General Asymptotics of Wiener Functionals and Application to Implied Volatilities

Abstract

In the present paper, we give an asymptotic expansion of probability density for a component of general diffusion models. Our approach is based on infinite dimensional analysis on the Malliavin calculus and Kusuoka-Stroock’s asymptotic expansion theory for general Wiener functionals (Kusuoka and Stroock, J. Funct. Anal. 99:1–74, 1991 [12]). The initial term of the expansion is given by the geodesic distance and we calculate it by solving Hamilton’s equation. We apply our approach to obtain asymptotic expansion formulae for implied volatilities in general diffusion models, e.g. CEV and SABR model.

Appendices

Appendix 1

In this section, we investigate some properties of functions defined in Sect. 1. First we consider $\varphi _n, ~n \ge 0$ defined by (18).

Lemma 7

The functions $\varphi _n$ have the following properties.

(1) $\varphi _n(x) > 0, ~x \ge 0.$

(2) $\lim _{x \rightarrow \infty } x^{n+1} \varphi _n(x) = n!.$

(3) $\displaystyle \sup _x \frac{\varphi _n(x)}{\varphi _1(x)} < \infty , \qquad n \ge 1.$

Proof

(1) is easy to check. We prove (2). Putting $y = xz$

$$ \varphi _n(x) = \int _0^{\infty } \exp ( - \frac{y^2}{2x^2}-y ) (\frac{y}{x})^n \frac{dy}{x} = \frac{1}{x^{n+1}} \int _0^{\infty } y^n \exp (-y - \frac{y^2}{2x^2}) dy $$

Then we have

$$ \lim _{x \rightarrow \infty } x^{n+1} \varphi _n(x) = \int _0^{\infty } y^n e^{-y} dy = n!. $$

(3) is an easy consequence of (1) and (2).$\square$

The following is easy to check.

Lemma 8

The functions $\{ \varphi _n \}$ satisfy the following recurrence relations.

$$\begin{aligned}&\varphi _{n+1}(x) = - x \varphi _n(x) + n \varphi _{n-1}(x), \\&\varphi '_n(x) = - \varphi _{n+1}(x). \end{aligned}$$

Example 3

$\varphi _i ~( 0 \le i \le 3)$ are given as follows:

$$\begin{aligned} \varphi _0(x)= & {} \exp (\frac{x^2}{2}) \int _x^{\infty } \exp (-\frac{z^2}{2})\textit{dz}, \\ \varphi _1(x)= & {} - x \varphi _0(x) + 1, \\ \varphi _2(x)= & {} (x^2+1) \varphi _0(x) - x, \\ \varphi _3(x)= & {} -(x^3 + 3x) \varphi _0(x) + x^2 + 2. \end{aligned}$$

Next we consider the function $h\in C^{\infty }([0,1] \times \mathbf {R}_+)$ defined by (53).

Lemma 9

The n-times differentiation of $\log h(t,y)$ with respect to t is given as follows. We define $\theta$ in (52).

$$\begin{aligned} \bigl ( \frac{\partial }{\partial t}\bigr )^n \log h(t, y) = \frac{1}{t^n} \theta _n( h(t, y) ), \quad t \in [0, 1], ~y > 0, \end{aligned}$$

where $\theta _n \in C_b[0, \infty ], ~n \ge 1$ are given inductively as follows:

$$\begin{aligned} \theta _1(x)= & {} \varphi _1(x), \\ \theta _{n+1}(x)= & {} n \theta _n(x) + \theta _n'(x) \theta _1(x)x. \end{aligned}$$

Proof

In the case $n=1$, since $f(h(t, y)) = t f(y),$ we have

$$ \frac{\partial h}{\partial t }(t, y) = \frac{f(h(t,y))}{t f'(h(t,y))}. $$

Since

$$ f'(x) = - (\frac{1}{x} + x + \frac{\varphi _2(x)}{\varphi _1(x)}) f(x) < 0, ~x > 0, $$

we have

$$ \theta _1(x) = \frac{f(x)}{xf'(x)} = \Bigl ( 1 + x^2 + x \frac{\varphi _2(x)}{\varphi _1(x)} \Bigr )^{-1} = \varphi _1(x). $$

It is easy to check that $\theta _1 \in C_b([0, \infty ])$ and $x \theta _1(x) \in C_b([0, \infty ])$. We have

$$ \frac{\partial }{\partial t } \log h(t,y) = \frac{1}{t} \theta _1(h(t,y)). $$

Since

$$\begin{aligned} \frac{\partial }{\partial t} \bigl ( \frac{1}{t^n} \theta _n(h(t,y)) \bigr ) = \frac{1}{t^{n+1}} \bigl ( -n \theta _n(h(t,y)) + \theta _n'(h(t,y)) \theta _1(h(t,y)) h(t,y) \bigr ), \end{aligned}$$

it is easy to prove our lemma.$\square$

Appendix 2

In this section, we summarize the main theorem in Kusuoka and Osajima [13]. See [13] for the definitions.

Let f,  g $\in \fancyscript{G}^{\infty }(\fancyscript{A}; {\mathbf R})$ and F $\in \fancyscript{G}^{\infty }(\fancyscript{A}; {\mathbf R}^N)$ be completely P-regular functions and Y be a compact subset in $\mathbf {R}^N.$ We assume the following.

(A1) There is an $\alpha >0$ such that

$$ \sup _{s \in (0,1]} s \log (\int _{\varTheta } \exp (\frac{(1+\alpha )f(s,\theta )}{s} )\mu _s(d\theta )) < \infty . $$

We define $e : \mathbf {R}^N \rightarrow [-\infty ,\infty ]$ by

$$ e(x) \equiv \inf \{\frac{\Vert h\Vert ^2}{2} - f(0,h) : F(0,h) = x \} , \qquad x \in {\mathbf R}^N . $$

We also assume the following.

(A2) For each $y \in Y$,

$$ M(y) \equiv \{h \in H; F(0, h) = y\} \not = \emptyset $$

and that

$$ e(y) = \frac{\Vert h(y) \Vert ^2}{2} - f(0, h(y)) $$

for precisely one $h(y) \in M(y).$

We assume moreover the following.

(A3) $T(y) \equiv DF(0, h(y))$ has rank N for every $y \in Y.$

Let $\pi (y)= T(y)^*(T(y)T(y)^*)^{-1}T(y),$ $y \in Y.$ $\pi (y)$ is an orthogonal projection in H. Let $\pi (y)^{\perp } = I_H - \pi (y).$ Then $\pi (y)^{\perp }$ is also an orthogonal projection in H onto $ker \; T(y).$ Let $V(y):H\times H \rightarrow \mathbf R$ be a bilinear form given by

$$\begin{aligned} \begin{array}{c} V(y)(h,h') \\ = D^2f(0,h(y))(\pi (y)^{\perp }h,\pi (y)^{\perp }h') \\ +\,(h(y) - Df(0,h(y)),T(y)^*(T(y)T(y)^*)^{-1}D^2F(0,h(y)) (\pi (y)^{\perp }h,\pi (y)^{\perp }h'))_H. \end{array} \end{aligned}$$

We assume the following furthermore.

(A4) For all $y \in Y$ and $h \in H \setminus \{ 0\}$

$$ V(y)(h,h) < \Vert h \Vert ^2 . $$

Finally we define

$$\begin{aligned} A(s,\theta )&= DF(s, \theta )DF(s, \theta )^* \\&= ((DF_i(s,\theta ), DF_j(s, \theta ))_{H})_{1 \leqq i,j \leqq N} \end{aligned}$$

and assume the following.

(A5) For any $p\in [1,\infty )$

$$ \varlimsup _{s\downarrow 0} s \log (\int _{\varTheta } |\det A(s,\theta ) |^{-p} \mu _s(d\theta )) \leqq 0. $$

Then Kusuoka-Stroock [12] proved the following.

Theorem 6

For each $s \in (0,1]$, a signed measure $P_s(\cdot )$ on $\mathbf {R}^N$ given by

$$\begin{aligned} P_s(\varGamma ) = \int _{F(s,\theta ) \in \varGamma } g(s,\theta )\exp \left( \frac{f(s,\theta )}{s}\right) \mu _s(d\theta ), ~\varGamma \in \fancyscript{B}({\mathbf {R}^N}), \end{aligned}$$

admits a smooth density $p_s(\cdot )$ with respect to Lebesgue’s measure. Moreover, there exist sequence $\{a_n\}_{n=0}^{\infty } \subseteq C(Y;\mathbf {R})$ and $\{K_n\}_{n=0}^{\infty } \subseteq (0,\infty )$ with the property that, for every $n \in \mathbf {N}$,

$$\begin{aligned} \Bigl |(2 \pi s)^{N/2} e^{e(y) / s}p_s(y;0) - \sum _{m=0}^n s^{m/2}a_m(y)\Bigr | \leqq K_n s^{(n+1)/2}, ~(s,y) \in (0,1] \times Y. \end{aligned}$$

The main theorem in Kusuoka-Osajima [13] is the following.

Theorem 7

e is smooth in the neighborhood of Y and

$$\begin{aligned} a_0(y) = (\det \nabla ^2 e (y) )^{1/2} {\det }_2 (I_H - B(y)) ^{-1/2}&\exp \Bigl ( \sum _{i=1}^N \frac{\partial e}{\partial y_i} (y) \fancyscript{A}F^i(0,h(y))\\&\qquad \quad + \fancyscript{A}f(0,h(y)) \Bigr ) \nonumber \end{aligned}$$

for $y\in Y,$ where

$$\begin{aligned}&B(y) \equiv \sum _{i=1}^N \frac{\partial e}{\partial y_i} (y) D^2 F^i(0,h(y)) + D^2 f(0,h(y)), \qquad y \in Y. \\ \end{aligned}$$

Here we identify a continuous symmetric bilinear form $B:H\times H \rightarrow {\mathbf R}$ with a bounded symmetric linear operator ${\tilde{B}}:H\rightarrow H$ given by

$$ ({\tilde{B}}h,k)_H = B(h,k), \qquad h,k \in H, $$

and $det_2$ is a Carleman-Fredholm determinant (c.f. Dunford and Schwartz [5] pp.1106).

Appendix 3

In this section, we discuss about the implied volatilities for the case $K < x_0^1$. We define the forward value of a put option of strike rate K and maturity T by

$$ P_{\varepsilon }(T, K) = E[(K - X^1_{\varepsilon }(T))^{+}] $$

Since we have put-call parity, the implied volatility of the put option is the same as the implied volatility of a call option with strike rate K and maturity T. Since

$$ P_{\varepsilon }(T, K) = E[(-X^1_{\varepsilon }(T) - (-K))_{+}] = E[(-(X^1_{\varepsilon }(T) - x_0^1) - (-(K - x_0^1) ))_{+}] $$

It is enough to discuss in the case $x_0^1 = 0.$

Let $x = (x^1, \ldots , x^n) \in \mathbf {R}^n$. We denote $\bar{x} = (-x^1, x^2, \ldots , x^n).$ We define $\bar{X}_{\varepsilon }(t) = {{\bar{X}}_{\varepsilon }(t)}$. Then we have

$$ d \bar{X}^i_{\varepsilon }(t) = \sum _{k=1}^d \varepsilon \bar{V}_k^i(t, \tilde{X}_{\varepsilon }(t)) \textit{dW}^k(t) + \bar{V}_0^i(t, \tilde{X}_{\varepsilon }(t))\textit{dt}, \quad 1 \le i \le N, $$

where

$$ \bar{V}_k^j (t, x) = {\left\{ \begin{array}{ll} -V_k^1(t, \bar{x}) &{} {(1 \le k \le d)}\\ V_k^j(t, \bar{x}) &{} {(1 \le k \le d, ~j \ne 1)}. \end{array}\right. } $$

Since the associated Riemaniann metric $\bar{g}^{\textit{ij}}(t,x) = \sum _{k=1}^d \bar{V}_k^i(t,x) \bar{V}_k^j(t,x)$ is given by

$$ \bar{g}^{11}(t,x) = g^{11}(t,x), \quad \bar{g}^{1i}(t,x) = -g^{1i}(t,x)~(i \ne 1), \quad \bar{g}^{\textit{ij}}(t,x) = g^{\textit{ij}}(t,x) ~(i,j \ne 1), $$

we have

$$ \bar{b}_1 = b_1, \quad \bar{b}_2 = -b_2, \quad \bar{b}_3 = b_3, \quad \bar{L} = L. $$

Therefore Theorems 1 and 3 still hold for $K < x_0^1.$