Journal of Computational Finance
ISSN:
14601559 (print)
17552850 (online)
Editorinchief: Christoph Reisinger
Need to know
 We obtain a formula to compute the vega index (sensitivity with respect to changes in volatility) for options depending on extrema.
 A onedimensional perturbed diffusion process is considered in this article.
 As a numerical application, we compute the vega index for options under the BlackScholes model perturbed with a CEV (constant elasticity of variance model) type perturbation.
 The values of the vega index obtained with the perturbed diffusion process and with nonperturbed BlackScholes model significantly differ.
Abstract
In this paper, we give a decomposition formula to calculate the vega index (sensitivity with respect to changes in volatility) for options with prices that depend on the extrema (maximum or minimum) and terminal value of the underlying stock price; this is assumed to follow a onedimensional perturbed diffusion process. As a numerical application, we compute the vega index for lookback, European and upin call options under the Black–Scholes model perturbed with a constant elasticity of variance modeltype perturbation. We compare these values with the standard nonperturbed Black–Scholes model, which, interestingly, turn out to be very different.
Introduction
1 Introduction
The Black–Scholes model is widely used by practitioners due to its simplicity and the existence of some explicit probability density functions which concern it. This model assumes a constant volatility; in option market data, however, we observe that the volatility is not constant. This phenomenon is often called “volatility smile”, after the shape of observed implied volatilities (see Dupire 1994; Gatheral 2006). For this reason, it is natural to consider a general model that may perform better than the Black–Scholes model. However, in general models, one knows neither the associated explicit density functions nor explicit formulas for option prices. Therefore, the risks involved in options, called Greeks, can only be computed through numerical approximations.
In this paper, we consider the sensitivity of the model to changes in the volatility for options depending on the extrema (maximum or minimum). This sensitivity is called the vega index, and calculating this index is the focus of our discussion. In our general model, the volatility is not constant; this makes the discussion complicated mathematically. We introduce a perturbation parameter to consider the directional derivatives for the diffusion coefficients to calculate the vega index. This problem in particular has been discussed by a number of authors. Fournié et al (1999), for instance, obtain a formula to calculate the vega index for options with payoffs that depend on the prices of the underlying at fixed times through Malliavin calculus. Other Greeks, such as the delta and gamma for options depending on the extrema, are discussed in Gobet and KohatsuHiga (2003). In both of these papers, integration by parts (IBP) formulas play an important role. In Hayashi and KohatsuHiga (2013), formulas of this type are used to prove the smoothness of the density function concerning the supremum of a multidimensional diffusion process satisfying some commutativity conditions on the diffusion coefficients. The authors obtain the formulas by means of the Garsia–Rodemich–Rumsey lemma (see Nualart 2006, Lemma A.3.1). In Bermin et al (2003), a formula to compute the vega index is obtained for options with payoffs that depend smoothly on the underlying (eg, an Asiantype option) by using Malliavin calculus.
However, the vega index for options depending on the extrema has not been considered yet, since the extrema of a diffusion process is not sufficiently smooth and is therefore difficult to treat from a mathematical point of view. In mathematical finance, various creditlinked and barriertype products have this kind of feature.
The main goal of this paper is to show a formula to calculate the vega index for options depending on the extrema of the underlying and obtain some financial conclusions about the properties of the vega index.
Technically, we consider a onedimensional perturbed stochastic differential equation (SDE) with timeindependent coefficients as the dynamics of an asset price under the pricing measure $P$. To deal with the extrema of such a diffusion process, we use the Lamperti transformation (see, for example, Karatzas and Shreve 1991, Exercise 5.2.20). That is, using the Girsanov theorem, we transform the SDE to a Stratonovichtype SDE without drift coefficients, which can then be expressed as a monotone transformation of a Wiener process. This method is different from the one considered in Gobet and KohatsuHiga (2003), where the Garsia–Rodemich–Rumsey lemma plays an important role.
Although the techniques used in Gobet and KohatsuHiga (2003) are quite interesting, the formulas obtained there have high computational complexity. However, the formula obtained in this paper is much simpler. By working under the new measure, we can express the extrema of our diffusion process in a simple fashion and calculate the directional derivatives. In addition, we use the duality formula of Malliavin calculus as it appears in Nualart (2006, p. 37) to obtain a formula that gives a better expression for the vega index to use numerical methods such as Monte Carlo simulation.
The formula of the vega index obtained in this paper has three components: the extrema and maturity features of options, and a byproduct of the Girsanov transformation. Our goal is to reveal some properties of the structure of these three components for realistic options through numerical studies.
In the numerical application, we consider the Black–Scholes model perturbed with a constant elasticity of variance (CEV) modeltype perturbation. Through the numerical studies, one can see that the decomposition of the vega index for options has some interesting properties. For example, the values of the vega index obtained with the perturbed diffusion process and with the nonperturbed Black–Scholes model significantly differ (see Table 1 and Figure 4). When we consider an upin call option, our Monte Carlo analysis shows that for the option with a lower barrier, the vega index is mostly conveyed by the maturity feature of the payoff, while for the option with a higher barrier, the extrema feature controls most of the vega index. We can see the existence of a barrier that determines which component in the decomposition is more important (see Figure 1). Moreover, we observe that for options with short maturity, we have to pay more attention to the change of the value of the vega index with respect to the maturity (see Figures 2 and 3).
This paper is organized as follows. In Section 2, we provide the mathematical result of the decomposition of the vega index. In Section 3, we carry out Monte Carlo simulations and obtain some results on the structure of the vega index, as mentioned in the previous paragraph. Section 3 concludes. In the online appendixes, we give some lemmas and proofs of our results.
Throughout the paper, we use ${C}_{b}^{k}(A,B)$ to denote the space of $B$valued $k$ times continuously differentiable functions defined on $A$ with bounded derivatives. For a differentiable function $F$ from ${\mathbb{R}}^{m}$ to $\mathbb{R}$, where $m\in \mathbb{N}$, we define
$${\partial}_{i}F(x):=\frac{\partial F}{\partial {x}_{i}}(x)$$ 
for $x\in {\mathbb{R}}^{m}$ and $1\le i\le m$. The letters $C$ and ${C}_{i}$ ,$i\in \mathbb{N}$, denote positive constants, which may depend on $f$, $p$, $x$ and $T$ (which will appear in this paper), and the values of $C$ and ${C}_{i}$ may change from line to line. We define ${\mathbb{R}}_{+}:=(0,\mathrm{\infty})$ and ${E}^{P}$ as the expectation under the probability measure $P$.
2 Main result: vega index for options depending on the extrema
Let $(\mathrm{\Omega},\mathcal{F},P)$ be a complete probability space that supports a onedimensional Wiener process $\{{W}_{t},t\in [0,T]\}$. For $\sigma ,\widehat{\sigma},b:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ and $x>0$, we consider the following perturbed onedimensional diffusion process:
$$\{\begin{array}{cc}\hfill \mathrm{d}{S}_{t}^{\epsilon}& =b({S}_{t}^{\epsilon})\mathrm{d}t+{\sigma}^{\epsilon}({S}_{t}^{\epsilon})\mathrm{d}{W}_{t},\hfill \\ \hfill {S}_{0}^{\epsilon}& =x,\hfill \end{array}$$  (2.1) 
where ${\sigma}^{\epsilon}$ is of the form ${\sigma}^{\epsilon}(z)=\sigma (z)+\epsilon \widehat{\sigma}(z)$, $\epsilon \in [0,1]$. We note that $\epsilon $ is the perturbation parameter and $\widehat{\sigma}$ is the direction of perturbation. $P$ denotes the equivalent martingale measure, where we assume that it exists and does not depend on $\epsilon $. For a function $f:{\mathbb{R}}^{2}\to \mathbb{R}$, we consider the quantity ${\mathrm{\Pi}}^{\epsilon}:={E}^{P}[f({\mathrm{max}}_{0\le t\le T}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon})]$. We assume the following hypotheses.
 (H1)

$\sigma ,\widehat{\sigma},b\in {C}_{b}^{2}({\mathbb{R}}_{+},{\mathbb{R}}_{+})$.
 (H2)

There exists ${\sigma}_{0}>0$ such that ${\sigma}^{\epsilon}(y)\ge {\sigma}_{0}y$ for all $y\in {\mathbb{R}}_{+}$ and $\epsilon \in [0,1]$.
 (H3)

There exist ${r}_{0}\in (0,1]$ and ${\sigma}_{1}>0$ such that ${\sigma}^{\epsilon}(y)\le {\sigma}_{1}y$ for all $y\in {\mathbb{R}}_{+}$ satisfying $$.
 (H4)

The function $b/{\sigma}^{\epsilon}$ is uniformly bounded.
 (H5)

$f\in {C}_{b}^{1}({\mathbb{R}}_{+}^{2},{\mathbb{R}}_{+})$.
Note that by (H1), for all $\epsilon \in [0,1]$, (2.1) has a unique strong solution, which we denote by ${S}^{\epsilon}=\{{S}_{t}^{\epsilon},t\in [0,T]\}$. In finance, ${\mathrm{\Pi}}^{\epsilon}$ defines a perturbed option price with payoff function $f$. We consider the quantity
$${\frac{\partial {\mathrm{\Pi}}^{\epsilon}}{\partial \epsilon}}_{\epsilon =0}:=\underset{\epsilon \searrow 0}{lim}\frac{{\mathrm{\Pi}}^{\epsilon}{\mathrm{\Pi}}^{0}}{\epsilon}$$ 
and call this the vega index of this option.
Our main result is the following theorem. It gives the decomposition formula for the vega index.
Theorem 2.1.
Assume the above hypotheses (H1)–(H5). Then, the following expression for the vega index is valid:
${{\displaystyle \frac{\partial {\mathrm{\Pi}}^{\epsilon}}{\partial \epsilon}}}_{\epsilon =0}$  $={E}^{P}\left[{\partial}_{1}f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T})\sigma \left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right){\displaystyle {\int}_{x}^{{\mathrm{max}}_{0\le t\le T}{S}_{t}}}{\displaystyle \frac{\widehat{\sigma}}{{\sigma}^{2}}}(y)\mathrm{d}y\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}+{E}^{P}\left[{\partial}_{2}f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T})\sigma ({S}_{T}){\displaystyle {\int}_{x}^{{S}_{T}}}{\displaystyle \frac{\widehat{\sigma}}{{\sigma}^{2}}}(y)\mathrm{d}y\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}+{E}^{P}[{\displaystyle {\int}_{0}^{T}}({\partial}_{1}f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T}){Y}_{\eta}{I}_{[0,\eta ]}(t)+{\partial}_{2}f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T}){Y}_{T})$  
$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\times \{{\displaystyle \frac{({b}^{\prime}\sigma b{\sigma}^{\prime}({\sigma}^{2}{\sigma}^{\u2033}/2))({S}_{t})}{{Y}_{t}}}\left({\displaystyle {\int}_{x}^{{S}_{t}}}{\displaystyle \frac{\widehat{\sigma}}{{\sigma}^{2}}}(y)\mathrm{d}y\right){\displaystyle \frac{((b\widehat{\sigma}/\sigma )+(\sigma {\widehat{\sigma}}^{\prime}/2))({S}_{t})}{{Y}_{t}}}\}\mathrm{d}t],$  (2.2) 
where $$
Before giving the proof of the theorem, let us state some remarks and some preparatory lemmas.
Remark 2.2.
 (A)
 (B)
For an irregular payoff function $f$, we are required to make limit arguments that depend on the situation in order to justify Theorem 2.1. Therefore, we do not discuss it in general; however, we do give an example of the application of Theorem 2.1 for an irregular $f$ in Section 3, and the limit argument is given in Appendix B (available online).
 (C)
The first term of (2.2) comes from the differentiation with respect to the maximum of the asset price; the second term is due to the asset price at maturity, and the third term comes from the change of measure. We call these three terms “extrema sensitivity”, “terminal sensitivity” and “drift sensitivity”, respectively.
 (D)
When we consider the change of measure
$$\frac{\mathrm{d}Q(\epsilon )}{\mathrm{d}P}:=\mathrm{exp}\left\{{\int}_{0}^{T}\left(\frac{\sigma ^{\epsilon}{}^{\prime}}{2}\frac{b}{{\sigma}^{\epsilon}}\right)({S}_{t}^{\epsilon})\mathrm{d}{W}_{t}\frac{1}{2}{\int}_{0}^{T}{\left(\frac{\sigma ^{\epsilon}{}^{\prime}}{2}\frac{b}{{\sigma}^{\epsilon}}\right)}^{2}({S}_{t}^{\epsilon})\mathrm{d}t\right\},$$ where we have defined $\sigma ^{\epsilon}{}^{\prime}:={({\sigma}^{\epsilon})}^{\prime}$, then, under $Q(\epsilon )$,
$${\widehat{W}}_{t}^{\epsilon}:={W}_{t}{\int}_{0}^{t}\left(\frac{\sigma ^{\epsilon}{}^{\prime}}{2}\frac{b}{{\sigma}^{\epsilon}}\right)({S}_{u}^{\epsilon})\mathrm{d}u,t\in [0,T],$$ is a onedimensional Wiener process. Note that due to the boundedness of $b/{\sigma}^{\epsilon}$ and $\sigma ^{\epsilon}{}^{\prime}$, Novikov’s condition is clearly satisfied. Then, under $Q(\epsilon )$, ${S}^{\epsilon}$ can be written as
$$\{\begin{array}{cc}\hfill \mathrm{d}{S}_{t}^{\epsilon}& =\frac{1}{2}{\sigma}^{\epsilon}\sigma ^{\epsilon}{}^{\prime}({S}_{t}^{\epsilon})\mathrm{d}t+{\sigma}^{\epsilon}({S}_{t}^{\epsilon})\mathrm{d}{\widehat{W}}_{t}^{\epsilon}={\sigma}^{\epsilon}({S}_{t}^{\epsilon})\circ \mathrm{d}{\widehat{W}}_{t}^{\epsilon},\hfill \\ \hfill {S}_{0}^{\epsilon}& =x,\hfill \end{array}$$ where $\circ \mathrm{d}{\widehat{W}}_{t}^{\epsilon}$ denotes the Stratonovich integral. Finally, we can write ${\mathrm{\Pi}}^{\epsilon}$ as follows:
$${\mathrm{\Pi}}^{\epsilon}={E}^{P}\left[f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon})\right]={E}^{Q(\epsilon )}\left[f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon})\frac{\mathrm{d}P}{\mathrm{d}Q(\epsilon )}\right].$$  (E)
Under $Q(\epsilon )$, ${S}^{\epsilon}$ is driven by ${\widehat{W}}^{\epsilon}$, and the distribution of ${\widehat{W}}^{\epsilon}$ does not depend on $\epsilon $. Indeed, for any element of the Borel $\sigma $field defined on the space of continuous functions, denoted by $A$, one has $Q(\epsilon )({\widehat{W}}_{\cdot}^{\epsilon}\in A)=P({W}_{\cdot}\in A)$.
Let $(\stackrel{~}{\mathrm{\Omega}},\stackrel{~}{\mathcal{F}},\stackrel{~}{Q})$ be another complete probability space, and let $\stackrel{~}{W}$ be a onedimensional Wiener process under $\stackrel{~}{Q}$. Then, ${\mathrm{\Pi}}^{\epsilon}$ can be written as
$${\mathrm{\Pi}}^{\epsilon}={E}^{\stackrel{~}{Q}}\left[f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon})\mathrm{exp}({X}_{T}^{\epsilon})\right],$$ where ${S}^{\epsilon}$ satisfies
$$\{\begin{array}{cc}\hfill \mathrm{d}{S}_{t}^{\epsilon}& ={\sigma}^{\epsilon}({S}_{t}^{\epsilon})\circ \mathrm{d}{\stackrel{~}{W}}_{t},\hfill \\ \hfill {S}_{0}^{\epsilon}& =x,\hfill \end{array}$$ (2.3) and
$${X}_{T}^{\epsilon}:={\int}_{0}^{T}\left(\frac{\sigma ^{\epsilon}{}^{\prime}}{2}\frac{b}{{\sigma}^{\epsilon}}\right)({S}_{t}^{\epsilon})\mathrm{d}{\stackrel{~}{W}}_{t}\frac{1}{2}{\int}_{0}^{T}{\left(\frac{\sigma ^{\epsilon}{}^{\prime}}{2}\frac{b}{{\sigma}^{\epsilon}}\right)}^{2}({S}_{t}^{\epsilon})\mathrm{d}t.$$ (2.4) We use the SDE of the form (2.3) to write down ${S}^{\epsilon}$ with only $\stackrel{~}{W}$ so that we can express ${\mathrm{max}}_{0\le t\le T}{S}_{t}^{\epsilon}$ in terms of ${\mathrm{max}}_{0\le t\le T}{\stackrel{~}{W}}_{t}$. From now on, we use the notation ${X}_{T}:={X}_{T}^{0}$.
We next introduce a function ${F}_{\epsilon}$, which is used to express the solution to (2.3) in an explicit form.
Definition 2.3 (The Lamperti transformation).
For $\epsilon \in [0,1]$, define ${F}_{\epsilon}:{\mathbb{R}}_{+}\to \mathbb{R}$ as
$${F}_{\epsilon}(z):={\int}_{1}^{z}\frac{1}{{\sigma}^{\epsilon}(y)}\mathrm{d}y.$$ 
Note that the inverse function ${F}_{\epsilon}^{1}$ exists because ${F}_{\epsilon}$ is a continuous monotone increasing function due to (H2). Further, it is clear that ${F}_{\epsilon}$ and ${F}_{\epsilon}^{1}$ are differentiable with respect to $z$. We have
$\frac{\partial {F}_{\epsilon}}{\partial z}}(z)$  $={\displaystyle \frac{1}{{\sigma}^{\epsilon}(z)}},$  
$\frac{\partial {F}_{\epsilon}^{1}}{\partial z}}(z)$  $={\displaystyle \frac{1}{(\partial {F}_{\epsilon}/\partial z)({F}_{\epsilon}^{1}(z))}}={\sigma}^{\epsilon}({F}_{\epsilon}^{1}(z)).$  (2.5) 
In this setup, one has the following result.
Theorem 2.4.
Under (H1)–(H2), there exists a unique strong solution to (2.3). Further, under $$
$${S}_{t}^{\epsilon}={F}_{\epsilon}^{1}({F}_{\epsilon}(x)+{\stackrel{~}{W}}_{t}).$$  (2.6) 
Therefore, one has
$$\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon}={F}_{\epsilon}^{1}\left({F}_{\epsilon}(x)+\underset{0\le t\le T}{\mathrm{max}}{\stackrel{~}{W}}_{t}\right).$$ 
Proof.
By (2.5) and applying Itô’s formula for the Stratonovich integral to ${F}_{\epsilon}^{1}({F}_{\epsilon}(x)+z)$, it is easy to see that (2.6) is a solution to (2.3). However, if there exists a solution to (2.3), then, again, by applying Itô’s formula for the Stratonovich integral to ${F}_{\epsilon}(z)$, the solution can be expressed by (2.6). Thus, one obtains the uniqueness of the solution.
The equality ${\mathrm{max}}_{0\le t\le T}{S}_{t}^{\epsilon}={F}_{\epsilon}^{1}({F}_{\epsilon}(x)+{\mathrm{max}}_{0\le t\le T}{\stackrel{~}{W}}_{t})$ follows from the monotonicity of ${F}_{\epsilon}^{1}(\cdot )$. ∎
Remark 2.5.
Although the representation of ${F}_{\epsilon}^{1}({F}_{\epsilon}(x)+{\stackrel{~}{W}}_{t})$ is clearly continuous in $(t,\epsilon )$, this does not imply the continuity of the solution to (2.3) in $\epsilon $, since the exceptional sets in which Itô’s formula does not hold may depend on $\epsilon $. To overcome this problem, we modify the solution to (2.3) to be continuous in $(t,\epsilon )$. This procedure will be performed in Appendix A.2 (available online).
The above representation is the key formula allowing us to obtain Theorem 2.1. Our next step is to state some results on the regularity of ${S}_{t}^{\epsilon}$ and ${X}_{T}^{\epsilon}$ with respect to $\epsilon $ and the exchange between ${E}^{\stackrel{~}{Q}}[\cdot ]$ and ${(\partial /\partial \epsilon )(\cdot )}_{\epsilon =0}$. The proof of the following four lemmas can be found in Appendix A (available online).
Lemma 2.6.
Let (H1)–(H2) be satisfied, and let $$
$${\frac{\partial}{\partial \epsilon}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon}\right)}_{\epsilon =0}=\sigma \left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right){\int}_{x}^{{\mathrm{max}}_{0\le t\le T}{S}_{t}}\frac{\widehat{\sigma}}{{\sigma}^{2}}(y)\mathrm{d}y,$$ 
almost surely.
Lemma 2.7.
Let (H1)–(H2) be satisfied, and let $$
$${Z}_{t}:={\frac{\partial {S}_{t}^{\epsilon}}{\partial \epsilon}}_{\epsilon =0}=\sigma ({S}_{t}){\int}_{x}^{{S}_{t}}\frac{\widehat{\sigma}}{{\sigma}^{2}}(y)\mathrm{d}y,$$  (2.7) 
for all $$
Lemma 2.8.
Let (H1)–(H3) be satisfied. Then, $$
${{\displaystyle \frac{\partial {X}_{T}^{\epsilon}}{\partial \epsilon}}}_{\epsilon =0}$  $={\displaystyle \frac{1}{2}}{\displaystyle {\int}_{0}^{T}}({\sigma}^{\u2033}({S}_{t}){Z}_{t}+{\widehat{\sigma}}^{\prime}({S}_{t}))\mathrm{d}{\stackrel{~}{W}}_{t}$  
$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{0}^{T}}{\displaystyle \frac{({b}^{\prime}\sigma )({S}_{t}){Z}_{t}b({S}_{t})({\sigma}^{\prime}({S}_{t}){Z}_{t}+\widehat{\sigma}({S}_{t}))}{{\sigma}^{2}({S}_{t})}}\mathrm{d}{\stackrel{~}{W}}_{t}$  
$\mathrm{\hspace{1em}\hspace{1em}}{\displaystyle \frac{1}{2}}{\displaystyle {\int}_{0}^{T}}\left({\displaystyle \frac{{\sigma}^{\prime}}{2}}{\displaystyle \frac{b}{\sigma}}\right)({S}_{t})({\sigma}^{\u2033}({S}_{t}){Z}_{t}+{\widehat{\sigma}}^{\prime}({S}_{t}))\mathrm{d}t$  
$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{0}^{T}}\left({\displaystyle \frac{{\sigma}^{\prime}}{2}}{\displaystyle \frac{b}{\sigma}}\right)({S}_{t}){\displaystyle \frac{({b}^{\prime}\sigma )({S}_{t}){Z}_{t}b({S}_{t})({\sigma}^{\prime}({S}_{t}){Z}_{t}+\widehat{\sigma}({S}_{t}))}{{\sigma}^{2}({S}_{t})}}\mathrm{d}t,$ 
almost surely.
From the above three lemmas, one can see the correspondence of the derivatives to (2.2). In fact, these three lemmas correspond to the derivatives of the maximum, the underlying at maturity and the change of measure with respect to $\epsilon $, respectively.
In addition to the above lemmas, we need the following lemma about the exchange between ${E}^{\stackrel{~}{Q}}[\cdot ]$ and ${(\partial /\partial \epsilon )(\cdot )}_{\epsilon =0}$.
Lemma 2.9.
Let (H1)–(H5) be satisfied. Then, we have the following equation:
${{\displaystyle \frac{\partial}{\partial \epsilon}}\left({E}^{\stackrel{~}{Q}}\left[f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon})\mathrm{exp}({X}_{T}^{\epsilon})\right]\right)}_{\epsilon =0}$  
$\mathrm{\hspace{1em}\hspace{1em}}={E}^{\stackrel{~}{Q}}\left[{{\displaystyle \frac{\partial}{\partial \epsilon}}\left(f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon})\mathrm{exp}({X}_{T}^{\epsilon})\right)}_{\epsilon =0}\right].$ 
Now, let us prove the main theorem.
Proof of Theorem 2.1.
As mentioned in Remark 2.2(E), we have
$${\mathrm{\Pi}}^{\epsilon}={E}^{\stackrel{~}{Q}}\left[f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon})\mathrm{exp}({X}_{T}^{\epsilon})\right],$$ 
where ${S}^{\epsilon}$ satisfies (2.3) and ${X}_{T}^{\epsilon}$ is given by (2.4). Due to Lemma 2.9, ${(\partial {\mathrm{\Pi}}^{\epsilon}/\partial \epsilon )}_{\epsilon =0}$ can be decomposed as
${{\displaystyle \frac{\partial {\mathrm{\Pi}}^{\epsilon}}{\partial \epsilon}}}_{\epsilon =0}$  $={E}^{\stackrel{~}{Q}}\left[{{\partial}_{1}f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon}){\displaystyle \frac{\partial}{\partial \epsilon}}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon}\right)\mathrm{exp}({X}_{T}^{\epsilon})}_{\epsilon =0}\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}+{E}^{\stackrel{~}{Q}}\left[{{\partial}_{2}f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon}){\displaystyle \frac{\partial {S}_{T}^{\epsilon}}{\partial \epsilon}}\mathrm{exp}({X}_{T}^{\epsilon})}_{\epsilon =0}\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}+{E}^{\stackrel{~}{Q}}\left[{f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon}){\displaystyle \frac{\partial}{\partial \epsilon}}(\mathrm{exp}({X}_{T}^{\epsilon}))}_{\epsilon =0}\right]$  
$=:(A)+(B)+(C).$ 
 ($A$)
 ($B$)
 ($C$)

By Lemma 2.7, we can express $Z$ as follows:
$${Z}_{t}=(\sigma G)({S}_{t}),$$  (2.8) 
where $G$ is defined as
$$G(z):={\int}_{x}^{z}\frac{\widehat{\sigma}}{{\sigma}^{2}}(y)\mathrm{d}y.$$ 
Due to Lemma 2.8, we get
$(C)$  $={E}^{\stackrel{~}{Q}}[f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T})\mathrm{exp}({X}_{T})\{{\displaystyle {\int}_{0}^{T}}({\displaystyle \frac{\sigma {\sigma}^{\u2033}G+{\widehat{\sigma}}^{\prime}}{2}}{\displaystyle \frac{{\sigma}^{2}{b}^{\prime}Gb(\sigma {\sigma}^{\prime}G+\widehat{\sigma})}{{\sigma}^{2}}})({S}_{t})\mathrm{d}{\stackrel{~}{W}}_{t}$  
$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{\displaystyle {\int}_{0}^{T}}({\displaystyle \frac{{\sigma}^{\prime}}{2}}{\displaystyle \frac{b}{\sigma}})({\displaystyle \frac{\sigma {\sigma}^{\u2033}G+{\widehat{\sigma}}^{\prime}}{2}}{\displaystyle \frac{{\sigma}^{2}{b}^{\prime}Gb(\sigma {\sigma}^{\prime}G+\widehat{\sigma})}{{\sigma}^{2}}})({S}_{t})\mathrm{d}t\}].$ 
We now use the Girsanov theorem. Define the change of measure by $\mathrm{d}\widehat{P}/\mathrm{d}\stackrel{~}{Q}:=\mathrm{exp}({X}_{T})$ and a $\widehat{P}$Wiener process $B$ as
$${B}_{t}:={\stackrel{~}{W}}_{t}+{\int}_{0}^{t}\left(\frac{{\sigma}^{\prime}}{2}\frac{b}{\sigma}\right)({S}_{u})\mathrm{d}u,t\in [0,T].$$ 
Then, $(C)$ can be written as
$$(C)={E}^{\widehat{P}}\left[f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T})\left\{{\int}_{0}^{T}\left(\frac{\sigma {\sigma}^{\u2033}G+{\widehat{\sigma}}^{\prime}}{2}\frac{{\sigma}^{2}{b}^{\prime}Gb(\sigma {\sigma}^{\prime}G+\widehat{\sigma})}{{\sigma}^{2}}\right)({S}_{t})\mathrm{d}{B}_{t}\right\}\right].$$ 
The duality formula of Malliavin calculus (see, for example, Nualart 2006, Equation (1.42)) yields
$$(C)={E}^{\widehat{P}}\left[{\int}_{0}^{T}{D}_{t}\left(f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T})\right)\left\{\left({b}^{\prime}\frac{b{\sigma}^{\prime}}{\sigma}\frac{\sigma {\sigma}^{\u2033}}{2}\right)G\left(\frac{b\widehat{\sigma}}{{\sigma}^{2}}+\frac{{\widehat{\sigma}}^{\prime}}{2}\right)\right\}({S}_{t})\mathrm{d}t\right],$$  (2.9) 
where ${D}_{t}$ denotes the Malliavin derivative operator with respect to the Wiener process $B$.
Further, from the chain rule of the Malliavin derivative (see, for example, Nualart 2006, Proposition 1.2.3), we have
$${D}_{t}\left(f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T})\right)={\partial}_{1}f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T}){D}_{t}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right)+{\partial}_{2}f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T}){D}_{t}({S}_{T}).$$ 
Due to Nualart (2006, Theorem 2.2.1) and Nakatsu (2013, Lemma 2), we obtain
$${D}_{t}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right)=\frac{{Y}_{\eta}}{{Y}_{t}}\sigma ({S}_{t}){I}_{[0,\eta ]}(t),$$ 
where ${Y}_{t}:=\partial {S}_{t}/\partial x$ and $\eta :=\underset{0\le t\le T}{\mathrm{arg}\mathrm{max}}{S}_{t}$, and
$${D}_{t}({S}_{T})=\frac{{Y}_{T}}{{Y}_{t}}\sigma ({S}_{t}).$$ 
This finishes the proof of (2.2). ∎
Remark 2.10.
In the calculation of $(C)$, it is clear that we can avoid the appearance of the derivative of $f$ without the duality formula and obtain
$$(C)={E}^{\widehat{P}}\left[f(\underset{0\le t\le T}{\mathrm{max}}{S}_{t},{S}_{T})\left\{{\int}_{0}^{T}\left(\frac{\sigma {\sigma}^{\u2033}G+{\widehat{\sigma}}^{\prime}}{2}\frac{{(\sigma )}^{2}{b}^{\prime}Gb(\sigma {\sigma}^{\prime}G+\widehat{\sigma})}{{\sigma}^{2}}\right)({S}_{t})\mathrm{d}{B}_{t}\right\}\right].$$  (2.10) 
However, we still prefer to avoid the stochastic integrals in (2.10) in order to obtain the stability of the Monte Carlo estimates.
Remark 2.11.
We can apply the above technique to obtain the representation of the vega index for options with payoffs that depend on the minimum of the asset. We define ${\stackrel{~}{\mathrm{\Pi}}}^{\epsilon}:={E}^{P}[f({\mathrm{min}}_{0\le t\le T}{S}_{t}^{\epsilon},{S}_{T}^{\epsilon})]$, and then we have the following formula for the vega index:
${{\displaystyle \frac{\partial {\stackrel{~}{\mathrm{\Pi}}}^{\epsilon}}{\partial \epsilon}}}_{\epsilon =0}$  $={E}^{P}\left[{\partial}_{1}f(\underset{0\le t\le T}{\mathrm{min}}{S}_{t},{S}_{T})\sigma \left(\underset{0\le t\le T}{\mathrm{min}}{S}_{t}\right){\displaystyle {\int}_{x}^{{\mathrm{min}}_{0\le t\le T}{S}_{t}}}{\displaystyle \frac{\widehat{\sigma}}{{\sigma}^{2}}}(y)\mathrm{d}y\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}+{E}^{P}\left[{\partial}_{2}f(\underset{0\le t\le T}{\mathrm{min}}{S}_{t},{S}_{T})\sigma ({S}_{T}){\displaystyle {\int}_{x}^{{S}_{T}}}{\displaystyle \frac{\widehat{\sigma}}{{\sigma}^{2}}}(y)\mathrm{d}y\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}+{E}^{P}[{\displaystyle {\int}_{0}^{T}}({\partial}_{1}f(\underset{0\le t\le T}{\mathrm{min}}{S}_{t},{S}_{T}){Y}_{\stackrel{~}{\eta}}{I}_{[0,\stackrel{~}{\eta}]}(t)$  
$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\partial}_{2}f(\underset{0\le t\le T}{\mathrm{min}}{S}_{t},{S}_{T}){Y}_{T}\left)\right\{{\displaystyle \frac{({b}^{\prime}\sigma b{\sigma}^{\prime}({\sigma}^{2}{\sigma}^{\u2033}/2))({S}_{t})}{{Y}_{t}}}\left({\displaystyle {\int}_{x}^{{S}_{t}}}{\displaystyle \frac{\widehat{\sigma}}{{\sigma}^{2}}}(y)\mathrm{d}y\right)$  
$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{\displaystyle \frac{((b\widehat{\sigma}/\sigma )+(\sigma {\widehat{\sigma}}^{\prime}/2))({S}_{t})}{{Y}_{t}}}\}\mathrm{d}t],$ 
where $\stackrel{~}{\eta}:=\underset{0\le t\le T}{\mathrm{arg}\mathrm{min}}{S}_{t}$. This formula provides possibilities for applications to creditlinked products.
3 Numerical experiment: structure of the vega index
In this section, we obtain some numerical results by using (2.2). We assume that the stock price follows the Black–Scholes model perturbed with a CEV modeltype perturbation and compute the value of the vega index. First, we compare the values of the vega index obtained in this setting with the one obtained in the standard (nonperturbed) Black–Scholes model. Second, we study the structure of the vega index of barrier options.
For convenience, we call the vega index obtained in (2.2) “LV vega” (LV stands for “local volatility”) and the vega in the Black–Scholes model “BS vega”.
The onedimensional model we will consider is specified as follows. Let $b(z)=0$, $\sigma (z)=\stackrel{~}{\sigma}z$ and
$$\widehat{\sigma}(z)=\{\begin{array}{cc}az\hfill & (z\le c),\hfill \\ \frac{a{c}^{1\beta}}{\beta}{z}^{\beta}+\left(1\frac{1}{\beta}\right)ac\hfill & (z>c),\hfill \end{array}$$  (3.1) 
where $a>0$, $$, $c>1$ and $\stackrel{~}{\sigma}>0$ are constants. This setting characterizes the Black–Scholes model perturbed with a CEVlike model. Note that the parameter “$a$” is the gradient and “$\beta $” is the convexity of the volatility surface. We introduce the parameter “$c$” so that $\widehat{\sigma}$ satisfies (H3).
Remark 3.1.
The function $\widehat{\sigma}(z)$ defined by (3.1) does not satisfy (H1), since ${\widehat{\sigma}}^{\u2033}(z)$ is not continuous at $z=c$. However, if $\sigma (z)=\stackrel{~}{\sigma}z$, $b(z)=0$ and $\widehat{\sigma}(z)$ is defined by (3.1), then we can prove that (2.2) holds for these $\sigma (z)$, $b(z)$ and $\widehat{\sigma}(z)$. Indeed, one can prove this by approximating $\widehat{\sigma}(z)$ with a sequence of smooth functions, which is defined in the same manner as (16) in Appendix B (available online).
To compute LV vega, we set the initial price of a stock $x=80$ with volatility $\stackrel{~}{\sigma}=0.3$. For the parameters of the perturbation function, we set $a=3$, $\beta =0.9$ and $c=50$. First, we consider options with strike $K=100$ and maturity $T=1$; then we shall change the value of maturity $T$. We set the number of partitions of the interval $[0,T]$ to $n={10}^{3}$ and the number of simulations to $N={10}^{6}$.
In Section 3.1, we consider the case of a payoff function that depends only on the maximum. In Section 3.2, we look at another case that depends on the maximum and the terminal value of the stock. In doing so, we study the three components of LV vega mentioned in Remark 2.2(C).
3.1 The case of a payoff function depending on only one component
We assume $f$ is of the form
$$f(y,z)=f(y)={(yK)}_{+},$$ 
where $K>x$. This option is called a lookback call option with strike $K$. In this case, we can show that (2.2) is valid with
$${f}^{\prime}(y):={I}_{(K,\mathrm{\infty})}(y),$$ 
although $f$ does not belong to ${C}_{b}^{1}({\mathbb{R}}_{+}^{2},{\mathbb{R}}_{+})$; namely, the following equation holds (recall that $\eta =\underset{0\le t\le T}{\mathrm{arg}\mathrm{max}}{S}_{t}$):
${{\displaystyle \frac{\partial {\mathrm{\Pi}}^{\epsilon}}{\partial \epsilon}}}_{\epsilon =0}$  $={\displaystyle \frac{1}{\stackrel{~}{\sigma}}}{E}^{P}\left[{I}_{(K,\mathrm{\infty})}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right)\underset{0\le t\le T}{\mathrm{max}}{S}_{t}{\displaystyle {\int}_{x}^{{\mathrm{max}}_{0\le t\le T}{S}_{t}}}{\displaystyle \frac{\widehat{\sigma}(y)}{{y}^{2}}}\mathrm{d}y\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}{\displaystyle \frac{\stackrel{~}{\sigma}}{2}}{E}^{P}\left[{I}_{(K,\mathrm{\infty})}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right)\underset{0\le t\le T}{\mathrm{max}}{S}_{t}{\displaystyle {\int}_{0}^{\eta}}{\widehat{\sigma}}^{\prime}({S}_{t})\mathrm{d}t\right].$  (3.2) 
The proof of (3.2) can be found in Appendix B (available online). Table 1 shows our numerical results.
LV vega  BS vega  

Lookback call option  171.523  60.145 
European call option  79.791  26.757 
In Table 1, LV vega for a European call option is obtained by replacing the payoff function $f(y,z)={(yK)}_{+}$ with $f(y,z)={(zK)}_{+}$ and ${\mathrm{max}}_{0\le t\le T}{S}_{t}$ with ${S}_{T}$ in (3.2).
We observe from Table 1 that there is a large difference between LV vega and BS vega, yet these two different models provide the same option price for an arbitrary payoff function. This difference may be crucial for traders of financial institutions, since once they trade an option, they start hedging procedures using the risks calculated at the same time with the option price. Therefore, if they use only the Black–Scholes model, they have much hedging (model) error in the case that the volatility surface changes in the direction previously indicated in (3.1). At this point, the difference between the lookback call option and the European call option with regard to the values of the vega index is large.
We will now study the extrema and maturity features of LV vega in detail with an example of a barriertype option.
3.2 The case of a payoff function depending on the extrema and the terminal value of the underlying
We assume $f$ is of the form
$$f(y,z)={I}_{(U,\mathrm{\infty})}(y){(zK)}_{+},$$ 
where $$. This option is called an upin call option with strike $K$ and barrier $U$. In this case, we have
${{\displaystyle \frac{\partial {\mathrm{\Pi}}^{\epsilon}}{\partial \epsilon}}}_{\epsilon =0}$  $={\displaystyle \frac{1}{\stackrel{~}{\sigma}}}{E}^{P}\left[{\delta}_{U}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right){({S}_{T}K)}_{+}\underset{0\le t\le T}{\mathrm{max}}{S}_{t}{\displaystyle {\int}_{x}^{{\mathrm{max}}_{0\le t\le T}{S}_{t}}}{\displaystyle \frac{\widehat{\sigma}(y)}{{y}^{2}}}\mathrm{d}y\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle \frac{1}{\stackrel{~}{\sigma}}}{E}^{P}\left[{I}_{(U,\mathrm{\infty})}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right){I}_{(K,\mathrm{\infty})}({S}_{T}){S}_{T}{\displaystyle {\int}_{x}^{{S}_{T}}}{\displaystyle \frac{\widehat{\sigma}(y)}{{y}^{2}}}\mathrm{d}y\right]$  
$\mathrm{\hspace{1em}\hspace{1em}}{\displaystyle \frac{1}{2}}{E}^{P}\left[{I}_{(U,\mathrm{\infty})}\left(\underset{0\le t\le T}{\mathrm{max}}{S}_{t}\right){({S}_{T}K)}_{+}{\displaystyle {\int}_{0}^{T}}{\widehat{\sigma}}^{\prime}({S}_{t})\mathrm{d}{B}_{t}\right]$  
$=:E(U,K)+T(U,K)+D(U,K),$  (3.3) 
where ${\delta}_{U}$ denotes the Dirac delta function at $U$. For drift sensitivity $D(U,K)$, we have used the expression of the form (2.10) to avoid the appearance of the delta function mentioned in Remark 2.10. Note that we can compute extrema sensitivity $E(U,K)$ and terminal sensitivity $T(U,K)$ explicitly from the explicit density function for $({\mathrm{max}}_{0\le t\le T}{S}_{t},{S}_{T})$.
As we stated in the previous subsection, in order to obtain the above equation we are required to use the limit arguments for $f$ that are employed in the proof of (3.2). This extension can be done using the same method in the proof of (3.2). Thus, we omit the proof.
In Figure 1, we plot the values of each sensitivity in LV vega against the barrier $U$. We observe that in the range of $U\le 130$, extrema sensitivities are smaller than terminal sensitivities, while the results are inverted if $U>130$. The existence of this critical barrier (say, ${U}^{*}$) is important, since in the case $U\le {U}^{*}$ we are required to pay more attention to the LV vega caused by the terminal feature than that caused by the maximum feature. The opposite occurs in the case $U>{U}^{*}$.
In Figure 2, we plot LV vega against the maturity $T$. The value of the barrier is fixed with $U=130$. We observe that extrema sensitivities are small for large $T$. The mathematical reasoning is that, for this option, extrema sensitivity becomes small as the probability of
$$ 
becomes small. (${S}_{t}^{\epsilon}$ denotes the solution to (2.1) with its perturbation parameter $\epsilon $.) Note that this probability is small for large $T$. From this result, we can conclude that extrema sensitivity is less important than terminal sensitivity and drift sensitivity for the option with long maturity. Thus, for barrier options with long maturity, we may ignore extrema sensitivity of the underlying when constructing a hedging strategy using LV vega.
Next, let us observe the standardized LV vega that is defined by LV vega divided by the maturity $T$. We set $U=130$. In Figure 3, we plot the values of standardized LV vega against the maturity in order to understand LV vega per unit of time. We observe that the growth of each sensitivity is sharp for small $T$ and almost linear for large $T$. This numerical result shows that we must be more careful about the LV vega of the options with short maturity than that of the options with long maturity. Moreover, from Figure 3, we observe that, for small $T$, the behavior of extrema sensitivity is the sharpest of the three sensitivities.
Finally, we observe the values of the vega index obtained in two different models. We compare LV vega with BS vega. The value of maturity is fixed with $T=1$. In Figure 4, we plot the values of LV vega and BS vega against the barrier $U$. The difference between LV vega and BS vega in Figure 4 (and Table 1) represents the importance of the selection of pricing models from the viewpoint of the vega index. We assume this difference is caused by the fact that, in the standard (nonperturbed) Black–Scholes model, only the parallel shift of the constant volatility is considered, while more flexible changes of the volatility are considered in the perturbed diffusion model.
4 Conclusion and final remarks
In this paper, we obtain a formula to calculate the vega index for options with payoff functions that may depend on the maximum or minimum of a onedimensional perturbed diffusion process. Our key technique is the Lamperti transformation, which enables us to calculate the directional derivatives with respect to the diffusion coefficient. This formula gives a decomposition of the vega index into three sensitivities: extrema sensitivity, terminal sensitivity and drift sensitivity. Numerical tests illustrate that there are some important relationships between extrema sensitivity and terminal sensitivity for realistic options.
The numerical result of a comparison of the vega index in perturbed and nonperturbed models tells us that the Black–Scholes model is very different from the onedimensional model dealt with in this paper, so far as the vega index is concerned. Today, some practitioners are using socalled stochastic volatility models, which deal with stochastic diffusion coefficients, to express the dynamics of economy (see, for example, Gatheral 2006). There are many difficulties in dealing with stochastic volatility models; however, computing the vega index for exotic options in stochastic volatility models is a challenging problem for the future. According to Brunick and Shreve (2013), when we consider some exotic options, there is a relationship between a onedimensional model and a stochastic volatility model. Thus, the results obtained in Brunick and Shreve (2013) may be applied to this problem.
Declaration of interest
The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.
Acknowledgements
The author would like to thank Arturo KohatsuHiga for helpful comments on this paper. The author is also grateful to Xiaoming Song for careful reading and comments to improve the paper. Finally, the author would like to thank the reviewer for a lot of important suggestions for the paper.
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