# Stochastic expansion for the pricing of Asian and basket options††thanks:
Date: March 10, 2024.

Fabien Le Floc’h<EMAIL_ADDRESS>
###### Abstract

We present closed analytical approximations for the pricing of basket options,
also applicable to Asian options with discrete averaging under the Black-
Scholes model with time-dependent parameters. The formulae are obtained by
using a stochastic Taylor expansion around a log-normal proxy model and are
found to be highly accurate for Asian options in practice as well as for
vanilla options with discrete dividends.

###### keywords:

Stochastic expansion, Asian option, basket option, Black-Scholes, arithmetic
averaging, cash dividends.

##  1 Introduction

The buyer of a vanilla basket option of strike $K$ and maturity $T$ on
underlying assets $(S_{i})_{i=1,...,n}$ receives at maturity $T$
$\max\left(\sum_{i=1}^{n}w_{i}S_{i}(T)-K\right)$ for a call and
$\max\left(K-\sum_{i=1}^{n}w_{i}S_{i}(T)\right)$ for a put. The weights
$w_{i}$ typically sum to one. The buyer of an Asian call option of strike $K$
and averaging dates $(t_{i})_{i=1,...,n}$ on an asset $S$ receives
$\max\left(\sum_{i=1}^{n}w_{i}S(t_{i})-K\right)$. Under the Black-Scholes
model with a term-structure of interest rates, dividend yields and
volatilities, Asian options with discrete arithmetic averaging can be reduced
to an equivalent vanilla basket option with a specific correlation structure.
Asian (basket) options are particularly popular among commodity traders, but
is also traded regularly in the foreign exchange and in the equity derivatives
spaces.

There is no exact closed analytic formula for the pricing of arithmetic Asian
or vanilla basket options. The problem has been widely studied. In the context
of the Black-Scholes model, early valuation techniques were based on
approximating the arithmetic mean by a geometric mean [17], or on moment
matching against a lognormal distribution as in Levy [12]. Later, Curran [4]
introduced a much more accurate pricing based on geometric conditioning which
has been more recently further improved, by exploring better integration
cutoffs and conditioning variables [5]. Ju proposed a third-order correction
for the Levy moment matching in [9], while Zhou [19] looked at matching four
moments with the log extended skew normal distribution. Beyond the Black-
Scholes model, but restricted to Asian options, Zhang and Oosterlee propose an
efficient method based on the Fourier cosine series for affine models with a
known characteristic function in [18].

Here, we consider a Black-Scholes setting with time-dependent parameters, and
propose a stochastic expansion using the geometric average as a proxy,
following the ideas of Etoré and Gobet, who applied their expansion to price
vanilla options with discrete dividends under the piecewise-lognormal model in
[6], with further refinements in [11]. We start by deriving the first, second
and third-order expansions and then evaluate their accuracy on a sample of
previously published cases. This allows for a comparison with other accurate
approximations.

##  2 Reductions to vanilla basket options

###  2.1 From Asian to basket

Let $F(0,t)$ be the forward price of the underlying asset from valuation time
to the date $t$. In the Black-Scholes model with time-dependent parameters we
have

$F(0,t)=S(0)e^{\int_{0}^{t}r(s)-q(s)\mathop{}\\!\kern 0.0pt\mathrm{d}s}\,,$

with $r,q$ being the instantateneous interest rate and dividend yield and
$S(t)$ the underlying asset spot at time $t$. Under the risk-neutral measure
$\mathbb{Q}$, the forward price follows

$\mathop{}\\!\kern 0.0pt\mathrm{d}F(t,T)=F(t,T)\sigma(t)\mathop{}\\!\kern
0.0pt\mathrm{d}W(t)\,,$

where $W$ is a $\mathbb{Q}$-Brownian motion.

The buyer of an arithmetic Asian option of strike $K$ and maturity $T$ on an
underlying $S$ with observation dates $(t_{i})_{i=1,...,n}$ with $t_{n}=T$,
receives at maturity $\max\left(\sum_{i=1}^{n}w_{i}S(t_{i})-K\right)$ for a
call and $\max\left(K-\sum_{i=1}^{n}w_{i}S(t_{i})\right)$ for a put. It may be
expressed as a basket option of strike $K$ and maturity $T$ on $n$ underlying
assets $(S_{i})_{i=1,...,n}$. The covariance between the assets must match the
covariance of $S$ between different observation times, this means111We use the
Itô isometry to obtain Equations 1 and 3.

(1) $\rho_{i,j}\sqrt{v_{i}v_{j}}=v_{i\wedge j}\,,$

where $\rho_{i,j}$ denotes the correlation between the assets $S_{i}$ and
$S_{j}$, $v_{i}=\int_{0}^{t_{i}}\sigma^{2}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s$ being the total variance up to time $t_{i}$ for the asset
$S$. Each underlying asset forward to $T$ must match the forwards of the
single asset to $t_{i}$, meaning

(2)
$F_{i}(0,T)=\mathbb{E}[S_{i}(T)]=\mathbb{E}[S(t_{i})]=F(0,t_{i})=S(0)e^{\int_{0}^{t_{i}}(r(s)-q(s))\mathop{}\\!\kern
0.0pt\mathrm{d}s}\,.$

Thus, the arithmetic Asian option on $S$ with observations $t_{i}$ corresponds
to a vanilla basket option on $S_{i}$ with forwards to maturity $F_{i}(0,T)$,
total variances $v_{i}$ and correlations $\rho_{i,j}$ for $i,j=1,...,n$.

###  2.2 Asian baskets

This may be extended to Asian baskets with payoff at maturity

$\max\left(\sum_{i=1}^{n}w_{i}\sum_{j=1}^{m}\mu_{j}S_{j}(t_{i})-K,0\right)\,,$

where $(\mu_{j})_{j=1...m}$ are basket weights and $S_{j}$ are the different
assets with correlation $\rho_{j,l}$. We may define $n\times m$ assets
$\hat{S}_{i+n(j-1)}$ with forward $F_{j}(t_{i})$, total variance
$\int_{0}^{t_{i}}\sigma_{j}^{2}(s)\mathop{}\\!\kern 0.0pt\mathrm{d}s$,
correlation

(3) $\hat{\rho}_{i+n(j-1),k+n(l-1)}=\rho_{j,l}\frac{\int_{0}^{t_{i}\wedge
t_{k}}\sigma_{j}(s)\sigma_{l}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}{\sqrt{\int_{0}^{t_{i}}\sigma_{j}^{2}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s\int_{0}^{t_{k}}\sigma_{l}^{2}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}}\,,$

for $(i,k)\in\\{1,...,n\\}^{2}$, $(j,l)\in\left\\{1,...,m\right\\}^{2}$. The
Asian basket option is equivalent to a vanilla basket option on
$\hat{S}_{i+n(j-1)}$ with weights $w_{i}\mu_{j}$.

###  2.3 Average strike options

A floating strike Asian (call) option paying at maturity $T$,
$\max\left(S(T)-k\sum_{i=1}^{n}S(t_{i})\right)$, with $k$ being a strike
percentage, can be expressed as an equivalent fixed strike Asian (put) option
[15]. Through this symmetry, the formulae given in this paper may be used to
price floating (also known as average) strike options.

##  3 Expansions

###  3.1 The proxy

An early technique, first explored by Vorst [17] to estimate the price of
Asian options with arithmetic averaging is to approximate it by the geometric
average option equivalent, because the geometric average of geometric Brownian
motions is another geometric Brownian motion and can thus be priced with the
usual Black-Scholes formula. This was extended to the more general cases of
vanilla basket options by Gentle [7].

Under the risk-neutral measure $\mathbb{Q}$, the forward price for each asset
follows

$\mathop{}\\!\kern
0.0pt\mathrm{d}F_{i}(t,T)=F_{i}(t,T)\sigma_{i}(t)\mathop{}\\!\kern
0.0pt\mathrm{d}W_{i}(t)\,,$

where $W_{i}$ is a $\mathbb{Q}$-Brownian motion, and we have
$\mathop{}\\!\kern 0.0pt\mathrm{d}W_{i}\mathop{}\\!\kern
0.0pt\mathrm{d}W_{j}=\rho_{i,j}\mathop{}\\!\kern 0.0pt\mathrm{d}t$. The payoff
at maturity of the basket option reads

$\displaystyle V(T)$
$\displaystyle=\left[\eta\left(\sum_{i=1}^{n}w_{i}S_{i}(T)-K\right)\right]^{+}$
$\displaystyle=\left(\sum_{i=1}^{n}w_{i}F_{i}(0,T)\right)\left[\eta\left(\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-{K}^{\star}\right)\right]^{+}=A\left[\eta\left(\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-{K}^{\star}\right)\right]^{+}$

with $\eta=\pm 1$ for a call (respectively a put) option and

$\displaystyle\tilde{a}_{i}=\frac{w_{i}}{A(T)}\,,\quad
A=\sum_{i=1}^{n}w_{i}F_{i}(0,T)\,,\quad{K}^{\star}=\frac{K}{A}\,,$
$\displaystyle
S_{i}^{\star}(t)=\frac{S(t)}{F(0,t)}=e^{-\frac{1}{2}\int_{0}^{t}\sigma_{i}^{2}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s+\int_{0}^{t}\sigma_{i}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}W_{i}(s)}\,.$

We approximate the arithmetic average by the geometric average leading to

(4) $V(0)\approx
B(0,T)A\mathbb{E}\left[\left(\eta\left(\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}-{K}^{\star}\right)\right)^{+}\\\
\right]$

with $a_{i}=\tilde{a}_{i}$ as a specific case.

Let

$G(T)=\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}=e^{\sum_{i=1}^{n}-\frac{1}{2}a_{i}\int_{0}^{T}\sigma_{i}^{2}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s+\int_{0}^{T}a_{i}\sigma_{i}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}W_{i}(s)}$

We have

(5) $\displaystyle m(T)=\mathbb{E}\left[\ln G(T)\right]$
$\displaystyle=-\frac{1}{2}\sum_{i=1}^{n}a_{i}v_{i}\,,$ (6)
$\displaystyle\tilde{\nu}^{2}(T)=\mathrm{Var}\left[\ln G(T)\right]$
$\displaystyle=\sum_{i,j=1}^{n}a_{i}a_{j}\rho_{i,j}\int_{0}^{T}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s\,.$

with $v_{i}=\int_{0}^{T}\sigma_{i}^{2}(s)\mathop{}\\!\kern 0.0pt\mathrm{d}s$.

Vorst [17], Gentle [7] propose to adjust the strike price of the geometric
average option such that the expectation of the term inside the max function
is preserved and use
$\tilde{K}^{\star}=K^{\star}-(1-\mathbb{E}\left[G(T)\right])$ instead of
$K^{\star}$. A different way to preserve the expectation consists in adjusting
the spot side instead of the strike side:

(7) $V(0)\approx
B(0,T)A\mathbb{E}\left[\left(\eta\left(\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}-{K}^{\star}\right)\right)^{+}\right]$

with $\alpha=\frac{1}{\mathbb{E}[G(T)]}$ and $B(0,T)$ is the discount factor
to the payment date. The latter is found to be more accurate in Borovykh [1].

Levy [12] matches not only the mean but also the variance of the arithmetic
average by a lognormal distribution. We have

(8)
$\nu_{A}^{2}=\ln\left(\mathrm{Var}\left[A(T)\right]\right)=\ln\left(\sum_{i,j=1}^{n}a_{i}a_{j}e^{\rho_{i,j}\int_{0}^{T}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}\right)\,..$

We may adjust the $a_{i}$ used in the geometric mean such that the variance of
the geometric mean matches exactly the one of the arithmetic mean by using

(9) $a_{i}=\tilde{a}_{i}\frac{\nu_{A}}{\tilde{\nu}}\,.$

Through Equation 6 and the definition of the factor $\alpha$, we obtain the
following formula for our Vorst Geometric (VG) and Vorst Levy (VL)
approximations, using respectively $a_{i}=\tilde{a}_{i}$ and $a_{i}$ as
defined by Equation 9:

(10) $\displaystyle V(0)\approx A\cdot\mathrm{Black}(1,K^{\star},{\nu}^{2},T)$

where $\mathrm{Black}(F,K,v,T)$ denotes the Black-76 formula applied to the
forward $F$, strike $K$ and total variance $v$ to maturity $T$ and
$\nu=\tilde{\nu}$ for the geometric approximation, $\nu=\nu_{A}$ for the Levy
approximation. We have

(11) $\mathrm{Black}(F,K,v,T)=\eta B(0,T)\left[F\Phi(\eta d_{1})-K\Phi(\eta
d_{2})\right]$

with $d_{1}=\frac{1}{\sqrt{v}}\left(\ln\frac{F}{K}+\frac{1}{2}v\right)$,
$d_{2}=d_{1}-\sqrt{v}$.

We will adopt the Vorst geometric formula with adjusted forward as proxy for
our stochastic Taylor expansion. The Vorst geometric formula with strike shift
leads to slightly more complicated equations. Interestingly, it results in
exactly the same approximation for expansions of order 1 and above.

The Vorst Levy formula corresponds exactly to the Levy approximation, but
defines a stochastic process based on the individual underlying asset
processes. This will allow to define the stochastic expansions based on this
alternative proxy as well.

###  3.2 First order expansion

Let $h(x)=\max(\eta x,0)$, we apply a Taylor expansion of order-1 on $h$:

(12)
$\mathbb{E}\left[B(0,T)h\left(\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-K^{\star}\right)\right]=\mathbb{E}\left[B(0,T)h\left(\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}-K^{\star}\right)\right]\\\
+\mathbb{E}\left[B(0,T)\left(\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}\right)h^{\prime}\left(\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}-K^{\star}\right)\right]\\\
+\mathrm{Error}_{2}(h)$

where
$\mathrm{Error}_{2}(h)\leq_{c}2\left(||\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}||_{3}\right)^{2}$.
We may rewrite the second term using a derivative with regards to $K^{\star}$,
and interchange derivation and expectation (see Etoré and Gobet [6]) to obtain

(13)
$\mathbb{E}\left[\left(\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}\right)h^{\prime}\left(\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}-K^{\star}\right)\right]\\\
=-\frac{\partial}{\partial
K^{\star}}\mathbb{E}\left[\left(\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}\right)h\left(\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}-K^{\star}\right)\right]$

Let $G^{\star}=\alpha G$, we have $\mathbb{E}\left[G^{\star}(T)\right]=1$ and
$G^{\star}$ may be thus expressed as

$G^{\star}(T)=e^{-\frac{1}{2}\int_{0}^{T}\beta(s)^{2}\mathop{}\\!\kern
0.0pt\mathrm{d}s+\int_{0}^{T}\beta(s)dW_{G}(T)}$

in probability, with $W_{G}$ being a Brownian motion and
$\beta^{2}(s)=\frac{\partial\nu^{2}(s)}{\partial s}$. We interpret
$\frac{G^{\star}(T)}{G^{\star}(0)}$ as a change of measure on
$\mathcal{F}_{T}$. Under the new induced measure $\mathbb{Q}^{G}$,
$\tilde{W}_{G}(T)=W_{G}(T)-\int_{0}^{T}\beta(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s$ is a Brownian motion. Then $G^{\star}(T)$ under
$\mathbb{Q}^{G}$ has the same law as $G^{\star}(T)e^{\nu^{2}(T)}$ under
$\mathbb{Q}$. Thus,

(14)
$\displaystyle\mathbb{E}\left[G^{\star}(T)h^{\prime}\left(G^{\star}(T)-K^{\star}\right)\right]$
$\displaystyle=-\frac{\partial}{\partial
K^{\star}}\mathbb{E}^{G}\left[h\left(\tilde{G}^{\star}(T)e^{\nu^{2}(T)}-K^{\star}\right)\right]\,.$

The right hand side corresponds to the derivative of a vanilla option price
under the Black model with forward $e^{\nu^{2}(T)}$, strike $K^{\star}$ and
total variance $\nu^{2}(T)$.

In similar fashion we use a change of measure
$\frac{S_{i}^{\star}(T)}{S_{i}^{\star}(0)}$ and define

$\tilde{W}_{i}(t)=W_{i}(t)-\int_{0}^{t}\sigma_{i}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s$

to obtain

(15)
$\displaystyle\mathbb{E}\left[S_{i}^{\star}(T)h^{\prime}\left(G^{\star}(T)-K^{\star}\right)\right]$
$\displaystyle=-\frac{\partial}{\partial
K^{\star}}\mathbb{E}^{i}\left[h\left(\tilde{G}^{\star}(T)e^{\sum_{j=1}^{n}a_{j}\int_{0}^{T}\rho_{i,j}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}-K^{\star}\right)\right]\,.$

We use Equation 14 and Equation 15 in Equation 13 to obtain the first order
expansion

(16) $V_{\textsf{VG1}}=A\cdot\mathrm{Black}(1,K^{\star},\nu^{2},T)\\\
+A\frac{\partial}{\partial
K^{\star}}\mathrm{Black}\left(e^{\nu^{2}},K^{\star},\nu^{2},T\right)-A\sum_{i=1}^{n}\tilde{a}_{i}\frac{\partial}{\partial
K^{\star}}\mathrm{Black}\left(e^{\bar{v}_{i}},K^{\star},\nu^{2},T\right)\,.$

with

(17) $\displaystyle\bar{v}_{i}=\sum_{l=1}^{n}a_{l}v_{i,l}\,,$
$\displaystyle\quad
v_{i,l}=\rho_{i,l}\int_{0}^{T}\sigma_{i}(s)\sigma_{l}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s\,.$

In particular, for an Asian, we have $v_{i,l}=\int_{0}^{t_{i}\wedge
t_{j}}\sigma^{2}(s)\mathop{}\\!\kern 0.0pt\mathrm{d}s$. For baskets, the
volatilities are taken to be constant equal to each underlying implied
volatility to maturity in practice and we have
$v_{i,l}=\rho_{i,l}\sqrt{v_{i}v_{l}}$.

The relevant derivatives of the Black-76 formula are given in Appendix A. The
formula with the Vorst Levy proxy is exactly the same, but uses different
$\nu$ and $a_{i}$.

###  3.3 Second order expansion

The second-order Taylor expansion involves the additional term

$\displaystyle\frac{1}{2}\frac{\partial}{\partial
K^{\star}}\mathbb{E}\left[\left(\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}\right)^{2}h\left(\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}-K^{\star}\right)\right]$

We decompose it into three terms. The first term involves only $G^{\star}$, we
perform two successive changes of measure around
$\frac{G^{\star}(T)}{G^{\star}(0)}$ to obtain

$\displaystyle\mathbb{E}\left[G^{\star}(T)^{2}h(G^{\star}(T)-K)\right]$
$\displaystyle=\mathbb{E}^{G}\left[\tilde{G}^{\star}e^{\nu^{2}(T)}h\left(\tilde{G}^{\star}e^{\nu^{2}(T)}-K^{\star}\right)\right]$
$\displaystyle=e^{\nu^{2}(T)}\mathbb{E}^{G^{2}}\left[h\left(\tilde{\tilde{G}}^{\star}e^{2\nu^{2}(T)}-K^{\star}\right)\right]\,.$

The second term involves the cross product $S_{i}^{\star}(T)G^{\star}(T)$. We
perform two successive changes of measure around
$\frac{G^{\star}(T)}{G^{\star}(0)}$ and
$\frac{S_{i}^{\star}(T)}{S_{i}^{\star}(0)}$ to obtain

$\mathbb{E}\left[S_{i}^{\star}(T)G^{\star}(T)h(G^{\star}(T)-K)\right]=\\\
\mathbb{E}^{i}\left[\tilde{G}^{\star}e^{\sum_{j=1}^{n}a_{j}\int_{0}^{T}\rho_{i,j}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}h\left(\tilde{G}^{\star}e^{\sum_{j=1}^{n}a_{j}\int_{0}^{T}\rho_{i,j}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}-K^{\star}\right)\right]=\\\
e^{\sum_{j=1}^{n}a_{j}\int_{0}^{T}\rho_{i,j}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}\mathbb{E}^{i,G}\left[h\left(\tilde{\tilde{G}}^{\star}e^{\nu^{2}(T)+\sum_{j=1}^{n}a_{j}\int_{0}^{T}\rho_{i,j}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}-K^{\star}\right)\right]\,.$

The third term involves $S_{i}^{\star}(T)S_{j}^{\star}(T)$. We perform two
successive changes of measure around
$\frac{S_{i}^{\star}(T)}{S_{i}^{\star}(0)}$ and
$\frac{S_{j}^{\star}(T)}{S_{j}^{\star}(0)}$ to obtain

$\mathbb{E}\left[S_{i}^{\star}(T)S_{j}^{\star}(T)h(G^{\star}(T)-K)\right]\\\
=\mathbb{E}^{i}\left[\tilde{S}_{j}^{\star}(T)e^{\int_{0}^{T}\rho_{i,j}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}h\left(\tilde{G}^{\star}e^{\sum_{l=1}^{n}a_{l}\int_{0}^{T}\rho_{i,l}\sigma_{i}(s)\sigma_{l}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}-K^{\star}\right)\right]\\\
=e^{\int_{0}^{T}\rho_{i,j}\sigma_{i}(s)\sigma_{j}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s}\mathbb{E}^{i,j}\left[h\left(\tilde{\tilde{G}}^{\star}e^{\sum_{l=1}^{n}a_{l}\left(\int_{0}^{T}\rho_{i,l}\sigma_{i}(s)\sigma_{l}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s+\int_{0}^{T}\rho_{j,l}\sigma_{j}(s)\sigma_{l}(s)\mathop{}\\!\kern
0.0pt\mathrm{d}s\right)}-K^{\star}\right)\right]\,.$

Our formula for the second order expansion thus reads

(18) $V_{\textsf{VG2}}=A\cdot\mathrm{Black}(1,K^{\star},\nu^{2},T)\\\
+A\frac{\partial}{\partial
K^{\star}}\mathrm{Black}\left(e^{\nu^{2}},K^{\star},\nu^{2},T\right)-A\sum_{i=1}^{n}\tilde{a}_{i}\frac{\partial}{\partial
K^{\star}}\mathrm{Black}\left(e^{\bar{v}_{i}},K^{\star},\nu^{2},T\right)\\\
+\frac{A}{2}e^{\nu^{2}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{2\nu^{2}},K^{\star},\nu^{2},T\right)+\frac{A}{2}\sum_{i=1}^{n}\tilde{a}_{i}e^{\bar{v}_{i}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{\nu^{2}+\bar{v}_{i}},K^{\star},\nu^{2},T\right)\\\
-A\sum_{i,j=1}^{n}\tilde{a}_{i}\tilde{a}_{j}e^{v_{i,j}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{\bar{v}_{i}+\bar{v}_{j}},K^{\star},\nu^{2},T\right)\,.$

###  3.4 Third order expansion

The third order expansion adds the term

$\displaystyle-\frac{1}{6}\frac{\partial}{\partial
K^{\star}}\mathbb{E}\left[\left(\sum_{i=1}^{n}\tilde{a}_{i}S_{i}^{\star}(T)-\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}\right)^{3}h\left(\alpha\prod_{i=1}^{n}S_{i}^{\star}(T)^{a_{i}}-K^{\star}\right)\right]$

In similar fashion as for the second order, we perform three consecutive
changes of measure involving $\frac{G^{\star}(0)}{G^{\star}(0)}$,
$\frac{S_{i}^{\star}(T)}{S^{\star}(0)}$. This leads to our third order
expansion Our formula for the second order expansion thus reads

(19) $\displaystyle V_{\textsf{VG3}}$
$\displaystyle=A\cdot\mathrm{Black}(1,K^{\star},\nu^{2},T)$
$\displaystyle+A\frac{\partial}{\partial
K^{\star}}\mathrm{Black}\left(e^{\nu^{2}},K^{\star},\nu^{2},T\right)-A\sum_{i=1}^{n}\tilde{a}_{i}\frac{\partial}{\partial
K^{\star}}\mathrm{Black}\left(e^{\bar{v}_{i}},K^{\star},\nu^{2},T\right)$
$\displaystyle+\frac{A}{2}e^{\nu^{2}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{2\nu^{2}},K^{\star},\nu^{2},T\right)$
$\displaystyle-A\sum_{i=1}^{n}\tilde{a}_{i}e^{\bar{v}_{i}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{\nu^{2}+\bar{v}_{i}},K^{\star},\nu^{2},T\right)$
$\displaystyle+\frac{A}{2}\sum_{i,j=1}^{n}\tilde{a}_{i}\tilde{a}_{j}e^{v_{i,j}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{\bar{v}_{i}+\bar{v}_{j}},K^{\star},\nu^{2},T\right)$
$\displaystyle+\frac{A}{6}e^{2\nu^{2}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{3\nu^{2}},K^{\star},\nu^{2},T\right)$
$\displaystyle-\frac{A}{2}\sum_{i=1}^{n}\tilde{a}_{i}e^{\nu^{2}+2\bar{v}_{i}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{2\nu^{2}+\bar{v}_{i}},K^{\star},\nu^{2},T\right)$
$\displaystyle+\frac{A}{2}\sum_{i,j=1}^{n}\tilde{a}_{i}\tilde{a}_{j}e^{\bar{v}_{i}+\bar{v}_{j}+v_{i,j}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{\nu^{2}+\bar{v}_{i}+v_{j}},K^{\star},\nu^{2},T\right)$
$\displaystyle-\frac{A}{6}\sum_{i,j,l=1}^{n}\tilde{a}_{i}\tilde{a}_{j}\tilde{a}_{l}e^{v_{i,l}+v_{j,l}+v_{i,j}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{\bar{v}_{i}+\bar{v}_{j}+\bar{v}_{l}},K^{\star},\nu^{2},T\right)\,,$

with $\bar{v}_{i}$, $v_{i,j}$ defined in Equation 17.

###  3.5 Symmetries

We may exploit the symmetries in the double and triple sums to reduce the
number terms being summed and speed up the corresponding algorithm. In
particular,

$\sum_{i,j=1}^{n}\tilde{a}_{i}\tilde{a}_{j}e^{v_{i,j}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{\bar{v}_{i}+\bar{v}_{j}},K^{\star},\nu^{2},T\right)=\sum_{i=1}^{n}\tilde{a}_{i}^{2}e^{v_{i,i}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{2\bar{v}_{i}},K^{\star},\nu^{2},T\right)\\\
+2\sum_{i=1}^{n}\sum_{j=1}^{i-1}\tilde{a}_{i}\tilde{a}_{j}e^{v_{i,j}}\frac{\partial^{2}}{\partial{K^{\star}}^{2}}\mathrm{Black}\left(e^{\bar{v}_{i}+\bar{v}_{j}},K^{\star},\nu^{2},T\right)\,.$

and

$\sum_{i,j,l=1}^{n}\tilde{a}_{i}\tilde{a}_{j}\tilde{a}_{l}e^{v_{i,l}+v_{j,l}+v_{i,j}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{\bar{v}_{i}+\bar{v}_{j}+\bar{v}_{l}},K^{\star},\nu^{2},T\right)=\\\
6\sum_{i=1}^{n}\sum_{j=1}^{i-1}\sum_{l=1}^{j-1}\tilde{a}_{i}\tilde{a}_{j}\tilde{a}_{l}e^{v_{i,l}+v_{j,l}+v_{i,j}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{\bar{v}_{i}+\bar{v}_{j}+\bar{v}_{l}},K^{\star},\nu^{2},T\right)\\\
+3\sum_{i=1}^{n}\sum_{j=1}^{i-1}\tilde{a}_{i}\tilde{a}_{j}^{2}e^{2v_{i,j}+v_{j,j}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{\bar{v}_{i}+2\bar{v}_{j}},K^{\star},\nu^{2},T\right)\\\
+3\sum_{i=1}^{n}\sum_{l=1}^{i-1}\tilde{a}_{i}^{2}\tilde{a}_{l}e^{2v_{i,l}+v_{i,i}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{2\bar{v}_{i}+\bar{v}_{l}},K^{\star},\nu^{2},T\right)\\\
+\sum_{i=1}^{n}\tilde{a}_{i}^{3}e^{3v_{i,i}}\frac{\partial^{3}}{\partial{K^{\star}}^{3}}\mathrm{Black}\left(e^{3\bar{v}_{i}},K^{\star},\nu^{2},T\right)\,.$

##  4 Numerical examples for Asian options

###  4.1 Weekly

We start by reproducing the example of Ju [9, Table 4] of a weekly averaging
call option of maturity 3 years with different strikes and volatilities in
Table 1.

Table 1: Prices of weekly averaging call options of maturity 3 years with different strikes $K$ and volatilities $\sigma$, with a constant interest rate $r=9\%$. $(\sigma,K)$ | MC | Ju | LB | Deelstra | VG1 | VG2 | VG3 | VL3
---|---|---|---|---|---|---|---|---
(0.05, 95) | 15.1197 | 15.1197 | 15.1197 | 15.1197 | 15.1197 | 15.1197 | 15.1197 | 15.1197
(0.05, 100) | 11.3069 | 11.3069 | 11.3069 | 11.3069 | 11.3069 | 11.3070 | 11.3069 | 11.3069
(0.05, 105) | 7.5561 | 7.5562 | 7.5561 | 7.5561 | 7.5561 | 7.5561 | 7.5561 | 7.5561
(0.10, 95) | 15.2163 | 15.2165 | 15.2161 | 15.2163 | 15.2159 | 15.2163 | 15.2163 | 15.2163
(0.10, 100) | 11.6390 | 11.6394 | 11.6388 | 11.6390 | 11.6387 | 11.6390 | 11.6390 | 11.6390
(0.10, 105) | 8.3911 | 8.3913 | 8.3909 | 8.3911 | 8.3908 | 8.3911 | 8.3911 | 8.3911
(0.20, 95) | 16.6342 | 16.6365 | 16.6322 | 16.6342 | 16.6317 | 16.6341 | 16.6342 | 16.6342
(0.20, 100) | 13.7626 | 13.7634 | 13.7607 | 13.7626 | 13.7600 | 13.7625 | 13.7626 | 13.7626
(0.20, 105) | 11.2146 | 11.2134 | 11.2128 | 11.2146 | 11.2118 | 11.2145 | 11.2146 | 11.2146
(0.30, 95) | 19.0145 | 19.0179 | 19.0089 | 19.0145 | 19.0058 | 19.0140 | 19.0144 | 19.0144
(0.30, 100) | 16.5766 | 16.5755 | 16.5712 | 16.5766 | 16.5675 | 16.5762 | 16.5766 | 16.5766
(0.30, 105) | 14.3830 | 14.3774 | 14.3775 | 14.3830 | 14.3733 | 14.3827 | 14.3830 | 14.3830
(0.40, 95) | 21.7269 | 21.7307 | 21.7148 | 21.7269 | 21.7056 | 21.7256 | 21.7268 | 21.7267
(0.40, 100) | 19.5738 | 19.5690 | 19.5614 | 19.5738 | 19.5516 | 19.5727 | 19.5737 | 19.5737
(0.40, 105) | 17.6110 | 17.5978 | 17.5979 | 17.6109 | 17.5878 | 17.6100 | 17.6109 | 17.6108
(0.50, 95) | 24.5527 | 24.5583 | 24.5296 | 24.5527 | 24.5106 | 24.5498 | 24.5524 | 24.5523
(0.50, 100) | 22.6115 | 22.6032 | 22.5871 | 22.6116 | 22.5679 | 22.6090 | 22.6113 | 22.6111
(0.50, 105) | 20.8241 | 20.8023 | 20.7980 | 20.8242 | 20.7791 | 20.8219 | 20.8239 | 20.8238
RMSE |  | 0.0068 | 0.0115 | 0.0000 | 0.0204 | 0.0011 | 0.0001 | 0.0002
MAE |  | 0.0218 | 0.0262 | 0.0001 | 0.0450 | 0.0029 | 0.0002 | 0.0004

It consists in 157 averaging dates, starting at the valuation date $t=0$ and
ending at the maturity date $t=T=3$ years. We add the prices for the lower
bound and refined conditioned moment matching lognormal approximation of [5]
(respectively named ”LB” and ”Deelstra”) as well as the first, second and
third order stochastic expansions named ”VLE-1”, ”VLE-2”, ”VLE-3”. The
reference prices are obtained by a quasi Monte-Carlo simulation with Brownian-
bridge variance reduction on 32 million paths.

The Deelstra lower bound is found to be more accurate than the Curran
geometric conditioning in [9]. We also report slightly lower errors for the Ju
approximation than what is presented in [9], because the reference prices used
in [9] are less precise than ours. The Ju approximation is slightly more
accurate than the Deelstra lower bound. The second order stochastic expansion
is significantly more accurate, and the third order expansion is of similar
accuracy as the more refined Deelstra approximation. The VL proxy leads to
slightly larger errors than the VG proxy.

###  4.2 Monthly

We then consider the example of an Asian option of maturity 1 year with
monthly averaging, on the HAL stock ticker, presented in [19]. The interest
rate is chosen to be $r=6\%$, the dividend yield $q=0.97\%$, the volatility is
$\sigma=41.33\%$ and the spot price is $S=30.78$. The 12 averaging dates are
evenly222We reproduce here the example of Zhou as is. It is a manufactured
example albeit on realistic market data, as, in practice, the averaging dates
are typically not exactly evenly spread out due to holidays and week-ends.
spread out. Figure 1 shows the error in price expressed in b.p. of the
underlying spot. The reference prices are obtained with a quasi Monte-Carlo
simulation on 128 million paths.

Figure 1: Error in the price of Asian options of maturity 1 year on HAL with
monthly averaging, for a range of spot moneyness.

The first order expansion is of similar accuracy as the Deelstra lower bound
approximation. The second order expansion is already very accurate, with
maximum error well below 0.1 b.p., much lower than the error of the Deelstra
lower bound approximation. In Zhou [19], their four moments matching is found
to be comparable but slightly less accurate than the Deelstra lower bound. The
third order and the Deelstra approximations are extremely accurate, below the
Monte-Carlo sampling error, as evidenced by the wiggles in the error profile,
with however a slightly larger error for the Deelstra approximation. Asian
options with weekly averaging lead to similar observations. Again, the VL
proxy leads to slightly larger errors than the VG proxy.

###  4.3 Yearly

Finally, we study the more extreme examples of Lord [13]: yearly averaging
call options of maturity 5 and 30 years. The 5 year options are on an asset of
spot price $S=100$, with interest rate $r=5\%$, no dividend and and a somewhat
large volatility $\sigma=50\%$ while the 30 year option use a volatility
$\sigma=25\%$. They respectively consist in 5 and 30 averaging dates. Tables 2
and 3 reproduce [13, Tables 7 and 9] and reports the error in basis points of
the underlying spot for three strikes $K$ of Asian moneyness

$M=\frac{K}{\sum w_{i}F(0,t_{i})}-1\,.$

We name LB(2,3) and Deelstra(2,3) the lower bound and conditioned lognormal
moment matching approximations in Deelstra et al. [5] using $\delta_{2}$ and
$f_{3}$ as defined in their paper. We also test the Deelstra(3,3) variant with
$\delta_{3}$ and $f_{3}$.

Table 2: Error in b.p. in the prices of yearly averaging call options of maturity 5 years with different strikes. D($i$,$j$) corresponds to the Deelstra($i$,$j$) approximation with $\delta_{i}$ and $f_{j}$ [5]. Strike | $M$ | Ju | LB(2,3) | D(2,3) | D(3,3) | VG1 | VG2 | VG3 | VL3
---|---|---|---|---|---|---|---|---|---
58.2370 | -0.5 | 5.99 | -7.93 | -0.24 | -0.43 | -10.96 | -0.20 | 0.19 | 0.31
116.4741 | 0.0 | -2.62 | -8.19 | -0.03 | 0.06 | -10.67 | -0.37 | -0.11 | -0.14
174.7111 | 0.5 | -9.27 | -10.41 | -0.21 | -0.04 | -10.27 | 0.45 | -0.29 | -0.23
Table 3: Error in b.p. in the prices of yearly averaging call options of maturity 30 years with different strikes. D($i$,$j$) corresponds to the Deelstra($i$,$j$) approximation with $\delta_{i}$ and $f_{j}$ [5]. Strike | $M$ | Ju | LB(2,3) | D(2,3) | D(3,3) | VG1 | VG2 | VG3 | VL3
---|---|---|---|---|---|---|---|---|---
118.9819 | -0.5 | 10.30 | -3.62 | 0.05 | -0.10 | -7.59 | -0.51 | 0.06 | 0.12
237.9638 | 0.0 | -2.43 | -14.04 | -0.29 | 0.05 | -7.95 | -0.43 | -0.04 | -0.08
356.9457 | 0.5 | -12.39 | -22.87 | -0.48 | 0.03 | -7.84 | 0.02 | -0.10 | -0.11

For the $(2,3)$ variant, the errors for LB and Deelstra match respectively
what is reported in [13] as ”LB_GA” and ”Curran2M+”. The maximum error of the
$(3,3)$ variant is slightly larger. The second and third order expansions are
found to be nearly as accurate as the Deelstra conditioned lognormal moment
matching approximation. The third order here does not appear to improve
significantly over the second order. Figure 2 shows that the third-order is
markedly more accurate if we widen the moneyness range. Again, the first order
expansion exhibits a similar error as the Deelstra lower bound approximation.

(a)  (b)

Figure 2: Error in the price of Asian options of maturity 5 years with yearly
averaging.

For yearly options of maturity 30 years, Table 3 shows the first order
expansion to be more accurate than the Ju and the LB(2,3) lower bound
approximations. The Deelstra(3,3) performs slightly better than the
Deelstra(2,3) on this example. The second order expansion is as accurate as
the Deelstra(2,3) approximation within the range of Asian moneyness [-0.5,
0.5] and becomes slightly less accurate for larger Asian moneyness (Figure 3).
The third order expansions VG3 or VL3 allow to further reduce the error, but
are still less accurate than the best Deelstra(3,3) approximation further away
from the money.

(a)  (b)

Figure 3: Error in the price of Asian options of maturity 30 years with yearly
averaging.

###  4.4 Two remaining observations

We keep the same setting as the maturity 5 years in [13] but change the
maturity to 1.1 year such that only two observations remain at $t=0.1$ and
$t=T=1.1$. Figure 4 plots the error for a wide range of strikes.

(a)  (b)

Figure 4: Error in the price of Asian options with observations at $t=0.1$ and
$t=1.1$ year.

The Ju approximation leads to a maximum error of 10 b.p., twice as large as
the Deelstra lower bound approximations. The first order expansion is slightly
less accurate than the Deelstra lower bound and the second order expansion is
significantly more accurate (with an maximum error of around 1 b.p.). The
third order expansion is nearly as accurate as the Deelstra conditioned moment
matching approximations.

##  5 Numerical examples for basket options

###  5.1 Accuracy on basket options

Krekel [10] tests various basket approximations in depth. We reproduce and
apply some of the more significant tests to our new expansions.

Table 4: Prices of vanilla basket options of maturity $T=5$ years, with homogeneous volatilities $\sigma_{i}=40\%$, varying the correlation $\rho_{i,j}$ for $i\neq j$.  $\rho$ | MC | Ju | MM3 | LB | Deelstra | VG1 | VG2 | VG3 | VL3
---|---|---|---|---|---|---|---|---|---
0.10 | 21.692 | 21.766 | 21.619 | 20.124 | 21.609 | 20.124 | 22.224 | 21.440 | 21.612
0.30 | 25.029 | 25.052 | 24.994 | 24.209 | 24.987 | 24.209 | 25.212 | 24.961 | 24.985
0.50 | 28.007 | 28.013 | 27.995 | 27.633 | 27.990 | 27.633 | 28.059 | 27.994 | 27.996
0.70 | 30.743 | 30.743 | 30.740 | 30.620 | 30.738 | 30.620 | 30.752 | 30.741 | 30.742
0.80 | 32.041 | 32.041 | 32.041 | 31.989 | 32.040 | 31.989 | 32.044 | 32.041 | 32.041
0.95 | 33.919 | 33.919 | 33.919 | 33.916 | 33.919 | 33.916 | 33.919 | 33.919 | 33.919
RMSE |  | 0.032 | 0.034 | 0.741 | 0.039 | 0.741 | 0.231 | 0.107 | 0.038
MAE |  | 0.073 | 0.073 | 1.568 | 0.083 | 1.568 | 0.532 | 0.252 | 0.081

The first test consists in a relatively long term option of maturity 5 years,
using the same volatility for each asset and the same cross correlation, but
varying this correlation from 10% to 95% (Table 4). We include the results of
a three moments matching to the shifted lognormal distribution (named MM3).

This very simple test already puts into evidence that the expansions on the VG
proxy are accurate for correlations larger than 50%, the third order being the
most accurate, but deteriorate as the correlation decreases to 10%. The
expansions on the VL proxy are accurate across the whole range of
correlations. Under the settings of this test, the first order expansion is
equivalent to the Deelstra lower bound approximation.

Table 5: Prices of vanilla basket options of maturity $T=5$ years, with homogeneous volatilities $\sigma_{i}=40\%$ and correlation $\rho_{i,j}=50\%$, for $i\neq j$, varying the strike $K$.  $K$ | MC | Ju | MM3 | LB | Deelstra | VG1 | VG2 | VG3 | VL3
---|---|---|---|---|---|---|---|---|---
50.00 | 54.310 | 54.310 | 54.273 | 54.158 | 54.300 | 54.158 | 54.345 | 54.290 | 54.289
60.00 | 47.481 | 47.482 | 47.446 | 47.270 | 47.468 | 47.270 | 47.524 | 47.459 | 47.459
70.00 | 41.522 | 41.525 | 41.491 | 41.257 | 41.507 | 41.257 | 41.572 | 41.501 | 41.502
80.00 | 36.351 | 36.355 | 36.327 | 36.041 | 36.335 | 36.041 | 36.404 | 36.332 | 36.334
90.00 | 31.876 | 31.881 | 31.858 | 31.530 | 31.859 | 31.530 | 31.930 | 31.860 | 31.862
100.00 | 28.007 | 28.013 | 27.995 | 27.633 | 27.990 | 27.633 | 28.059 | 27.994 | 27.996
110.00 | 24.660 | 24.667 | 24.654 | 24.266 | 24.644 | 24.266 | 24.710 | 24.651 | 24.653
120.00 | 21.762 | 21.769 | 21.761 | 21.356 | 21.746 | 21.356 | 21.808 | 21.756 | 21.758
130.00 | 19.249 | 19.256 | 19.252 | 18.837 | 19.234 | 18.837 | 19.291 | 19.246 | 19.248
140.00 | 17.065 | 17.072 | 17.071 | 16.652 | 17.052 | 16.652 | 17.102 | 17.065 | 17.066
150.00 | 15.164 | 15.171 | 15.173 | 14.753 | 15.151 | 14.753 | 15.196 | 15.165 | 15.167
RMSE |  | 0.005 | 0.021 | 0.347 | 0.015 | 0.347 | 0.045 | 0.014 | 0.013
MAE |  | 0.007 | 0.036 | 0.413 | 0.017 | 0.413 | 0.053 | 0.022 | 0.021

With a fixed correlation of $50\%$, the third order expansion has the same
accuracy as the refined Deelstra approximation across a range of strikes
(Table 5). The Ju approximation is the most accurate on this example where the
volatilities are, again, homogeneous.

Next we consider at-the-money options, with a constant correlation at 50% but
vary the homogeneous volatility from 5% to 100%. In practice, volatilities
larger than 50% are exceptional for long term options. This test however puts
in evidence the limits of applicability of some of the approximations. The
expansions become inaccurate for large volatility $\sigma_{i}>50\%$ for a
maturity of 5 years (Table 8). The third order expansion clearly diverges as
the volatility is increased. This is true for the two proxies, the VL proxy
being more accurate in general for $\sigma_{i}\leq 50\%$ but worse for larger
volatilities.

Inhomogeneous volatilities (Table 6) decrease the accuracy of the expansions
slightly, but not significantly so, in contrast to the Ju approximation. This
explains why the Ju approximation is not so good for Asian options.

Table 6: Prices of vanilla basket options of maturity $T=5$ years, with inhomogeneous volatilities $\sigma_{2}=\sigma_{3}=\sigma_{4}=\sigma$ but $\sigma_{1}=100\%$ and correlation $\rho_{i,j}=50\%$, for $i\neq j$, varying the volatility $\sigma$. $\sigma$ | MC | Ju | MM3 | LB | Deelstra | VG1 | VG2 | VG3 | VL3
---|---|---|---|---|---|---|---|---|---
0.05 | 19.403 | 35.591 | 18.513 | 19.451 | 19.462 | 16.579 | 17.854 | 18.687 | 19.251
0.10 | 21.092 | 36.189 | 18.646 | 20.838 | 20.998 | 18.822 | 19.934 | 20.542 | 20.836
0.15 | 23.001 | 36.925 | 18.812 | 22.602 | 23.042 | 21.263 | 22.286 | 22.751 | 22.757
0.20 | 25.304 | 37.801 | 19.016 | 24.694 | 25.400 | 23.836 | 24.823 | 25.209 | 24.987
0.30 | 30.553 | 39.967 | 19.567 | 29.518 | 30.572 | 29.186 | 30.225 | 30.541 | 30.164
0.40 | 36.023 | 42.664 | 20.379 | 34.722 | 35.980 | 34.601 | 35.841 | 36.031 | 35.806
0.50 | 41.493 | 45.836 | 21.608 | 39.959 | 41.411 | 39.920 | 41.538 | 41.270 | 41.283
0.60 | 46.733 | 49.393 | 23.566 | 45.047 | 46.746 | 45.036 | 47.264 | 45.719 | 45.907
0.70 | 51.961 | 53.214 | 26.925 | 49.880 | 51.901 | 49.878 | 52.998 | 48.465 | 48.679
0.80 | 56.792 | 57.171 | 33.186 | 54.394 | 56.802 | 54.394 | 58.733 | 47.745 | 47.711
1.00 | 65.354 | 64.932 | 53.650 | 62.324 | 65.627 | 62.324 | 70.201 | 15.447 | 9.938
RMSE |  | 9.527 | 15.633 | 1.610 | 0.079 | 2.081 | 1.726 | 15.351 | 16.964
MAE |  | 16.139 | 25.006 | 3.086 | 0.217 | 3.086 | 4.791 | 49.963 | 55.416

###  5.2 Asian basket options on realistic market data

Previously, we looked at the accuracy using simple, manufactured market data
in order to assess the loss of accuracy when a single parameter was changing.
In this section, we consider Asian basket options on the more realistic market
data of Beisser [5]. The basket consists in 5 stocks, with different forwards,
volatilities and correlations. The averaging is monthly over the last 5
months. Figure 5 shows the error in basis points

$\frac{V_{\textsf{approx}}-V_{\textsf{MC}}}{\sum_{j=1}^{m}\mu_{j}S_{j}(0)}\times
10000\,,$

for three different maturities across a range of forward moneyness

$\frac{K}{\sum_{i=1}^{n}\sum_{j=1^{m}}w_{i}\mu_{j}F_{j}(0,t_{i})}-1\,.$

(a)  (b)  (c)

Figure 5: Error in the price of Asian basket options across a range of
strikes, for three different maturities.

Although not displayed, the first order expansion is found to have nearly the
same error profile as the Deelstra lower bound approximation. The higher order
expansions are accurate for the 6 months and 1 year maturity, but degrade with
the 5 years maturity, where the Deelstra conditioned moment matching leads to
the smallest error. This is not too surprising in light of the previous tests:
the volatility is kept the same for the long term option and for the short
term option, so the total variance is significantly larger in the long term
case, which degrades the expansions accuracy. If we scale the volatility by
the square root of the time to maturity, the error of the 5 years option price
becomes very similar to the one of the 1 year option.

##  6 Numerical examples for vanilla options with cash dividends

Healy [8] shows that the problem of pricing vanilla option under the piecewise
constant lognormal model, where the underlying asset jumps from the cash
dividend amount at the ex-dividend date, can be reduced to a vanilla basket
option pricing problem under the Black-Scholes model. We may thus use our new
expansions for basket options to price vanilla options with cash dividends and
assess their accuracy against the stochastic expansions of Etoré, Gobet and Le
Floc’h [6, 11] and other discrete dividends approximations.

In Table 7, we reproduce the example of a 7-year option, with varying yearly
dividend amounts, first presented in [16] and in [6].

Table 7: Price of a call option of maturity 7 years, on an asset with spot price $S=100$ paying yearly dividends of respective amounts 6, 6.5, 7, 7.5 8, 8, 8, starting at $t_{1}=0.9$, with $r=6\%$ and $\sigma=25\%$.  Strike | FDM | VL3 | VL2 | LL3 | EG3 | GS | Deelstra
---|---|---|---|---|---|---|---
70 | 27.21395 | 27.21392 | 27.21367 | 27.21769 | 27.20498 | 27.26457 | 27.21395
100 | 19.48229 | 19.48226 | 19.48181 | 19.48478 | 19.47843 | 19.51519 | 19.48245
130 | 14.13026 | 14.13023 | 14.12969 | 14.13139 | 14.12933 | 14.15233 | 14.13046
MRE | N/A | 2E-6 | 4E-5 | 1E-4 | 3E-4 | 2E-3 | 1E-5
Figure 6: Error in the price of vanilla options of maturity 10 years with
semi-annual dividends for a range of strikes.

The reference, named ”FDM” is the price obtained by the TR-BDF2 finite
difference method using 10,000 space steps and 3,650 time-steps and is
accurate to the fifth decimal. We also give the values of the third order
expansion from Etoré and Gobet [6] named ”EG3”, the third order expansion from
Le Floc’h [11] named ”LL3” and the approximation from Gocsei and Sahel [14]
named ”GS”.

Figure 6 reproduces the example of a call option of maturity 10 years, on a
stock of price $S=100$, paying dividends of amount $2$ on a semi-annual
schedule, with the first dividend starting one day from the valuation date at
$t_{1}=1/365$ presented in [14]. The interest rate is taken to be $r=3\%$ and
the volatility $\sigma=25\%$.

On those two examples, the third order expansion on the Levy proxy is the most
precise, and the second order expansion is more accurate than the third order
expansions of Etoré, Gobet and Le Floc’h.

##  7 Conclusion

For Asian options, the second order expansion on the Vorst Geometric proxy is
already very accurate, better than the third order formula of Ju or the
Deelstra lower bound approximations. The third order expansion is comparable
in accuracy to the more sophisticated Deelstra conditioning with moment
matching. The stochastic expansions have the advantage of requiring no
numerical solver and no numerical integration; they thus allow for a
straightforward stable calculation of the greeks, and facilitate a robust
implementation. The Vorst Levy proxy leads to a slightly worse accuracy of the
prices.

The expansions do not fare as well on basket options. In particular for very
large (not so realistic) volatilities, the expansions misbehave, especially
the third order expansion. The expansions on the Vorst Geometric proxy are
also not so accurate under mild correlations. The Vorst Levy proxy remediates
this latter issue. In the context of commodity options, the assets are
typically futures on the same commodity but of different maturities, and have
a correlation close to one [3]. The payoff also often involves the arithmetic
average of those futures prices: the contract is effectively an Asian basket
option. Given such a correlation structure, the expansions would behave very
well on those options.

When applied to price vanilla European options with cash dividends under the
piecewise-lognormal model (by reducing the problem to a vanilla basket option
pricing problem), the expansions are found to be the most precise when
compared to previously published approximations tailored to the specific case
of cash dividends.

Further research may apply similar expansions to Asian spread options, using a
double geometric approximation as proxy (see also [2]). Other possible
extensions would be for Asian options under discrete dividends assumptions,
Asian options under stochastic volatility models with a reasonably fast
pricing of European options such as Schöbel-Zhu.

## References

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##  Appendix A Derivatives of the Black-76 formula towards the strike

The following formulae are used to compute the expansions in practice:

(20) $\displaystyle\frac{\partial}{\partial K}\mathrm{Black}(F,K,v,T)$
$\displaystyle=-\eta B(0,T)\Phi(\eta d_{2})\,,$ (21)
$\displaystyle\frac{\partial^{2}}{\partial K^{2}}\mathrm{Black}(F,K,v,T)$
$\displaystyle=\frac{B(0,T)}{K\sqrt{v}}\phi(d_{2})\,,$ (22)
$\displaystyle\frac{\partial^{3}}{\partial K^{3}}\mathrm{Black}(F,K,v,T)$
$\displaystyle=\frac{B(0,T)}{K^{2}\sqrt{v}}\left(\frac{\phi(d_{2})}{\sqrt{v}}-1\right)\,,$

where $\Phi$, $\phi$ are the cumulative normal distribution function and the
normal probability density function.

##  Appendix B Additional tables

Table 8: Prices of vanilla basket options of maturity $T=5$ years, with homogeneous volatilities $\sigma_{i}=\sigma\%$ and correlation $\rho_{i,j}=50\%$, for $i\neq j$, varying the volatility $\sigma$. $\sigma$ | MC | Ju | MM3 | LB | Deelstra | VG1 | VG2 | VG3 | VL3
---|---|---|---|---|---|---|---|---|---
0.05 | 3.526 | 3.526 | 3.526 | 3.525 | 3.526 | 3.525 | 3.526 | 3.526 | 3.526
0.10 | 7.050 | 7.050 | 7.050 | 7.043 | 7.050 | 7.043 | 7.050 | 7.050 | 7.050
0.15 | 10.570 | 10.570 | 10.570 | 10.548 | 10.569 | 10.548 | 10.570 | 10.570 | 10.570
0.20 | 14.083 | 14.083 | 14.083 | 14.032 | 14.082 | 14.032 | 14.085 | 14.083 | 14.083
0.30 | 21.078 | 21.080 | 21.079 | 20.912 | 21.072 | 20.912 | 21.091 | 21.078 | 21.078
0.40 | 28.009 | 28.013 | 27.995 | 27.633 | 27.990 | 27.633 | 28.059 | 27.994 | 27.996
0.50 | 34.826 | 34.843 | 34.730 | 34.147 | 34.796 | 34.147 | 34.986 | 34.737 | 34.750
0.60 | 41.488 | 41.519 | 41.044 | 40.412 | 41.450 | 40.412 | 41.881 | 41.070 | 41.119
0.70 | 47.940 | 47.967 | 46.471 | 46.390 | 47.907 | 46.390 | 48.768 | 46.363 | 46.502
0.80 | 54.128 | 54.095 | 50.433 | 52.050 | 54.120 | 52.050 | 55.705 | 48.888 | 49.139
1.00 | 65.354 | 64.932 | 53.650 | 62.324 | 65.627 | 62.324 | 70.201 | 15.447 | 9.938
RMSE |  | 0.145 | 3.745 | 1.281 | 0.069 | 1.281 | 1.547 | 15.155 | 16.782
MAE |  | 0.478 | 11.760 | 3.086 | 0.217 | 3.086 | 4.791 | 49.963 | 55.416