\Psi}(K)=0$$ | (2.5) |

on the intervals $[L,x)$ and $(x,U]$. The function $\mathrm{\Psi}(K)$ is ${C}^{2}$-smooth everywhere on $[L,U]$, except for one point, $K=x$, where it is only continuous.

###### Assumption 2.5.

The function $\mathrm{\Psi}\mathit{}\mathrm{(}K\mathrm{)}$ vanishes at the two endpoints $L$ and $U$:

$\underset{K\to L}{lim}\mathrm{\Psi}(K)=\underset{K\to U}{lim}\mathrm{\Psi}(K)=0.$ |

Call option prices $C(K)$ should satisfy the following boundary conditions:

$C(K)=\{\begin{array}{cc}0\hfill & \text{for}K\to \mathrm{\infty},\hfill \\ (x-L)\hfill & \text{for}K\to 0.\hfill \end{array}$ |

Assumption 2.5 is a manifestation of these conditions for large parameter $U$ and small $L$. In the proposition below, we present the unique solution to (2.1) given Assumptions 2.2 and 2.5.

###### Proposition 2.6.

There is a unique solution $C\mathit{}\mathrm{(}K\mathrm{)}$ to (2.1) for $K\mathrm{\in}\mathrm{[}L\mathrm{,}U\mathrm{]}$ for the LVG volatility function defined in Assumption 2.2:

$$C(K)=\mathrm{\Psi}(K)+{(x-K)}^{+},$$ |

where $\mathrm{\Psi}\mathit{}\mathrm{(}K\mathrm{)}$ satisfies Assumption 2.5 and is equal to

$$\mathrm{\Psi}(K)=\sum _{i=1}^{4}{\psi}_{i}(K){\mathrm{?}}_{\{K\in {I}_{i}\}},$$ |

and

${\psi}_{1}(K)$ | $={\displaystyle \frac{1}{{\beta}_{x}}}{\left({\displaystyle \frac{{\sigma}_{1}}{{\sigma}_{x}}}\right)}^{{q}_{1}}{\displaystyle \frac{1-{\mu}_{1}}{1-{\mu}_{1}{({\sigma}_{1}/{\sigma}_{x})}^{Q}}}{\mathrm{e}}^{p(K-{\nu}_{1})}\left({\displaystyle \frac{1-{\mathrm{e}}^{2p(L-K)}}{1-{\mathrm{e}}^{2p(L-{\nu}_{1})}}}\right),$ | ||

${\psi}_{2}(K)$ | $={\displaystyle \frac{1}{{\beta}_{x}}}{\left({\displaystyle \frac{{\alpha}_{2}(K)}{{\sigma}_{x}}}\right)}^{{q}_{1}}{\displaystyle \frac{1-{\mu}_{1}{({\sigma}_{1}/{\alpha}_{2}(K))}^{Q}}{1-{\mu}_{1}{({\sigma}_{1}/{\sigma}_{x})}^{Q}}},$ | ||

${\psi}_{3}(K)$ | $={\displaystyle \frac{1}{{\beta}_{x}}}{\left({\displaystyle \frac{{\alpha}_{3}(K)}{{\sigma}_{x}}}\right)}^{{r}_{1}}{\displaystyle \frac{1-{\mu}_{2}{({\sigma}_{2}/{\alpha}_{3}(K))}^{R}}{1-{\mu}_{2}{({\sigma}_{2}/{\sigma}_{x})}^{R}}},$ | ||

${\psi}_{4}(K)$ | $={\displaystyle \frac{1}{{\beta}_{x}}}{\left({\displaystyle \frac{{\sigma}_{2}}{{\sigma}_{x}}}\right)}^{{r}_{1}}{\displaystyle \frac{1-{\mu}_{2}}{1-{\mu}_{2}{({\sigma}_{2}/{\sigma}_{x})}^{R}}}{\mathrm{e}}^{s(K-{\nu}_{2})}\left({\displaystyle \frac{1-{\mathrm{e}}^{2s(U-K)}}{1-{\mathrm{e}}^{2s(U-{\nu}_{2})}}}\right).$ |

The coefficients in the above expression are given by

$$p=\sqrt{\frac{2}{{\sigma}_{1}^{2}{t}^{\star}}},s=\sqrt{\frac{2}{{\sigma}_{2}^{2}{t}^{\star}}},Q=\sqrt{1+\frac{8}{{\gamma}_{2}^{2}{t}^{\star}}},R=\sqrt{1+\frac{8}{{\gamma}_{3}^{2}{t}^{\star}}},$$ | ||

$${q}_{1/2}=\frac{1}{2}\pm \frac{1}{2}Q,{r}_{1/2}=\frac{1}{2}\pm \frac{1}{2}R,$$ | ||

$$\begin{array}{cc}\hfill {\mu}_{1}& =\frac{{\gamma}_{2}{q}_{1}(1-{\mathrm{e}}^{2p(L-{\nu}_{1})})-{\sigma}_{1}p(1+{\mathrm{e}}^{2p(L-{\nu}_{1})})}{{\gamma}_{2}{q}_{2}(1-{\mathrm{e}}^{2p(L-{\nu}_{1})})-{\sigma}_{1}p(1+{\mathrm{e}}^{2p(L-{\nu}_{1})})},\hfill \\ \hfill {\mu}_{2}& =\frac{{\gamma}_{3}{r}_{1}(1-{\mathrm{e}}^{2s(U-{\nu}_{2})})-{\sigma}_{2}s(1+{\mathrm{e}}^{2s(U-{\nu}_{2})})}{{\gamma}_{3}{r}_{2}(1-{\mathrm{e}}^{2s(U-{\nu}_{2})})-{\sigma}_{2}s(1+{\mathrm{e}}^{2s(U-{\nu}_{2})})},\hfill \end{array}$$ | ||

$${\beta}_{x}=\frac{{\gamma}_{2}{q}_{1}}{{\sigma}_{x}}\left(\frac{1-{\mu}_{1}{q}_{2}{({\sigma}_{1}/{\sigma}_{x})}^{Q}/{q}_{1}}{1-{\mu}_{1}{({\sigma}_{1}/{\sigma}_{x})}^{Q}}\right)-\frac{{\gamma}_{3}{r}_{1}}{{\sigma}_{x}}\left(\frac{1-{\mu}_{2}{r}_{2}{({\sigma}_{2}/{\sigma}_{x})}^{R}/{r}_{1}}{1-{\mu}_{2}{({\sigma}_{2}/{\sigma}_{x})}^{R}}\right).$$ |

Parameters ${\gamma}_{2}$ and ${\gamma}_{3}$ can be calculated from the set $\omega $ as in Assumption 2.2. Parameterizations of the type given in Proposition 2.6 generally have five unknown parameters corresponding to the elements of $\omega $. A proof of Proposition 2.6 can be found in the online appendix; in this proof, it is also shown that if the subintervals ${I}_{2}$ and ${I}_{3}$ vanish, the solution is identical to the solution given by Carr and Nadtochiy (2017, p. 23) for $R=1$.

### 2.4 Limits

Let us consider the case when the boundary points $L$ and $U$ go to zero and infinity, respectively. This choice of parameters may be motivated by situations when modeling assets that can theoretically take any value on the positive real axis, such as exchange rates or stocks.

Letting $L$ decrease toward zero can be done by simple substitution. Letting $U$ increase toward infinity requires us to solve two simple limits on the local solution on interval ${I}_{4}$:

$\underset{U\to \mathrm{\infty}}{lim}{\psi}_{4}(K)={\displaystyle \frac{1}{{\beta}_{x}}}{\left({\displaystyle \frac{{\sigma}_{2}}{{\sigma}_{x}}}\right)}^{{r}_{1}}{\displaystyle \frac{1-{lim}_{U\to \mathrm{\infty}}({\mu}_{2})}{1-{lim}_{U\to \mathrm{\infty}}({\mu}_{2}){({\sigma}_{2}/{\sigma}_{x})}^{R}}}{\mathrm{e}}^{s({\nu}_{2}-K)}.$ |

In the derivation above, it is assumed that the limit for ${\mu}_{2}$ exists. This limit can be shown as

$\underset{U\to \mathrm{\infty}}{lim}{\mu}_{2}=\underset{U\to \mathrm{\infty}}{lim}{\displaystyle \frac{{\gamma}_{3}{r}_{1}-{\sigma}_{2}s-{\mathrm{e}}^{2s(U-{\nu}_{2})}({\gamma}_{3}{r}_{1}+{\sigma}_{2}s)}{{\gamma}_{3}{r}_{2}-{\sigma}_{2}s-{\mathrm{e}}^{2s(U-{\nu}_{2})}({\gamma}_{3}{r}_{2}+{\sigma}_{2}s)}}={\displaystyle \frac{({\gamma}_{3}{r}_{1}+{\sigma}_{2}s)}{({\gamma}_{3}{r}_{2}+{\sigma}_{2}s)}}.$ | (2.6) |

### 2.5 Discussion

There is an issue related to the structure of the function $\alpha (K)$ from Assumption 2.2; $\alpha (K)$ is assumed to be constant for $K\in {I}_{1}$ and $K\in {I}_{4}$, while in the Black–Scholes model the diffusion function increases linearly from zero. Using a constant LVG volatility in the tails will cause the tail probabilities to become unrealistically large for $K\ll x$ and unrealistically small for $K\gg x$. In Section 3.4, we study the impact of this effect. Evidently, implied volatility smiles generated by the proposed model will start decreasing for very large strikes. A related (but much more powerful) effect can be seen in the normal model for asset returns, where volatility decreases as a function of strike. Here this effect is much less pronounced, due to the gamma process introduced in the model, which causes tail probabilities in the return to increase.

The parameterization of European call option prices in Proposition 2.6 uses lower and upper bounds $L$ and $U$. In the numerical calculations in this paper, these parameters will be set to zero and infinity for simplicity. In practice, other choices of boundary points may be considered.

## 3 Calibration

In this section, the closed-form solution from Proposition 2.6 is calibrated to volatility smiles from the FX options market. The calibration scope is restricted to independent volatility smiles with a fixed time to maturity. The data used in these calibrations consists of daily Reuters quoted EURUSD and EURSEK volatility smiles for 208 days between May 20, 2013 and March 12, 2014.

### 3.1 Two submodels

The option price parameterization in Proposition 2.6 can be considered as having either three or five unknown parameters. The parameter set $\omega $ uniquely determines an LVG volatility function. The set $\omega $ is defined as $\omega :=\{{\sigma}_{1},{\sigma}_{x},{\sigma}_{2},{\nu}_{1},{\nu}_{2}\}$, where the ${\sigma}_{1}$, ${\sigma}_{2}$ and ${\sigma}_{x}$ are the LVG volatilities in the two subintervals $K\in {I}_{1}$, $K\in {I}_{4}$, and at $K=x$. The parameters ${\nu}_{1}$ and ${\nu}_{2}$ correspond to the two border points where the LVG volatility function goes from being constant to being linear. These two positive parameters are assumed to satisfy the criteria $$ and $$. Logically, these values should be within the range of calibration points, since this is the region where our degrees of freedom are needed to fit the market-quoted option prices. The first model explored in this section is a five-parameter model, denoted model 5P. For this model, the calibration algorithm is allowed to use the entire parameter set $\omega $, including ${\nu}_{1}$ and ${\nu}_{2}$, as degrees of freedom. The second model, denoted model 3P, uses values for ${\nu}_{1}$ and ${\nu}_{2}$ derived directly from the market-quoted strikes. FX options market quotes generally contain volatilities for ATM, two risk-reversal spreads and two butterfly spreads or strangles. By using linear identities (see, for example, Clark 2011), market quotes can be transformed to volatilities at five absolute strikes:^{2}^{2}These five strikes correspond to two puts, two calls and at a delta neutral straddle. See Clark (2011) for more details on market-quotation conventions for FX options.

$${K}_{10\mathrm{\Delta}\mathrm{PUT}},{K}_{25\mathrm{\Delta}\mathrm{PUT}},{K}_{\mathrm{DNS}},{K}_{25\mathrm{\Delta}\mathrm{CALL}},{K}_{10\mathrm{\Delta}\mathrm{CALL}}.$$ |

This list is in order of magnitude. The second of our proposed models, model 3P, uses the values

$$\begin{array}{cc}\hfill {\nu}_{1}& =\frac{{K}_{10\mathrm{\Delta}\mathrm{PUT}}+{K}_{25\mathrm{\Delta}\mathrm{PUT}}}{2},\hfill \\ \hfill {\nu}_{2}& =\frac{{K}_{10\mathrm{\Delta}\mathrm{CALL}}+{K}_{25\mathrm{\Delta}\mathrm{CALL}}}{2}.\hfill \end{array}\}$$ | (3.1) |

Hence, the calibration for model 3P uses only three degrees of freedom. Although other static values for ${\nu}_{1}$ and ${\nu}_{2}$ could be considered, our tests show that the values from (3.1) produce smooth and well-fitted implied volatility smiles (see Figure 1).

### 3.2 Optimization problem

The residual minimization will be performed using a least-squares approach. In order to obtain a numerically stable calibration, put options are used for $$ and call options are used for $K\ge x$. This corresponds to solving the following optimization problem:

$\underset{\omega \in {\mathbb{R}}_{+}^{5}}{\mathrm{min}}{\displaystyle \frac{1}{2}}\left[{\displaystyle \sum _{i=1}^{2}}{|{P}_{\omega}^{\mathrm{A}}({K}_{i})-{P}^{\mathrm{M}}({K}_{i})|}^{2}+{\displaystyle \sum _{i=3}^{5}}{|{C}_{\omega}^{\mathrm{A}}({K}_{i})-{C}^{\mathrm{M}}({K}_{i})|}^{2}\right],$ |

such that $$, $$, $$, $$ and $$, where the option prices marked $\mathrm{A}$ come from the parameterization in Section 2.3, using the parameter set $\omega $. The prices marked $\mathrm{M}$ are mids coming from market-quoted volatilities. The problem formulation corresponds to minimizing the square of the residual norm, provided that the set $\omega $ is used to generate the option price function. As the intrinsic value for both put options with $K\le x$ and call options with $K\ge x$ is zero, we can describe $\mathrm{\Psi}(K)$ from Proposition 2.6 as

$$ |

Therefore, the optimization problem can be redefined using only $\mathrm{\Psi}(K)$:

$$ | (3.2) |

such that $$, $$, $$, $$, $$ and

$$ |

To perform the optimization, we propose using a modified Levenberg–Marquardt algorithm (Fletcher 1987).

###### Remark 3.1.

Note that no weights are used in the suggested objective function. We did not find that adding weights and modifying the objective function to use relative errors had any positive effect on the calibration result or stability of the calibrated parameters. To optimize performance, we chose to calibrate the model to prices, as opposed to implied volatilities. Any numerical optimization routine can potentially lead to calibration risk, and we cannot always guarantee that optimization finds the global minimum of the objective function. The quantification of such risk and its impact on derivatives pricing lies beyond the scope of this paper. We refer the reader to, for example, Cont (2006) and Deryabin (2012).

### 3.3 Results

In Table 1, average absolute errors from market mids in implied volatility basis points are shown, together with the average calibration time. This example considers options maturing in three months. The market data was collected from Reuters. Evidently, model 5P is more accurate than model 3P and calibrates almost exactly to market-quoted mids. This is usually considered to be an advantage, but it can also cause some problems. In the case of a currency pair such as the EURSEK, input data is often of poor quality. Market-quoted mids may not follow a pattern to which it is possible to fit a well-behaved curve. In these cases, we found a risk of model 5P generating irregular volatility smiles with local maximums. This effect was not observed when calibrating model 3P.

Average absolute errors from mid | ||||||

10$?$PUT | 25$?$PUT | ATM | 25$?$CALL | 10$?$CALL | $?$ (ms) | |

EURUSD/5P | 1.62 | 1.48 | 0.52 | 0.41 | 0.59 | 2.17 |

EURUSD/3P | 4.92 | 3.10 | 0.63 | 2.66 | 5.10 | 0.995 |

EURSEK/5P | 0.99 | 0.29 | 0.82 | 5.13 | 9.23 | 5.22 |

EURSEK/3P | 1.46 | 1.01 | 2.98 | 8.06 | 10.80 | 1.04 |

Model 3P calibrates faster than model 5P. Neither of the two submodels misses the bid–ask spread for any quote in the calibration sample. On considering the total test coverage of 3328 different volatility smiles, each consisting of five points, we conclude that both models tend to deliver volatility smiles that fit market quotes within the bid–ask spread. Note that we have not cleaned up the Reuters volatility surfaces. In particular, we encountered several cases of negative butterflies. This was done on purpose to test the model’s limitations. When the surface is regular, the fit for both models is further enhanced. A graphical example of calibrated volatility smiles can be seen in Figure 1. In this example, we consider eight different maturities, from one week to one year. The calibration of these smiles is done using model 3P. The crosses in the figure correspond to market mids. While quoted FX volatility surfaces quickly become more sparse with increasing time to maturity, in our example we have extrapolated the price functions into a rectangular grid.

### 3.4 Extreme situations

In this section, we investigate how the option price parameterization behaves for some extreme situations that are problematic from a theoretical perspective. The first situation concerns asymptotic volatility behavior for very large strikes. As concluded in Section 2.5, the proposed model may generate unrealistically small asymptotic tail volatilities due to the constant LVG volatility for large strikes. Calibration of the model confirms this behavior, but suggests that it is present only at unrealistically large moneyness values. The example in Figure 2 shows the effect on tail volatilities for the EURUSD options from Figure 1 that will mature in six months. In this example, the effect is observable and starts at $K/F\approx 5$. Between $K/F=5$ and $K/F=10$ the volatility decreases by around 0.007, which corresponds to a drop of roughly 5%.

A second situation concerns the delta for options that are extremely in-the-money. In the Black–Scholes model for put options, ${\mathrm{\Delta}}_{\mathrm{P}}=-1$ when the value of the underlying asset vanishes (ie, $x=0$). The corresponding relation for call options is that ${\mathrm{\Delta}}_{\mathrm{C}}=1$ when the strike vanishes (ie, $K=0$). Ideally, the model in this paper should exhibit these attributes.

In the construction of the closed-form solution in Proposition 2.6, we imposed the boundary conditions

$\underset{K\to L}{lim}\mathrm{\Psi}(K)=\underset{K\to U}{lim}\mathrm{\Psi}(K)=0.$ |

If boundary conditions are also imposed on the $\mathrm{\Delta}$, the nontrivial solution space for (2.1) vanishes. Hence, theoretically, the $\mathrm{\Delta}$ will not satisfy the Black–Scholes boundary point $\mathrm{\Delta}$ behavior. However, numerical testing shows that the error usually is smaller than ${10}^{-10}$.

## 4 Maturity interpolation

In this section, we explore ways of extending the model proposed in Section 2 to allow interpolation between maturities. Specifically, we will explore the possibilities of performing interpolation between closed-form solutions of (2.1) in such a way that the absence of calendar spread arbitrage is ensured. The proposition below implies that arbitrage will appear if and only if the function $\mathrm{\Psi}(K)=\mathrm{\Psi}(K,{t}^{*})$ from Proposition 2.6 is decreasing with increasing time to maturity ${t}^{*}$.

###### Proposition 4.1.

Let ${\mathrm{\Psi}}_{i}\mathit{}\mathrm{(}K\mathrm{)}$, $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$, be the time values of call options from Definition 2.4, with time to maturity ${t}^{\mathrm{\star}}\mathrm{=}{\tau}_{i}$. Further, let $$ be a sequence of increasing maturities. Then the condition

${\mathrm{\Psi}}_{i+1}(K)\ge {\mathrm{\Psi}}_{i}(K),i=1,\mathrm{\dots},N,\mathit{\text{for all}}K\in [L,U]$ | (4.1) |

is the necessary and sufficient condition for the absence of calendar spread arbitrage between the two maturities ${\tau}_{i}$ and ${\tau}_{i\mathrm{+}\mathrm{1}}$ in the interval $K\mathrm{\in}\mathrm{[}L\mathrm{,}U\mathrm{]}$.

###### Proof.

Option prices contain calendar spread arbitrage if a call option with time to maturity ${\tau}_{i+1}$ is priced cheaper than a call option with the same strike but a shorter time to maturity ${\tau}_{i}$. For a discrete set of maturities, this condition corresponds to

$$ |

Since the intrinsic value is independent of the time to maturity, the condition

$C(K,{\tau}_{i+1})-C(K,{\tau}_{i})={\mathrm{\Psi}}_{i+1}(K)-{\mathrm{\Psi}}_{i}(K)$ |

is satisfied. From put–call parity it follows that this condition is also true for put options. This proposition can also easily be proved in a setting with continuous maturities. Differentiating the solution in Proposition 2.6 with respect to maturity $\tau $ yields

$$\frac{\partial C}{\partial \tau}(K,\tau )=\frac{\partial \mathrm{\Psi}}{\partial \tau}(K,\tau )\mathit{\hspace{1em}}\text{for all}K\in [L,U].$$ |

However, in this paper only the discrete maturity case will be considered. ∎

### 4.1 LVG variance

Since an option price function exists for every parameter set $\omega \in {\mathbb{R}}_{+}^{5}$, prices for intermediate maturities can be found by producing new parameter sets. This approach requires a technique to ensure the absence of arbitrage between maturities. When developing such a technique, we will make use of a new variable.

###### Definition 4.2.

The LVG variance $V(K,\tau )$ is the product of the time to maturity and the square of the corresponding LVG volatility function:

$V(K,{\tau}_{i})={\tau}_{i}{\alpha}^{2}(K,{\tau}_{i})\mathit{\hspace{1em}}\text{for all}K\in [L,U],{\tau}_{i}0\text{and}i=0,1,\mathrm{\dots},N,$ | (4.2) |

where $\alpha (K,{\tau}_{i})$ is the LVG volatility function calibrated to market-quoted option prices at maturity ${\tau}_{i}$.

Since only a finite number of strikes are considered, it is trivial to perform the numerical comparison between the price functions from Proposition 2.6. Hence, it is easy to test the criteria in Proposition 4.1. However, our aim is to produce conditions on the relation between parameter sets and the corresponding maturities that ensure Proposition 4.1 is satisfied. In order to simplify this problem, we use the following theorem.

###### Theorem 4.3.

Let $V\mathit{}\mathrm{(}K\mathrm{,}{\tau}_{\mathrm{1}}\mathrm{)}$ and $V\mathit{}\mathrm{(}K\mathrm{,}{\tau}_{\mathrm{2}}\mathrm{)}$ be two LVG variance functions as in Definition 4.2. Let nonnegative functions ${\mathrm{\Psi}}_{\mathrm{1}}\mathit{}\mathrm{(}K\mathrm{)}$ and ${\mathrm{\Psi}}_{\mathrm{2}}\mathit{}\mathrm{(}K\mathrm{)}$ be solutions to the corresponding homogeneous equations

$\frac{V(K,{\tau}_{1})}{2}}{\displaystyle \frac{{\partial}^{2}{\mathrm{\Psi}}_{1}(K)}{\partial {K}^{2}}$ | $={\mathrm{\Psi}}_{1}(K),$ | ||

$\frac{V(K,{\tau}_{2})}{2}}{\displaystyle \frac{{\partial}^{2}{\mathrm{\Psi}}_{2}}{\partial {K}^{2}}}(K)$ | $={\mathrm{\Psi}}_{2}(K),$ |

on intervals $\mathrm{[}L\mathrm{,}x\mathrm{)}$ and $\mathrm{(}x\mathrm{,}U\mathrm{]}$, with the boundary conditions

$\underset{K\to L}{lim}{\mathrm{\Psi}}_{1}(K)$ | $=\underset{K\to L}{lim}{\mathrm{\Psi}}_{2}(K)=0,$ | ||

$\underset{K\to U}{lim}{\mathrm{\Psi}}_{1}(K)$ | $=\underset{K\to U}{lim}{\mathrm{\Psi}}_{2}(K)=0.$ |

Assume further that the functions ${\mathrm{\Psi}}_{\mathrm{1}}\mathit{}\mathrm{(}K\mathrm{)}$ and ${\mathrm{\Psi}}_{\mathrm{2}}\mathit{}\mathrm{(}K\mathrm{)}$ are continuous for $K\mathrm{\in}\mathrm{[}L\mathrm{,}U\mathrm{]}$, and satisfy

$\underset{K\to {x}^{-}}{lim}{\displaystyle \frac{\partial {\mathrm{\Psi}}_{1}(K)}{\partial K}}$ | $=1+\underset{K\to {x}^{+}}{lim}{\displaystyle \frac{\partial {\mathrm{\Psi}}_{1}(K)}{\partial K}},$ | ||

$\underset{K\to {x}^{-}}{lim}{\displaystyle \frac{\partial {\mathrm{\Psi}}_{2}(K)}{\partial K}}$ | $=1+\underset{K\to {x}^{+}}{lim}{\displaystyle \frac{\partial {\mathrm{\Psi}}_{2}(K)}{\partial K}}.$ |

Then, if the two LVG variance functions satisfy

$V(K,{t}_{1})\le V(K,{t}_{2})\mathit{\hspace{1em}}\mathit{\text{for all}}K\in [L,U],$ |

the two solutions ${\mathrm{\Psi}}_{\mathrm{1}}\mathit{}\mathrm{(}K\mathrm{)}$ and ${\mathrm{\Psi}}_{\mathrm{2}}\mathit{}\mathrm{(}K\mathrm{)}$ will satisfy

${\mathrm{\Psi}}_{1}(K)\le {\mathrm{\Psi}}_{2}(K)\mathit{\hspace{1em}}\mathit{\text{for all}}K\in [L,U].$ |

A direct effect of this theorem is that increasing LVG variance is a sufficient condition for Proposition 4.1 and for the absence of calendar spread arbitrage. Theorem 4.3 is related to the classical majorant method in the theory of ordinary differential equations.

###### Proof of Theorem 4.3.

Consider two solutions ${\mathrm{\Psi}}_{1}(K)$ and ${\mathrm{\Psi}}_{2}(K)$ from the LVG model at maturities ${\tau}_{1}$ and ${\tau}_{2}$. The functions will be solutions to the boundary-value problem:

$$\begin{array}{cc}\hfill \frac{1}{2}V(K,{\tau}_{1}){\mathrm{\Psi}}_{1}^{\mathrm{\prime \prime}}(K)& ={\mathrm{\Psi}}_{1}(K)\hfill \\ \hfill \frac{1}{2}V(K,{\tau}_{2}){\mathrm{\Psi}}_{2}^{\mathrm{\prime \prime}}(K)& ={\mathrm{\Psi}}_{2}(K)\hfill \end{array}\}\mathit{\hspace{1em}}\text{for all}K\in [L,U],$$ |

with the boundary conditions

${\mathrm{\Psi}}_{1}(L)={\mathrm{\Psi}}_{2}(L)={\mathrm{\Psi}}_{1}(U)={\mathrm{\Psi}}_{2}(U)=0.$ |

Further, ${\mathrm{\Psi}}_{1}(K)$ and ${\mathrm{\Psi}}_{2}(K)$ are continuous nonnegative functions that are ${C}^{2}$ at all points except those of $x$, where the following conditions are satisfied:

$\underset{K\to {x}^{+}}{lim}{\mathrm{\Psi}}_{1}^{\prime}(K)+1$ | $=\underset{K\to {x}^{-}}{lim}{\mathrm{\Psi}}_{1}^{\prime}(K),$ | ||

$\underset{K\to {x}^{+}}{lim}{\mathrm{\Psi}}_{2}^{\prime}(K)+1$ | $=\underset{K\to {x}^{-}}{lim}{\mathrm{\Psi}}_{2}^{\prime}(K).$ |

Define the function

$$G(K)={\mathrm{\Psi}}_{2}(K)-{\mathrm{\Psi}}_{1}(K).$$ |

Since

$$\underset{K\to {x}^{+}}{lim}{\mathrm{\Psi}}_{2}^{\prime}(K)-{\mathrm{\Psi}}_{1}^{\prime}(K)=\underset{K\to {x}^{-}}{lim}{\mathrm{\Psi}}_{2}^{\prime}(K)-{\mathrm{\Psi}}_{1}^{\prime}(K),$$ |

$G(K)$ will be ${C}^{2}$ on $[L,U]$. If we assume $$ for all $K\in [L,U]$, then

$$ |

is satisfied. Hence, $G(K)$ is a ${C}^{2}$ function that has to be concave at all points in $[L,U]$, where it is negative. A consequence of these properties is that if $G(K)$ is negative at any point $K\in [L,U]$, the boundary conditions ${\mathrm{\Psi}}_{1}(L)={\mathrm{\Psi}}_{2}(L)={\mathrm{\Psi}}_{1}(U)={\mathrm{\Psi}}_{2}(U)=0$ cannot be satisfied. We conclude that $$ for any $K\in [L,U]$ contradicts our assumptions. Since $V(K,{\tau}_{1})=V(K,{\tau}_{2})\u27f9{\mathrm{\Psi}}_{1}(K)={\mathrm{\Psi}}_{2}(K)$, the implication

$$V(K,{\tau}_{1})\le V(K,{\tau}_{2})\mathit{\hspace{1em}}\text{for all}K\in [L,U]\u27f9{\mathrm{\Psi}}_{1}(K)\le {\mathrm{\Psi}}_{2}(K)\mathit{\hspace{1em}}\text{for all}K\in [L,U]\hspace{1em}$$ |

is satisfied. ∎

### 4.2 Conditions for the absence of calendar spread arbitrage

Let ${\tau}_{i}>0$, $i=1,\mathrm{\dots},M$, be an increasing sequence of market-quoted maturities. Let

${\omega}_{i}=\{{\sigma}_{(1,i)},{\sigma}_{(x,i)},{\sigma}_{(2,i)},{\nu}_{(1,i)},{\nu}_{(2,i)}\},i=1,\mathrm{\dots},M,$ | (4.3) |

denote a calibrated parameter set that uniquely determines an option price function at maturity ${\tau}_{i}$. Let us investigate which conditions need to be satisfied for ensuring that the LVG variance is increasing in maturity. It is sufficient to look at this problem in a two-maturity setting:

$${\tau}_{i+1}{\alpha}^{2}(K,{\tau}_{i+1})\ge {\tau}_{i}{\alpha}^{2}(K,{\tau}_{i})\mathit{\hspace{1em}}\text{for all}K\in [L,U]\u27f9\sqrt{\frac{{\tau}_{i}}{{\tau}_{i+1}}}\le \frac{\alpha (K,{\tau}_{i+1})}{\alpha (K,{\tau}_{i})}.\mathit{\hspace{1em}}$$ | (4.4) |

Let us define the function ${\mathrm{\Phi}}_{i+1}(K):=\alpha (K,{\tau}_{i+1})/\alpha (K,{\tau}_{i})$. Ensuring that

$\underset{K\in [L,U]}{\mathrm{min}}{\mathrm{\Phi}}_{i+1}(K)\ge \sqrt{{\displaystyle \frac{{\tau}_{i}}{{\tau}_{i+1}}}}$ |

is then equivalent to ensuring that the condition in (4.4) is satisfied. We will make some simplifying assumptions concerning the structure of the LVG volatility functions. Recall that in Section 2.2 we factorized our space $K\in [L,U]$ into four subintervals, ${I}_{1}$, ${I}_{2}$, ${I}_{3}$ and ${I}_{4}$. The point ${\nu}_{1}$ represents the border between ${I}_{1}$ and ${I}_{2}$, and the point ${\nu}_{2}$ represents the border between ${I}_{3}$ and ${I}_{4}$.

###### Remark 4.4.

We assume that the size of the subintervals ${I}_{2}$ and ${I}_{3}$ from (2.2) increases with maturity. Note that this also means that the value of ${\nu}_{1}$ is decreasing and ${\nu}_{2}$ is increasing with maturity. Since FX option quotes become sparser with increasing maturity, this assumption is always satisfied for model 3P from Section 3, where the values of ${\nu}_{1}$ and ${\nu}_{2}$ are derived from market quotes. Because of this feature, and the fact that overfitting seems to be more likely for model 5P, in the remainder of this paper we will assume that model 3P is being used. All figures involving maturity interpolation are generated using model 3P.

Denoting by ${I}_{j}^{i}$, $i=1,\mathrm{\dots},M,j=1,\mathrm{\dots},4$, each subinterval $j$ at time to maturity ${\tau}_{i}$, we will split up the problem into six different subintervals:

$K\in \{{I}_{1}^{i+1},{I}_{1}^{i}\cap {I}_{1}^{i+1},{I}_{2}^{i},{I}_{3}^{i},{I}_{4}^{i}\cap {I}_{4}^{i+1},{I}_{4}^{i+1}\}.$ | (4.5) |

Figure 3 shows a graphical presentation of the subintervals. The union of these six subintervals is the entire range $[L,U]$. In accordance with Assumption 2.2, ${\mathrm{\Phi}}_{i+1}(K)$ will be constant for $K\in {I}_{1}^{i+1}$, $K\in {I}_{4}^{i+1}$, and linear for $K\in {I}_{1}^{i}\cap {I}_{1}^{i+1}$, $K\in {I}_{4}^{i}\cap {I}_{4}^{i+1}$, and the ratio of two linear functions for $K\in {I}_{2}^{i}$ and $K\in {I}_{3}^{i}$. Since ${\mathrm{\Phi}}_{i+1}(K)$ will be a positive continuous function with this structure, we can conclude that its minimum value can be found at one of the following points: $K={\nu}_{(1,i+1)}$, $K={\nu}_{(1,i)}$, $K=x$, $K={\nu}_{(2,i)}$ or $K={\nu}_{(2,i+1)}$. This can be expressed as

$$\underset{K\in [L,U]}{\mathrm{min}}{\mathrm{\Phi}}_{i+1}(K)=\mathrm{min}(\frac{{\sigma}_{(1,i+1)}}{{\sigma}_{(1,i)}},\frac{{\sigma}_{(2,i+1)}}{{\sigma}_{(2,i)}},\frac{{\sigma}_{(x,i+1)}}{{\sigma}_{(x,i)}},\frac{\alpha ({\nu}_{(1,i)},{\tau}_{i+1})}{{\sigma}_{(1,i)}},\frac{\alpha ({\nu}_{(2,i)},{\tau}_{i+1})}{{\sigma}_{(2,i)}}).$$ | (4.6) |

Let us first have a look at the last two elements of ${\mathrm{min}}_{K\in [L,U]}{\mathrm{\Phi}}_{i+1}(K)$:

$\frac{\alpha ({\nu}_{(1,i)},{\tau}_{i+1})}{{\sigma}_{(1,i)}}}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\displaystyle \frac{\alpha ({\nu}_{(2,i)},{\tau}_{i+1})}{{\sigma}_{(2,i)}}}.$ |

Since $\alpha ({\nu}_{(1,i)},{\tau}_{i+1})$ and $\alpha ({\nu}_{(2,i)},{\tau}_{i+1})$ are elements of neither ${\omega}_{i}$ or ${\omega}_{i+1}$, finding sufficient conditions for them is more complicated than for the other points.

###### Remark 4.5.

Note that for all volatility smiles in our data set the calibrated LVG volatility function had a structure similar to the example in Figure 3, satisfying the conditions ${\sigma}_{x}>{\sigma}_{1}$ and ${\sigma}_{x}>{\sigma}_{2}$. While these conditions are not necessary to guarantee the absence of arbitrage in a single smile, they simplify the constraints needed for the absence of calendar spread arbitrage when interpolating in maturity.

Assuming that the two conditions from Remark 4.5 are satisfied for all maturities, the minimum of ${\mathrm{\Phi}}_{i+1}(K)$ cannot be at these two points since $\alpha ({\nu}_{(1,i)},{\tau}_{i+1})>{\sigma}_{(1,i+1)}$ and $\alpha ({\nu}_{(2,i)},{\tau}_{i+1})>{\sigma}_{(2,i+1)}$.

Linear conditions are easily found for the first three elements in the vector in (4.6). We can now formulate a condition for the absence of calendar spread arbitrage between two price functions.

###### Condition 4.6.

Let ${\omega}_{i}$ and ${\omega}_{i\mathrm{+}\mathrm{1}}$ be parameter sets found through calibration to market-quoted volatility smiles at maturity ${\tau}_{i\mathrm{+}\mathrm{1}}\mathrm{>}{\tau}_{i}$. Assume that the parameters satisfy

$${\sigma}_{(x,i)}>{\sigma}_{(1,i)},{\sigma}_{(x,i)}>{\sigma}_{(2,i)},{\sigma}_{(x,i+1)}>{\sigma}_{(1,i+1)},{\sigma}_{(x,i+1)}>{\sigma}_{(2,i+1)},$$ | ||

$$ |

Then the following three linear conditions are sufficient to ensure absence of arbitrage between the two maturities ${\tau}_{i}$ and ${\tau}_{i\mathrm{+}\mathrm{1}}$:

$$\frac{{\sigma}_{(1,i+1)}}{{\sigma}_{(1,i)}}\ge \sqrt{\frac{{\tau}_{i}}{{\tau}_{i+1}}},\frac{{\sigma}_{(x,i+1)}}{{\sigma}_{(x,i)}}\ge \sqrt{\frac{{\tau}_{i}}{{\tau}_{i+1}}},\frac{{\sigma}_{(2,i+1)}}{{\sigma}_{(2,i)}}\ge \sqrt{\frac{{\tau}_{i}}{{\tau}_{i+1}}}.$$ | (4.7) |

We can now easily introduce these linear parameter constraints into the optimization problem from (3.2):

$$ | (4.8) |

such that

$${\sigma}_{(1,i+1)}\ge {\sigma}_{(1,i)}\sqrt{\frac{{\tau}_{i}}{{\tau}_{i+1}}},{\sigma}_{(x,i+1)}\ge {\sigma}_{(x,i)}\sqrt{\frac{{\tau}_{i}}{{\tau}_{i+1}}},$$ | |||

$${\sigma}_{(2,i+1)}\ge {\sigma}_{(2,i)}\sqrt{\frac{{\tau}_{i}}{{\tau}_{i+1}}},\nu (1,i+1)\le \nu (1,i),\nu (2,i+1)\ge \nu (2,i),$$ | |||

and | |||

$${\sigma}_{(x,i)}\ge {\sigma}_{(1,i)},{\sigma}_{(x,i)}\ge {\sigma}_{(2,i)},{\sigma}_{(x,i+1)}\ge {\sigma}_{(1,i+1)},{\sigma}_{(x,i+1)}\ge {\sigma}_{(2,i+1)}.$$ |

Note that the constraints added to the optimization problem above are linear and will not significantly slow down the computation compared with the optimization problem (3.2). However, since these constraints add restrictions to the calibration, they could increase the residuals of the curve fit. This is problematic since increasing LVG variance is sufficient, but not necessary, for the absence of arbitrage. Our results imply that parameter sets found without these additional restrictions rarely violate Condition 4.6. When calibrating the 3328 market smiles from Section 3 independently using (3.2), decreasing LVG variance was only observed on a few occasions, none of which corresponded to real calendar spread arbitrage.

### 4.3 Interpolation

Let $$ be two market-quoted maturities with known parameter sets ${\omega}_{1}$ and ${\omega}_{2}$. The next step toward generating intermediate option price functions is to add a new parameter set $\omega $ for a maturity $\tau $ satisfying $$. In this section, we will interpolate linearly between parameter sets, a method producing implied volatility surfaces that are continuous in maturity. In practice, however, using, for example, monotone piecewise cubic interpolation (Fritsch and Carlson 1980) may be more appropriate, especially if the objective is to generate Dupire local volatility surfaces, as in Section 5. It is easily shown using Condition 4.6 that linear interpolation in the parameter space ensures an increasing LVG variance for intermediate maturities. This will also be the case with other monotone interpolation techniques.

We will present two different algorithms to generate new option price functions on intermediate maturities. Let ${\tau}_{1}$ and ${\tau}_{N}$, $$, denote two market-quoted maturities with corresponding calibrated parameter sets ${\omega}_{1}$ and ${\omega}_{N}$. The algorithms below evaluate the region between two parameter sets, but are trivially expandable to cover an entire surface.

Let $N$ be the number of requested maturity points. Assume that the grid of option prices is rectangular. This means that there exist factors ${K}_{\mathrm{min}}$ and ${K}_{\mathrm{max}}$ that are common for all maturities. Since option prices are represented by continuous functions that are well defined in an arbitrarily large region, the existence of such factors can be ensured by using the same $[L,U]$ for all maturities.

- (1)
Find parameter sets ${\omega}_{1}$ and ${\omega}_{N}$ for two consecutive market-quoted volatility smiles with maturities $$ by solving the optimization problem in (4.8).

- (2)
Create a maturity grid $$.

- (3)
For $$, generate new intermediate parameter sets by linear interpolation: $\omega ({\tau}_{i+1})={\omega}_{N}(1-\lambda )+{\omega}_{i}\lambda $, where $\lambda =({\tau}_{N}-{\tau}_{i+1})/({\tau}_{N}-{\tau}_{i})$.

The main problem with the approach outlined above is that Condition 4.6 is sufficient, but not necessary, for the absence of arbitrage. An alternative approach is to solve a least-squares problem when arbitrage is introduced between two maturities. This would involve solving a minimization problem with nonlinear constraints. The optimization problem can be formulated as

$$ | (4.9) |

such that $$ for all $K\in [{K}_{\mathrm{min}},{K}_{\mathrm{max}}]$.

This optimization problem needs to be solved each time the calendar spread arbitrage is found. Note that solving (4.9) takes significantly more time than the original calibration. Using the same notation as above, an algorithm could be formulated as follows.

- (1)
Generate independent option price functions for maturities ${\tau}_{1}$ and ${\tau}_{N}$ by using (3.2).

- (2)
If ${\mathrm{\Psi}}_{1}(K)>{\mathrm{\Psi}}_{N}(K)$ for any $K\in [{K}_{\mathrm{min}},{K}_{\mathrm{max}}]$, generate a new parameter set for maturity ${\tau}_{N}$ by using (4.9).

- (3)
For $$, generate a new intermediate parameter set by linear interpolation: $\omega ({\tau}_{i+1})={\omega}_{N}(1-\lambda (\tau ))+{\omega}_{i}\lambda (\tau )$, where $\lambda (\tau )=({\tau}_{N}-{\tau}_{i+1})/({\tau}_{N}-{\tau}_{i})$.

The strength of the least-squares approach is that, unlike the LVG variance approach, no unnecessary constraints are imposed on the LVG volatility functions. We may imagine that the LVG variance approach would be more effective for illiquid currency pairs, where bad model fits are common, while the least-squares approach is better suited for liquid, well-behaved currency pairs. An interpolated LVG volatility surface is shown in Figure 4. This surface is based on calibrations to eight different EURUSD volatility smiles with maturities ranging from one week to one year. New parameter sets have been generated by linear interpolation. The result is a surface consisting of piecewise linear LVG volatility functions for fifty equally distributed maturities.

### 4.4 Results

Volatility surfaces produced using the methods in this paper will be twice continuously differentiable in the strike dimension, while the smoothness in the maturity dimension depends on the choice of parameter-interpolation technique. If we use piecewise cubic interpolation, the resulting implied volatility surface will also be continuously differentiable in the maturity dimension. As a consequence of this differentiability, the model can be used to produce continuous representations of delta and gamma. A calibrated and heavily extrapolated implied volatility surface on EURUSD options is shown in Figure 5. The crosses in the figure correspond to market-quoted mids. This surface allows arbitrage-free pricing of European options for strikes in the region $K/F\in [0.8,1.2]$.

## 5 Additional applications

In addition to pricing European options, there are a variety of applications for arbitrage-free option price surfaces that are smooth and allow significant extrapolation. In this section, the option price parameterizations are used for producing Dupire local volatility surfaces and for pricing variance swaps.

### 5.1 Dupire local volatility

Let us define a grid $j=1,2,\mathrm{\dots},M$ and $i=1,2,\mathrm{\dots},N$. $C({K}_{j},{\tau}_{i})$ will then correspond to a European call option with time to maturity ${\tau}_{i}>0$ and strike ${K}_{j}>0$. Using the finite-difference approximation of the Dupire equation used by Andreasen and Huge (2011), a local volatility function can be derived from an option price surface by

$\sigma ({K}_{j},{\tau}_{i})={\displaystyle \frac{1}{{K}_{j}}}\sqrt{{\displaystyle \frac{2{(\mathrm{\Delta}K)}^{2}}{\mathrm{\Delta}\tau}}{\displaystyle \frac{C({K}_{j},{\tau}_{i+1})-C({K}_{j},{\tau}_{i})}{C({K}_{j-1},{\tau}_{i})+C({K}_{j+1},{\tau}_{i})-2C({K}_{j},{\tau}_{i})}}}.$ | (5.1) |

According to earlier theoretical results, $\sigma ({K}_{j},{\tau}_{i})$ will be real, since neither butterfly or calendar spread arbitrage should exist in the option price surface. As mentioned in Section 4.3, the Dupire local volatility surfaces constructed from our proposed model are continuous in both dimensions if the parameter sets used in the construction of option prices are interpolated using a piecewise cubical interpolation (see Figure 6). In Figure 7, the parameters have instead been interpolated linearly, giving a local volatility surface discontinuous in the maturity dimension.

As a comparison, a EURUSD local volatility surface constructed using the assumption of a piecewise constant LVG volatility function can be seen in Figure 8. It can be seen that, while the local volatility surface under our model is regular and continuous, this is in sharp contrast to local volatilities emerging from models using the original assumption of the LVG volatility function by Carr and Nadtochiy (2017).

### 5.2 Valuation of variance swaps

In this section, another useful application of the proposed model is presented: valuation of variance swaps. After extrapolation, arbitrage-free explicit solutions for prices of European options enable fast valuation of this type of instrument. The variance swap is an over-the-counter traded derivative that exposes the buyer directly to market volatility. As concluded by Carr and Lee (2009), variance swaps are popular instruments in the FX market. The payoff of the variance swap is the difference of the realized variance during the lifetime of the contract, ${\sigma}_{\mathrm{R}}^{2}(t,T)$, and some set swap rate, ${\sigma}_{\mathrm{S}}^{2}(T)$. Generally, the contract uses a fair-value swap rate, where the swap rate is picked in such a way that the contract has no value at inception. Let $\mathrm{VS}(t,T)$, $t\in [0,T]$ denote the risk premium of a variance swap with maturity $T$ at time $t$. The risk premium can then be calculated as

$$\mathrm{VS}(t,T)=?[{\sigma}_{\mathrm{R}}^{2}(t,T)\mid {\mathcal{F}}_{t}]-{\sigma}_{\mathrm{S}}^{2}(T),$$ | (5.2) |

where $\mathrm{VS}(0,T)=0\u27f9?[{\sigma}_{\mathrm{R}}^{2}(0,T)\mid {\mathcal{F}}_{0}]={\sigma}_{\mathrm{S}}^{2}(T)$.

This calls for a technique to determine the fair swap rate. Carr and Lee (2009), among others, use a Taylor expansion to show that the fair swap rate can be found by solving the integral

${\sigma}_{\mathrm{S}}^{2}({\tau}_{i})={\displaystyle \frac{2}{{\tau}_{i}}}{\displaystyle {\int}_{0}^{\mathrm{\infty}}}{\displaystyle \frac{{\mathrm{\Psi}}_{i}(K)}{{K}^{2}}}\mathrm{d}K,$ | (5.3) |

where ${\mathrm{\Psi}}_{i}(K)$ is the price of European out-of-the-money options with strike $K$ and maturity ${\tau}_{i}$. The above integral may be solved analytically using the explicit expression for $\mathrm{\Psi}(K)$ from Proposition 2.6. However, we will not investigate this here.

Note that the function $\mathrm{\Psi}(K)$ goes to zero at the point $K=0$. However, it is easily shown that

$$\underset{K\to 0}{lim}\frac{\mathrm{\Psi}(K)}{{K}^{2}}=\mathrm{\infty}.$$ |

We will not analyze the nature of the divergence of $\mathrm{\Psi}(K)/{K}^{2}$ at the point $K=0$ thoroughly. Tests show that a computer implementation yields

$$\frac{\mathrm{\Psi}(K)}{{K}^{2}}\approx 0\mathit{\hspace{1em}}\text{for}K={M}_{\epsilon},$$ |

where ${M}_{\epsilon}\approx 2.22\times {10}^{-16}$. Hence, the divergence only becomes problematic at a single point, $K=0$. Therefore, this obstacle will be solved pragmatically by defining the function $\mathrm{\Omega}(K)$ as

${\mathrm{\Omega}}_{i}(K)=\{\begin{array}{cc}0\hfill & \text{if}K=0,\hfill \\ {\displaystyle \frac{{\mathrm{\Psi}}_{i}(K)}{{K}^{2}}}\hfill & \text{if}K0.\hfill \end{array}$ |

The fair swap rate of a variance swap at maturity ${\tau}_{i}$ can then be computed as

${\sigma}_{\mathrm{S}}^{2}({\tau}_{i})={\displaystyle \frac{2}{{\tau}_{i}}}{\displaystyle {\int}_{0}^{\mathrm{\infty}}}{\mathrm{\Omega}}_{i}(K)\mathrm{d}K.$ | (5.4) |

Figure 9 displays the fair swap rate for EURUSD variance swaps with maturities between one week and one year. Note the saddle in the swap rate between the one- and two-month expiries. This can be attributed to these two expiries being traded at very similar ATM volatilities in the example surface (see Figure 1).

## 6 Conclusion

In this paper, we proposed a fast and arbitrage-free way of constructing FX volatility surfaces. The main result of Section 2 is a closed-form solution of the local variance gamma version of the Dupire equation. This method guarantees the absence of butterfly and vertical spread arbitrage. The main difference between our approach and those in the literature is the assumption regarding the underlying LVG volatility function. We proposed a continuous piecewise linear function that is constant for very low and very high strikes.

In Section 3, we showed that the model calibrates accurately to Reuters quoted EURUSD and EURSEK volatility smiles. Further, the model calibrates at a remarkable speed; in an implementation using a Levenberg–Marquardt algorithm, the three-parameter model had an average calibration time of around one millisecond for both currency pairs in the calibration sample.

In Section 4, we proposed a technique for interpolating between market-quoted maturities. The main result here was that we provided sufficient conditions for the absence of calendar spread arbitrage by introducing the concept of increasing LVG variance. When monotone piecewise cubical interpolation was used, the constructed volatility surface was arbitrage-free and ${C}^{1}$ in maturity and ${C}^{2}$ in strike. Further, the model supported strike extrapolation at an arbitrary level.

In Section 5, we used the model to produce Dupire local volatility surfaces, which were positive and continuous over the entire surface. We also showed how the model can be used for quickly producing smooth fair swap rates for FX variance swaps.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper. The views expressed are those of the authors rather than of their respective institutions.

## Acknowledgements

We thank Rommael Luzac and Fredrik Berntsson for supporting the work. We also thank the anonymous referees for their detailed comments that helped us significantly improve the quality of the paper.

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