#
Probability Density of Lognormal Fractional SABR Model^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model Specification

#### 2.1. The Model

#### 2.2. Malliavin Calculus with Respect to Brownian Motion

**Lemma**

**1.**

#### 2.3. Bridge Representation for the Joint Density

**Theorem**

**1.**

**Remark**

**1.**

**Proof.**

**Corollary**

**1.**

## 3. Expansion around Deterministic Path

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 4. Small Time Approximation of Option Price and Implied Volatility

**Lemma**

**2.**

**Proof.**

**Remark**

**4.**

**Theorem**

**3.**

**Remark**

**5.**

## 5. A Heuristic Large Deviation Principle

**Theorem**

**4.**

**Proof.**

**Remark**

**6.**

**Proposition**

**1**

**.**Let $k=log\left(\right)open="("\; close=")">\frac{K}{{s}_{0}}$ be the log moneyness. The implied volatility ${\sigma}_{\mathrm{BS}}(k,t)$ in a small time to expiry has the asymptotics

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Technical Proofs

#### Appendix A.1. Error Analysis

- (i)
- for any $r\ge 0$, $\underset{(\xi ,y)\in {\mathbb{R}}^{2}}{sup}\left(\right)open="|"\; close="|">{\xi}^{r}\widehat{f}(\xi ,y)\infty$;
- (ii)
- for any $r\ge 0$ and $p>0$, ${\int \left|\xi \right|}^{r}\underset{y\in \mathbb{R}}{sup}{\left|\widehat{f}(\xi ,y)\right|}^{p}d\xi <\infty$.

- ${J}_{1}$: Choosing ${p}_{1}>0$ such that $\frac{q{q}_{1}}{2}>1$, by Hölder’s inequality, the Burkholder-Davis-Gundy inequality, Jensen’s inequality and a change of variables, we obtain Notice that$$\begin{array}{ccc}\hfill {J}_{1}& \le & {\left|\rho \right|}^{q}{y}_{0}^{q}\int {\left|\xi \right|}^{q}{\left(\right)}^{\mathbb{E}}\frac{1}{{p}_{1}}\hfill & {\left(\right)}^{\mathbb{E}}\end{array}d\xi $$$$\begin{array}{ccc}\hfill & & \underset{t\to {0}^{+}}{lim\; sup}\int {\left|\xi \right|}^{q}{\left(\right)}^{\mathbb{E}}\frac{1}{{p}_{1}}\hfill & {\left(\right)}^{{\int}_{0}^{1}}\frac{1}{{q}_{1}}\\ d\xi \end{array}$$
- ${J}_{2}$ and ${J}_{4}$: The asymptotic behavior of ${J}_{2}$ and ${J}_{4}$ is the same as that of ${t}^{q}{L}_{1}$, and hence, ${J}_{2},{J}_{4}=O\left({t}^{q}\right)$ as $t\to {0}^{+}$.
- ${J}_{3}$: By using the same technique to ${J}_{1}$, we have$${J}_{3}\le {\left|\rho \right|}^{q}{y}_{0}^{q}{t}^{\frac{q}{2}}\int {\left|\xi \right|}^{q}{\left(\right)}^{\mathbb{E}}{\left(\right)}^{\mathbb{E}}$$

#### Appendix A.2. Laplace Asymptotic Formula

**Lemma**

**A1**

**.**Let $\mathcal{C}$ be a closed and convex set in ${\mathbb{R}}^{2}$ with a nonempty and smooth boundary $\partial \mathcal{C}$. Suppose that $\theta (t,x):={\theta}_{0}\left(x\right)+{t}^{\alpha}{\theta}_{1}\left(x\right)+{t}^{2\alpha}{\theta}_{2}\left(x\right)$, with $0\le 2\alpha <1$, has continuous second-order partial derivatives in $x\in \mathcal{C}$, and, for every t sufficiently small, the function $\theta (t,x)$ is locally convex in $\mathcal{C}$ and attains its minimum uniquely at ${x}^{*}\left(t\right)\in \partial \mathcal{C}$. Moreover, there is ${\u03f5}_{0}>0$ such that for any $0<\u03f5<{\u03f5}_{0}$, there exist ${t}_{0}$ and $\delta >0$ for which

**Proof.**

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**Figure 1.**The contour plots. Parameters $\rho =-0.7$, $\nu =1$, ${y}_{0}=1$, $t=0.5$. $H=0.75$ on the

**right**; $H=0.25$, on the

**left**.

**Figure 2.**The plot on the

**left**shows the approximate implied volatility curves versus logmoneyness with time to expiry $t=1$ produced by (20) (in blue) the SABR Formula (3) (in red). Parameters are set as $\rho =-0.06867$, $\nu =0.5778$, ${a}_{0}=0.13927$. The plot on the

**right**shows the difference between the two curves.

**Figure 3.**The implied volatility curves for $t=0.01$ on the

**left**, $t=1$ on the

**right**. Parameters are set as $\rho =-0.06867$, $\nu =0.5778$, ${a}_{0}=0.13927$. $H=0.1$ in red, $H=0.3$ in orange, $H=\frac{1}{2}$ in green, $H=0.7$ in blue, $H=0.9$ in purple.

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**MDPI and ACS Style**

Akahori, J.; Song, X.; Wang, T.-H.
Probability Density of Lognormal Fractional SABR Model. *Risks* **2022**, *10*, 156.
https://doi.org/10.3390/risks10080156

**AMA Style**

Akahori J, Song X, Wang T-H.
Probability Density of Lognormal Fractional SABR Model. *Risks*. 2022; 10(8):156.
https://doi.org/10.3390/risks10080156

**Chicago/Turabian Style**

Akahori, Jiro, Xiaoming Song, and Tai-Ho Wang.
2022. "Probability Density of Lognormal Fractional SABR Model" *Risks* 10, no. 8: 156.
https://doi.org/10.3390/risks10080156