text:
The decline and fall of the British empire
When the Great Exhibition opened in 1851, Britain was the greatest imperial power in history. At the launch of the Festival of Britain in 1951, that empire was crumbling before the nation's eyes. Denis Judd explains why the intervening century saw a slump in the nation's fortunes
This article was first published in the July 2011 issue of BBC History Magazine
1851: The Great Exhibition
The Great Exhibition of 1851 was Britain's glittering shop window and showcase for the world's attention and admiration. The first and greatest industrial power, the greatest imperial power, and the greatest naval power was, in effect, showing off its extraordinary achievements and at the same time advertising its manufacturing and industrial wares.
Although William Morris and others were to react negatively to the mass production of everyday utensils, furniture and textiles as depressingly lacking in beauty and originality, the tide could not be turned.
Within the Crystal Palace some 100,000 objects were displayed – taking up ten miles of space – the work of 15,000 contributors. Over half the display came from Britain and its empire, but other nations were invited to participate. In fact, the event was tactfully entitled "The Great Exhibition of the Works of Industry of All Nations".
The largest foreign contributor was France, which produced a particularly impressive array of textiles, and was an increasingly worrying competitor in Britain's overseas markets. Russia was another significant contributor, even though its exhibits arrived late, having been held up by ice in the Baltic Sea.
For the vast majority of the British visitors, however, it was sufficient to bask in the reflected glow of the amazing diversity, quality and inventiveness of Britain's industrial and manufacturing output. There was, as Queen Victoria wrote in her diary, "every conceivable invention".
Yet within a few years, there were signs that the nation that had so triumphantly mounted the Great Exhibition was far from infallible. Not only was domestic reform urgently needed to combat inequalities and deprivation, but the Crimean War (1854–56) became a byword for military inefficiency and the 1857–58 Indian Rebellion shook British rule from the Punjab to Bengal.
Indeed, during each decade that linked the Great Exhibition with the Festival of Britain a century later, there is ample evidence that the self-confident superpower of 1851 was in slow decline.
1860s: Colonial uprisings
The decade opened in 1860 with the outbreak of the bitterly fought Second Maori War in New Zealand. The struggle lasted for 12 years and required the intervention of British regular troops to achieve victory. The Maoris' stubborn resistance meant that they were eventually admitted to the franchise as well as enjoying a far higher status in colonial society than, for example, the Aboriginals of Australia.
Another colonial struggle erupted in 1865 with the Jamaica, or Morant Bay, Rebellion, when freed slaves, protesting at their impoverishment and lack of equality, were brutally crushed by Governor Eyre. The uprising's ringleaders were hanged and many blacks shot or flogged, and their villages burnt down. Then, in 1867, Fenian groups carried out acts of violence in London and Manchester.
In the same year the British North America Act created the Dominion of Canada with full internal self-government. Earlier, Karl Marx had set up the First Communist International in London in 1864.
The decade closed with the opening of the Suez Canal in 1869 (a development that was to draw Britain far deeper into the affairs of Egypt and the Middle East) and with the Red River Rebellion in Canada.
1870s: Loss of life
Disraeli's premiership (1874–80) ended with two military humiliations: the massacre of British troops at the battle of Isandlhwana at the outbreak of the 1879 Zulu War and, in the same year, an invasion of Afghanistan that got off to an equally disastrous start.
Disraeli did, however, succeed in purchasing controlling shares in the Suez Canal Company, easing British worries over who would control the vital quick route to the east. And, for good measure, he got parliamentary approval to bestow the new title 'Empress of India' upon Queen Victoria in 1876.
At home, working class discontent was addressed by the legalisation of Trade Unions in 1871, and the introduction of the secret ballot for elections a year later.
In 1873 the appalling loss of life during the Bengal famine put in doubt Britain's claim that the welfare of the queen's Indian subjects was central to British rule and to the 'civilising mission' itself.
1880s: Rifts at home, defeats abroad
One of the most traumatic events of the decade was the defeat of the Irish Home Rule bill in 1886 and the resulting permanent split in the dominant Liberal party. The defection of the Liberal unionists led by Joseph Chamberlain, the Duke of Devonshire and others made it far more difficult for the Liberals to win elections and in 1895 resulted in a Liberal unionist coalition with the Conservatives. The defeat of the bill also meant that the Irish Question remained unsolved, and that the continuing controversy threatened to destabilise the United Kingdom. Indeed, some politicians began to call for 'Home Rule All Round' – that is, for each of the four countries of the Union.
The decade had begun with the assassination of the chief secretary for Ireland, Lord Frederick Cavendish, and his deputy, TH Burke, in Phoenix Park, Dublin, by a Fenian splinter group. In South Africa, the British were soundly beaten by the Afrikaners in the Transvaal war of 1880-81, and were forced to restore that country's independence. In the Sudan, General Gordon was killed in Khartoum by Mahdist rebels. At home the decade ended with the crippling London dock strike of 1889.
1890s: The rise of Germany
The 1890s ended with the triple humiliation of 'Black Week' in the South African (or Boer) War of 1899–1902 – in December 1899 when British forces suffered three significant defeats at the hands of the Boer civilian army. Despite Britain's military victory two and a half years later, the conflict had exposed the British army's shortcomings and led to the deaths of over 30,000 Afrikaner civilians in various refugee camps, also known as concentration camps.
The 1890s was a decade when Britain's global supremacy seemed more threatened than ever, in particular by the rapidly rising power of the German empire. In the final years of the century, Germany offered Paul Kruger's Transvaal moral support in its clash with Britain, and muscled in on the African partition – for example, by dividing up the Cameroons with Britain. Then, in 1898, it announced a massive naval building programme.
In 1893 Cecil Rhodes's drive to conquer what was to become Southern Rhodesia (modern Zimbabwe) precipitated the serious Matabele uprising. At home, Keir Hardie became the first Labour MP in 1892.
The Second Irish Home Rule bill was thrown out by the House of Lords in 1893. Even Queen Victoria's elaborate Diamond Jubilee celebrations of 1897 prompted a poem by Kipling, Recessional, which seemed to predict national decline.
1900s: The end of isolation
In 1902 Britain signed a treaty of alliance with the rapidly rising 'other island empire' power of Japan. The treaty was renewed and strengthened in 1905 to guarantee either nation coming to the other's aid if attacked.
In 1904 Britain concluded an 'entente' with the old enemy France. This was not an alliance but an agreement to lessen bilateral tensions at certain points of imperial and international rivalry, and, more vaguely, a commitment to future co-operation. The 'entente' with France inevitably led to a similar understanding with Russia, France's formal ally.
These arrangements showed that Britain's era of 'splendid isolation' was over, and that the 19th-century's superpower feared that it could no longer afford to go it alone.
The decade contained other sobering events: the death of the much-adored, totemic queen-empress Victoria in 1901; the Treaty of Vereeniging, which ended the bitter Boer War but included terms blatantly aimed at accommodating the defeated Afrikaners to some sort of partnership with the British in the new South Africa; and the launch of Sinn Féin in 1905 and the Muslim League in 1906 (following the unpopular partition of Bengal).
The general election of 1906 delivered not merely an antiunionist landslide and a huge Liberal overall majority but also sent some 29 Labour MPs of various affiliations to the Commons. Three years later, Lloyd George's controversial 'socialistic' People's Budget ignited a ferocious debate over taxation and led to a constitutional crisis that resulted in the limiting of the delaying powers of the House of Lords.
Amid this furore the two general elections of 1910 resulted in two 'hung parliaments'.
1910–19: Under attack
The First World War, inevitably, dominated this decade, and once more exposed serious failings in Britain's top military leadership, as well as producing embarrassing administrative messes like the notorious 'shell shortage' of 1915.
The abortive 1916 Easter Uprising in Dublin was a warning that the demands of Irish nationalism must soon be met, especially after the failure to implement the Third Irish Home Rule bill in 1913–14. Perhaps most significant of all was the need from 1915 on to negotiate huge loans from the United States to finance Britain's war effort. At a stroke, the great creditor nation of the 19th century had become an international debtor.
Before the outbreak of war there had been a plethora of serious internal conflicts: not merely the 1909–11 constitutional crisis, but also the growing tide of suffragette agitation, sometimes leading to acts of calculated violence and even death threats aimed at cabinet members. A serious strike launched by a 'triple alliance' of coal, dock and transport workers in 1912 served to confirm suspicions that the old order was under a sustained attack.
For some, the outbreak of war was almost a relief from these tensions and provided a sense that the nation could now restore its usually adequate working relationships.
1920s: Ruptured union
The controversial 1921 Anglo-Irish treaty partitioning Ireland provided at last a partial, though bloody, resolution of the Irish Question. It led to a civil war between the pro and anti-treaty factions, and ultimately the grudging acceptance of the Dominion of the Irish Free State by republicans as a temporary expedient.
This was the first rupture in the union of the United Kingdom that had been established in 1801. But the decade contained plenty of other indicators of British decline. The 1926 imperial conference finally defined the constitutional status of the Dominions, acknowledging their right to full internal self-government and leaving it up to them as to whether they went along with British foreign policy. This marked the end of any hope that the empire might retain a fully coherent and binding set of external policies. Indeed, it would be left to the Dominions to decide whether they wished to take Britain's side in any future war.
The round table conferences on Indian constitutional reform, beginning in 1930, eventually guaranteed Dominion status to Britain's 'Jewel in the Crown', a concession that many Indian nationalists hoped would lead to complete independence. The 1929 Wall Street Crash heralded the onset of the Great Depression. At home, the 1922 Glasgow hunger march to London indicated the impact of economic decline on Britain's traditional industrial heartlands. The first minority Labour government was formed in 1924. Two years later the General Strike, though in the end a failure, awoke fears of socialist revolution and endemic class warfare.
1930s: Appeasing Hitler
The 1930s are indelibly associated with the failed policy of appeasement. It was an indicator of Britain's decline that Neville Chamberlain's government felt unable to offer a more robust response to Hitler's policy of aggrandisement in Europe.
Interwar cuts in defence spending meant that Britain needed time to rearm if it was to seriously challenge Nazi aggression. In addition, the horrific losses of the First World War, in which over a million British and empire troops died, were fresh in the memory, and many hoped to avoid a similar second conflict with Germany.
Nonetheless, the Munich Agreement of 1938, with its shameful betrayal of the integrity of Czechoslovakia, was a cynical piece of diplomacy. As popular as it initially was, appeasement failed to achieve Chamberlain's boast of "peace in our time". It also led to the resignation of the foreign secretary Anthony Eden in 1938.
Other signs of Britain's reduced standing can be found in the 1931 Statute of Westminster which put into statutory form the concessions made to the Dominions at the 1926 imperial conference. Indian nationalism also forced the government to introduce the 1935 Government of India Act, which in effect gave India home rule and Dominion status.
Although the Great Depression was lessening by the mid-1930s, the chaos it caused had encouraged the formation of the National Government (a coalition of the three main parties) in 1931. The growth of British fascism under the leadership of Oswald Mosley was another sign of unsettled times, even though the resistance to the Blackshirts, perhaps best demonstrated by the left's victory in the 'Battle of Cable Street' in Stepney in 1936, stopped the movement from gaining mass support. The abdication crisis of 1936 also indicated that times had changed, when the playboy king Edward VIII chose to renounce the throne for the sake of a twice-divorced American woman with a dubious past.
The decade ended with the government guaranteeing Poland's frontiers – a move that led to the outbreak of war with Germany on 3 September 1939.
1940s: Empire out, welfare state in
Despite the heroics of the Battle of Britain, the Blitz, El Alamein and D-Day, one of the most significant episodes of the war was the Atlantic Charter of 1941. This was an agreement that was largely the initiative of US president Roosevelt, with Britain's premier, Winston Churchill, the reluctant signatory.
Aimed at the reordering of the world when the war ended, the charter spoke of "the right of all people to choose the form of government under which they live". Despite Churchill's attempt to pretend otherwise, this could only mean that empires like Britain's should devolve power as soon as reasonably possible. And the fact that Churchill was prepared to sign it reveals Britain's determination – desperation even – to keep on the right side of America in the months leading up to Pearl Harbor.
The war provided other humiliations for Britain: the fall of Singapore in 1942 seemed almost a death knell for the far eastern empire; the Japanese conquest of Hong Kong and Burma were heavy blows; Gandhi and the Congress party's 'Quit India' movement in 1942 brought chaos.
When, in 1947, India and Pakistan gained independence, some must have recalled the words of the Viceroy Lord Curzon in 1902: "As long as we rule India we are the greatest power in the world. If we lose it we shall drop straight way to a third-rate power". As if in confirmation, in 1948 Ceylon and Burma became independent and Palestine was abandoned.
In 1949 the Irish Free State became a republic and left the Commonwealth; India also became a republic, but stayed in the organisation. The Labour government pushed through a raft of radical domestic reforms. It did indeed seem, as the historian AJP Taylor once said, as if: "Imperial greatness was on the way out; the welfare state was on the way in."
1951: The Festival of Britain
Herbert Morrison, Labour minister and a former leader of the London County Council, said of the event that he had done so much to create: "The festival is the British showing themselves to themselves – and the rest of the world."
A collective pat on the back, the festival was different in many ways to the Great Exhibition of a century before. Above all, it was a celebration of the nation's recovery from the perils and privations of the Second World War. It also regenerated London's semi-decaying industrial South Bank, making it the site of the main exhibitions which explored the British identity, the British landscape, British science and industry. There was the Dome of Discovery, the Royal Festival Hall and the slender, eye-catching Skylon, one of the best-remembered displays.
A large fun fair was set up in Battersea Park, and in Poplar in east London the new Lansbury Estate was designed to show off all the latest ideas about urban architecture, town planning and community living. A Trinidadian steel band came to play – a first in the capital, and a hint of future population changes, with the beginning of Caribbean immigration a few years later. Throughout the summer of 1951 the festival permeated much of British life. By September over eight million people had visited the South Bank exhibition.
Many of the displays looked to the future, and the focus was kept firmly on the liberating power of science and technology. Apart from providing, as some claimed, 'a tonic for the nation', the Festival of Britain transformed London's South Bank for ever.
What happened next…
Just five years later the Suez Crisis cruelly demonstrated Britain's fall from great power status. The plot to invade Egypt with France and to take back the newly nationalised Suez Canal from the control of Colonel Nasser's revolutionary regime in Cairo by military force was a lamentable failure.
The USA refused to support the invasion; Commonwealth countries like India disapproved; there was a near catastrophic run on the pound; and mass demonstrations in Britain showed how much opposition there was to the attempted coup.
Whereas only a few years before Britain would have been able to cow a country like Egypt with ease, it now had to slink out of the Suez Canal zone, humiliated and widely reviled.
Professor Denis Judd's book Empire: The British Imperial Experience from 1765 to the Present (IB Tauris, 2011) will be issued in a revised edition later this year.
More on: United Kingdom
The Library | The first British Empire
The 'Father of History' and India in the British empire
El Alamein: The British empire's last hurrah
1914: Why Britain had to go to War


Janice Martin achieves virtuosity as a violinist, pianist, guitarist, soprano and perhaps most impressively, boasts the title as the world's only aerial violinist.
GALLIPOLIS — Is there no better season for dancing than spring?
Janice Martin will put a spring in your step when she joins forces with The Ohio Valley Symphony at 7:30 p.m. Saturday at the Ariel-Ann Carson Dater Performing Arts Centre in downtown Gallipolis. Led by Maestro Ray Fowler, the OVS and Martin will present a variety of virtuosic music for the audience.
Martin achieves virtuosity as a violinist, pianist, guitarist, soprano and boasts the title as the world's only aerial violinist. Janice's multi-dimensional artistry has disarmed classical, popular and jazz audiences alike while creating ardent fans who span a broad demographic.
In 2009, she wowed millions of viewers on the popular NBC series "America's Got Talent" with her aerial violin and vocal performance, and her playing has been featured on television projects for the Discovery and History channels, as well on the big screen in the Miramax film "Paperclips." She is a frequent guest performer with Cirque de la Symphonie, an "orchestrated aerial dance spectacular," and is a headline act in Branson, Mo.
For her accomplishments, the Stradivari Society awarded her the honorary use of the 1708 Burstein/Bagshawe Stradivarius violin.
Martin and the OVS will perform the Tchaikovsky Piano Concerto, Sarasate's violin showpiece, Carmen Fantasy, and vocal selections from Die Fledermaus, La Traviata, Les Miserables, Sound of Music, West Side Story and more.
Join Steven Huang, director of orchestras at Ohio University and OVS program annotator, for a pre-concert chat to hear more about the music and the musicians. Held in the third-floor Ariel Chamber Theatre, the pre-concert talks are free, interactive, informal and begin at 6:45 p.m. Stick around for a post concert reception to meet the musicians and have some treats.
Tickets for The Ohio Valley Symphony's concerts are $24 for adults, $22 seniors and $12 for students. The price includes the pre-concert chat, the concert and the reception immediately following. Tickets are available on the website at arieltheatre.org or ohiovalleysymphony.org or at the box office at 428 Second Ave.
Funding for The Ohio Valley Symphony is provided in part by the Gallia County Medical Society and the Ann Carson Dater Endowment.
http://aimmedianetwork.com/wp-content/uploads/sites/42/2016/03/web1_StringIntoSpring-1.jpgJanice Martin achieves virtuosity as a violinist, pianist, guitarist, soprano and perhaps most impressively, boasts the title as the world's only aerial violinist.


text:
\section{Introduction}

Neural networks (NNs) have found successful applications such as
image understanding, handwriting recognition, speech recognition, and drug discovery
\cite{hinton, ng, jugen, deng, Ramsundar}.
This success is due to the universal approximation capability of NNs,
approximating the function accurately without assuming any mathematical relationship between the input and the output of a function.
Whereas no mathematical relationships are found for the foregoing applications,
strong governing differential equations are presented in many fields such as finance, fluid dynamics, quantum mechanics, and diffusion phenomena.
For these problems, the approximation capabilities of NNs can be improved
when the differential relationships are included in the modeling.

Neural networks with sufficiently smooth activation functions can approximate a function and its derivatives \cite{Cybenko, HORNIK1990, GALLANT1992129};
they are also differentiable because the composition of differentiable functions is differentiable.
Thus, we can obtain the gradient and Hessian of differentiable NNs, and use these derivatives to make the NNs themselves satisfy the differential equations of the problems of interest.
To obtain accurate derivatives of NNs,
we calculate the gradient and Hessian of NNs using automatic differentiation \cite{Nocedal, tf, raissi2018},
which is an exact method based on the chain rule for differentiating compositions of functions.
Since the differentiation of NNs plays a key role in the present work,
activation functions should be sufficiently smooth as well as effective for our applications.
Although any smooth activation function makes NNs differentiable,
we need the activation functions which show good practical performance; we compare several activation functions in terms of estimation accuracy for the value, gradient, and Hessian.

We need suitable benchmark problems to examine the effectiveness of our differential NN.
We consider financial option pricing because the option pricing problem has a widely known mathematical relationship, the Black-Scholes equation \cite{Black1973}, and the derivatives of option price, Greeks, are extensively studied since they have many practical applications such as hedging or speculating the future asset price \cite{Hull, Shreve2004, glasserman}.
Hence, the performance of the proposed differential NNs can be easily tested
using the experiments on option pricing.

Most multidimensional options have no general closed-form solution for pricing
and they are priced by numerical approximation techniques.
Various numerical methods have therefore been developed to solve this problem such as  finite differences methods (FDMs), Fourier methods, and Monte Carlo (MC) simulations \cite{Hull, Shreve2004,  Carr1999, glasserman}.
FDMs approximate the discrete version of Black-Scholes partial differential equation (PDE);
MC methods use the generated sequence of stock prices governed by the stochastic differential equation (SDE).
The present work combines the information of the PDE and SDE,
and trains NNs using --
(a) the option price generated by the SDE and (b) the constraint of the Black-Scholes equation.
As a result, the proposed method becomes mesh-free method like MC simulations, and does not suffer from the curse of dimensionality associated with high-dimensional FDMs.
Furthermore, Greeks are easily computed by differentiating the neural network
because the proposed method represents the solution of Black-Scholes equation in the form of
a neural network.

NNs have also been applied for option pricing problems
\cite{Gencay2001, Kohler2010, Sirignano2018,Wei2017, Becker2019}.
Our approach relates to methods of differentiating NNs used in \cite{Sirignano2018, raissi2018}.
We use automatic differentiation method to compute the gradient and Hessian of NNs,
while \cite{Sirignano2018} use a MC method to approximate the Hessian;
automatic differentiation is an exact operation and not a kind of approximation methods used in MC methods and FDMs \cite{Nocedal, tf}.
The present work generates training data using the SDE which gives more realistic evolution of asset prices
while  \cite{Sirignano2018}  use random samples from the region where the function is defined.
\cite{raissi2018} uses automatic differentiation method to compute the gradient of NNs
and SDE to generate training data;
but, the loss function for training NN does not contain PDE and the Hessian of NNs are not used.
In addition, we adopt sufficiently smooth activation function softplus \cite{softplus} for the efficient differentiation of NNs based on the performance comparison experiments,
while \cite{Sirignano2018} use tanh and \cite{raissi2018} uses sine as an activation function.

The remainder of the paper is organized as follows.
We discuss the related works on multidimensional option pricing and numerical approximation methods in Section 2.
The proposed method is developed for learning stochastic differential equations and the Black-Scholes equation for option pricing in Section 3.
The validity of the proposed method is demonstrated in Section 4, and conclusions are drawn in Section 5.

\section{Price and Greeks of multidimensional options}

A financial option is a contract between the seller and the buyer (holder of the option).
A European call option gives the holder the right but no obligation
to buy the risky assets at expiration time $T$ for a price that is agreed on now, the strike price $K$.
The option itself has a price because it give the holder a right.
Let the stock price $S_i(t)$ be a geometric Brownian motion,
\begin{equation}
dS_i(t)= r S_i(t) dt + \sigma_i S_i(t) d\tilde W_i(t), \label{eq:sde}
\end{equation}
where $t$ denotes time, $r$ risk-free interest rate,
$\sigma_i$ the volatility of the asset price, and $\tilde W_i$ a Brownian motion under a risk-neutral probability measure with
covariance $Cov(\tilde W_i (t), \tilde W_j (t)) = \rho_{ij} t$.
The correlated Brownian motion $\tilde W_i (t)$ can be expressed as $\sqrt{t} L_i Z$
where $Z$ is a standard $n$-dimensional normally distributed vector
and $L_i$ is the $i$th row of $L$ ;
$L$ represents any matrix such that $\Sigma = LL^\prime$ where the $(i,j)$ component of $\Sigma$ is $\rho_{ij}$.
We choose $L$ to be the Cholesky factor of  $\Sigma$.
Then the stock prices between time $t_1$ and $t_2$ are related by
\begin{equation}
S_i(t_2)= S_i(t_1)  \exp ((r-\frac{1}{2} \sigma_i^2)(t_2 - t_1)+ \sigma_i \sqrt{t_2 - t_1} L_i Z)\label{eq:stock}
\end{equation}
\cite{glasserman, Shreve2004, Hull}.

\subsection{Exchange option}\label{sec:exchange}
An exchange option gives the holder the right to exchange one asset for another in given time
in the future and it is commonly seen in the energy market.
The payoff of exchange option is
\begin{equation}
	\max(S_1(T) - S_2(T), 0).\label{eq:exop}
\end{equation}
\cite{Margrabe1978} developed an analytical solution formula
for the price of this option.
The price of exchange option at time $t$ is given by
\begin{equation}
u(t,S_1, S_2) = S_1 N (d_1) - S_2 N(d_2), \label{exCall}
\end{equation}
where
$ \sigma = \sqrt{ \sigma_1^2 + \sigma_2^2 - 2 \rho_{12} \sigma_1 \sigma_2},
\quad d_1= \frac{1}{\sigma \sqrt{T-t}} \left[\ln{\left(\frac{S_1}{S_2}\right)} + \frac{\sigma^2}{2} (T-t)  \right],
\quad d_2= d_1 - \sigma \sqrt{T-t}$,
and $N(x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} \mathrm e^{-\frac{1}{2}z^2} dz. $

Greeks are the sensitivities of the option price to the movement of various parameters.
They are used for risk management;
the risk in a short position in an option is offset by holding delta units of each underlying asset, where the delta is the partial derivative of the option price with respect to the current price of
that underlying asset. Sensitivities with respect to other parameters are also
widely used to measure and manage risk.
The Greeks of exchange option is obtained by differentiating Eq.\eqref{exCall}.
Let $\phi (x) = \frac{ 1}{\sqrt{2\pi} } \exp(-\frac{ x^2}{2})$,
differentiating Eq.\eqref{exCall} by $S_1$ and using the property $\phi (d_1) / \phi (d_2) = S_2 / S_1$
gives the delta
\begin{equation}
\Delta = \frac{\partial u}{\partial S_1} = N(d_1).\label{exdelta}
\end{equation}
In some other hedging strategy, we need to hedge away the risk due to the
changes of underlying asset's delta; the gamma is defined by
\begin{equation}
\Gamma = \frac{\partial \Delta}{\partial S_1} = \frac{\partial^2 u}{\partial S_1^2}
= \frac{ \phi (d_1)}{S_1 \sigma \sqrt{T-t} }.\label{exgamma}
\end{equation}
The time decay of the value for an option is called theta, given by
\begin{equation}
\Theta = \frac{\partial u}{\partial t}
=-\frac{\sigma}{4\sqrt{T-t} }\left[S_1 \phi (d_1) + S_2 \phi (d_2)\right].\label{extheta}
\end{equation}

\subsection{Basket option}
A European basket call option gives the holder to buy a group of underlying assets
at the same time.
The price of the option at maturity are given by
\begin{equation}
C(T,S_1(T), S_2(T), \ldots, S_n(T)) = \max( \sum_{i=1}^{n} w_i S_i(T) - K,0)\label{eq:BasCall}
\end{equation}
where $w_i$ denotes the quantity of $i$th asset in basket option contracts.
Since this option has no exact formula for price and Greeks,
these values are approximated by numerical methods such as
finite difference methods (FDMs) and Monte Carlo (MC) simulations \cite{Hull, Shreve2004, glasserman}.
While FDMs are accurate in low dimensional problems, they become infeasible in higher dimensions due to increased number of grid points and numerical instability.
MC methods are versatile for handling general type of options, multi-asset, and path dependent problems.
But, the MC method converges slow at rate $O(1/\sqrt{M})$, where $M$ denotes the number of random samples.

FDMs approximate the price of option, $u(t,S_1,S_2, \ldots, S_n)$, on the basis of the Black-Scholes equation \cite{Hull, Shreve2004},
\begin{equation}
	\frac{\partial u}{ \partial t } + \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n}
\sigma_i \sigma_j \rho_{i,j}  S_i S_j \frac{\partial^{2} u}{\partial S_i \partial S_j}
	+ r  \sum_{i=1}^{n} S_i \frac{\partial u}{\partial S_i} =	r u, \label{eq:mBS}
\end{equation}
where $r$ denotes risk-free interest rate,
and $(\sigma_i \sigma_j \rho_{i,j})_{1 \le i,j \le n}$ denotes the covariance matrix of the stock prices.
FDMs discretize the stock price and time dimensions
and use the option values at the time of expiration.
We use central difference equation to obtain the first and second derivatives with respect to stock price, and forward difference equation to obtain time derivatives.
The Greeks at time zero are obtained using the solved $u(\cdot)$.
The delta is given by
\begin{equation}
\Delta_i = \frac{u(0,S_1, \ldots, S_i + \delta_s, \ldots, S_n)-u(0,S_1, \ldots, S_i - \delta_s, \ldots, S_n)}{2\delta_s},
\end{equation}
where $\delta_s$ denotes a stock price discretization unit.
The gamma is given by
\begin{equation}
\Gamma_i = \frac{u(0,\ldots, S_i + \delta_s, \ldots)-2u(0,\ldots, S_i , \ldots)+u(0,\ldots, S_i -\delta_s, \ldots)}{\delta_s^2}
\end{equation}
The theta is given by
\begin{equation}
\Theta =  \frac{u(\delta_t,S_1, \ldots, S_n)-u(0,S_1, \ldots, S_n)}{\delta_t},
\end{equation}
where $\delta_t$ denotes a time discretization unit.

\begin{table}[t]
\centering
\begin{tabular}{|c|c|}
\hline
 Greeks & Expectation of the following formulas\\ \hline
 \multicolumn{2}{c}{Exchange option} \\ \hline
$\Delta$ & $e^{-rT} \frac{S_1(T)}{S_1(0)}  \mathbbm{1} \{S_1(T) > S_2(T) \} $  \\
\hline
$\Theta$ & $e^{-rT} [ S_1(T) ( \frac{\sigma_1^2}{2} - \frac{\sigma_1 L_1 Z}{2 \sqrt{T} } )
- S_2(T) ( \frac{\sigma_2^2}{2} - \frac{\sigma_2 L_2 Z}{2 \sqrt{T} } ) ] \mathbbm{1} \{S_1(T) > S_2(T) \}$  \\
\hline
$\Gamma$ & $e^{-rT}  \frac{ ( Z^{\prime} L^{-1} A^{-1} )_1 }{\sqrt{T} }  \frac{ S_2 (T) }{ S_1^2(0)}
 \mathbbm{1} \{(S_1(T) > S_2(T) \} $  \\ \hline

 \multicolumn{2}{c}{Basket option} \\ \hline
$\Delta_i$ & $e^{-rT} \frac{S_i(T)}{S_i(0)}  w_i \mathbbm{1} \{ \sum_{i=1}^{n} w_i S_i(T) > K \}$  \\ \hline
$\Theta$ & $ e^{-rT} [  -rK + \sum_{i=1}^{n} w_i S_i(T) ( \frac{\sigma_i^2}{2} - \frac{\sigma_i L_i Z}{2 \sqrt{T} } ) ]
 \mathbbm{1} \{ \sum_{i=1}^{n} w_i S_i(T) > K \} $  \\ \hline
$\Gamma_i$ & $e^{-rT}  \frac{ ( Z^{\prime} L^{-1} A^{-1} )_i }{\sqrt{T} S_i^2(0) }
\{ w_i S_i(T) - \sum_{i=1}^{n} w_i S_i(T) + K \}$ \\
&  $ \times \mathbbm{1} \{ \sum_{i=1}^{n} w_i S_i(T) > K \} $  \\ \hline
\end{tabular}
\caption{Greeks estimation by MC simulation.}
\label{tab:Greeks}
\end{table}

MC methods generate multiple simulated paths of stock price by Eq.~\eqref{eq:stock}.
The price of options is obtained by averaging the payoff as follows.
\begin{equation}
V(t) =  E_Q [ e^{-r(T-t)}V(T) | \mathcal F(t)], \label{eq:mcPrice}
\end{equation}
where $V(T)$ is the payoff function,
$\mathcal F(t)$ is a filteration for the Brownian motion $\tilde W_i (t)$
in Eq.\eqref {eq:sde} and $Q$ is the risk-neutral measure \cite{Hull, Shreve2004, glasserman}.
The Greeks estimation is a practical challenge in MC methods.
In the present work, the pathwise derivative estimate is used for calculating the delta and theta,
and combination of likelihood ratio method and pathwise derivative estimate is used for calculating gamma.
The Greeks at time zero are given by averaging the formulas in Table ~\ref{tab:Greeks}.
Derivations of these formulas are described in Appendix.

\section{Learning stochastic differential equations and the Black-Scholes equation}

\subsection{Learning stochastic differential equations}
We approximate the price of option $u(t,S_1,S_2, \ldots, S_n)$ in Eq.\eqref{eq:mBS}
using a neural network.
Expiry time $T$ is divided into $N(=200)$ equal intervals $t_0=0 < t_1 < t_2 < \cdots < t_N=T$.
Multiple stock price $S(t_k) =(S_1 (t_k), S_2 (t_k), \ldots, S_n (t_k)) $ is generated by Eq.~\eqref{eq:stock}
\begin{equation}
S_i(t_k)= S_i(t_{k-1})  \exp ((r-\frac{1}{2} \sigma_i^2)(t_k - t_{k-1}) + \sigma_i d \tilde W_i(t_k))\label{eq:istock}
\end{equation}
with an initial stock price $S(t_0)=S(0)$ and $d \tilde W_i(t_k) = \sqrt{t_k - t_{k-1}} L_i Z\label{eq:dw}$.
The price of option $u(t_k,S(t_k))$ is approximated by the neural network $\mathcal N(t, S)$
\begin{align}
& \mathcal N^1 (t_k,S(t_k)) = f [ w^1 (t_k, S(t_k))  + b^1 ], \nonumber \\
& \mathcal N^2 (t_k,S(t_k)) = f [ w^2 \mathcal N^1 (t_k,S(t_k))  + b^2 ], \nonumber \\
& \cdots \label{eq:C}, \\
& \mathcal N(t_k,S(t_k)) = \mathcal N^l (t_k,S(t_k)) =  w^l \mathcal N^{l-1} (t_k,S(t_k)) + b^l , \nonumber
\end{align}
where $f$ is an activation function, $w^l$ and $b^l$ parameters of $l$th layer of the neural network. In the present work, the number of layers $l$ is five,  the number nodes 35 for layer $l=1,2,3,4$, and one in the last layer $l=5$.

To train the neural network $\mathcal N(t, S)$, we need to know the target $u(t, S)$.
Although the $u(\cdot)$ is of course unknown at $t<T$,
this function satisfies the following SDE when the stock price follows Eq.\eqref{eq:sde}  \cite{Hull, Shreve2004, glasserman}.
\begin{equation}
du = ( u_t + \sum_{i=1}^{n} r S_i  u_{S_i}  +  \frac{1}{2} S^\prime H S  )dt +
\sum_{i=1}^{n} \sigma_i S_i  u_{S_i} d\tilde W_i, \label{eq:uSDE}
\end{equation}
where $u_t = \frac{\partial u}{\partial t}, S^\prime = (S_1,S_2, \ldots, S_n), H_{ij} = \rho_{ij}\sigma_i \sigma_j \frac{\partial^2 u}{\partial S_i \partial S_j}$
and $u_{S_i}= \frac{\partial u}{\partial S_i}$.
Based on this SDE and the derivatives of the neural network $\mathcal N(t, S)$,
we generate the following $\tilde u(t_k, S(t_k))$ that approximates $u(t_k, S(t_k))$.

\begin{align}
\tilde u(t_k, S(t_k)) &= \mathcal N(t_{k-1}, S(t_{k-1}))  + \{ \mathcal N_t(t_{k-1}, S(t_{k-1})) + \nonumber \\
& \sum_{i=1}^{n} r S_i(t_{k-1}) \mathcal N_{S_i}(t_{k-1}, S_i(t_{k-1}))  +  \frac{1}{2} S(t_{k-1})^\prime H_{k-1} S(t_{k-1})  \} dt_k \nonumber \\
& + \sum_{i=1}^{n} \sigma_i S_i(t_{k-1}) \mathcal N_{S_i}(t_{k-1}, S_i(t_{k-1})) d\tilde W_i (t_k),\label{eq:Ctilde}
\end{align}
where  $\mathcal N_t$ and $\mathcal N_{S_i}$ respectively denote partial derivatives of the neural network $\mathcal N(t, S)$ with respect to $t$ and $S_i$,
$(H_{k-1})_{ij} = \rho_{ij}\sigma_i \sigma_j \frac{\partial^2 \mathcal N(t_{k-1}, S(t_{k-1}) )}{\partial S_i \partial S_j}$, $dt_k = t_k - t_{k-1}$ and
$d \tilde W_i(t_k) = \sqrt{t_k - t_{k-1}} L_i Z$.
This $\tilde u(t_k, S(t_k))$ is used to train $\mathcal N(t_k, S(t_k))$
with the following loss function which is minimized during training the neural network.
\begin{equation}
L_{SDE} = \sum_{k=1}^N [ \tilde u(t_k, S(t_k)) - \mathcal N(t_k, S(t_k))  ]^2.\label{eq:lossC}
\end{equation}

\subsection{Learning the Black-Scholes equation}\label{sec:Loss}
We enforce the neural network $\mathcal N(t, S)$ to
satisfy the Black-Scholes equation in Eq.~\eqref{eq:mBS}
by minimizing the following representation of Black-Scholes equation in the form of a neural network
\begin{align}
L_{BS} = & \sum_{k=1}^m [ \mathcal N_t(t_k, S(t_k))  + \sum_{i=1}^{n} r S_i(t_k) \mathcal N_{S_i}(t_k, S_i(t_k)) \nonumber  \\
& +  \frac{1}{2} S(t_k)^\prime H_k S(t_k)  - r \mathcal N(t_k, S(t_k))  ]^2.\label{eq:lossD}
\end{align}

We need to differentiate the neural network with respect to variables $t_k$ and $S(t_k)$
for the calculation of Eq.~\eqref{eq:Ctilde} and Eq.~\eqref{eq:lossD}.
This differentiation is performed by using automatic differentiation.
Automatic differentiation is an exact differentiation method, not an approximation method,
so that it gives accurate differential results.
It apply the chain rule for differentiating compositions of a set of elementary functions for which derivatives
are known exactly \cite{Nocedal, tf, raissi2018}.
Since software libraries like Tensorflow \cite{tf} already provide operations for automatic differentiation,
we used the tensorflow.GradientTape operation in TensorFlow to calculate the gradient of neural network; tensorflow.GradientTape operation is performed to the gradient of NN for the calculation of the Hessian of NN.

At expiration time $T$, true $u(T,S_1,S_2, \ldots, S_n)$ is given by the payoff function.
To match the option value at maturity time $T$, the third loss function is
\begin{equation}
L_T =  [ \mathcal N(T, S(T)) - h(T)  ]^2 * w_T,\label{eq:lossT}
\end{equation}
where $h(T)$ denotes the payoff function such as
Eq.\eqref{eq:exop} and Eq.\eqref{eq:BasCall},
and $w_T$ denotes the weight for accurate approximation at $T$, $w_T = N / 20 (=10.)$ as a default weight in this work.
Finally, we use the following loss function that penalizes
the neural network output deviations from the stochastic differential equation,
the neural network differential structure deviations from Black-Scholes equation,
and the neural network output deviations from the payoff function at expiration time $T$,
\begin{equation}
Loss = L_{SDE} + L_{BS} + L_T\label{eq:loss}.
\end{equation}

\subsection{Training and estimation procedures of the neural network}

We call our neural network SDBS, because it is based on the stochastic differential equation and the Black-Scholes equation. We summarize the training procedure of SDBS in Algorithm 1.
For updating the parameters of SDBS (line 12 in Algorithm 1),
we use a stochastic gradient descent method, adam, that is based on adaptive estimation of first-order and second-order moments \cite{adam}.
It is commonly observed that a monotonically decreasing learning rate results in a better performing model.
The present work uses a PolynomialDecay schedule in Tensorflow and the learning rate is linearly decayed from
$10^{-3}$ to $10^{-7}$. As the final trained model, we choose the model which has the minimum $Loss$ during training iterations.

\begin{algorithm}
	\caption{Training of SDBS.}

	\begin{algorithmic}[1]
	 	\State \textbf{Input: }  $S(t_0)$, $T, \sigma_i, r, \rho_{ij}, N$, nEpoch
	 	\State Calculate $L$ representing $\Sigma = LL^\prime$, $t_0 = 0$
		\For {epoch=1, 2, $\cdots$, nEpoch}
		\State Calculate $\mathcal N, \mathcal N_t, \mathcal N_{S_i}, H$ at $(t_0, S(t_0))$

		\For {$k=1, 2, \cdots, N$}
		\State $t_k = t_{k-1} + T/N$
		\State Calculate $\tilde u(t_k, S(t_k))$ by Eq.~\eqref{eq:Ctilde}
		\State Calculate $S(t_k)$ by Eq.~\eqref{eq:istock} and
		\State Calculate $\mathcal N, \mathcal N_t, \mathcal N_{S_i}, H$ at $(t_k, S(t_k))$
		\EndFor

		\State Calculate $Loss$ by Eq.~\eqref{eq:loss}
		\State Update the parameters of neural network by minimizing $Loss$

		\If {$Loss$ is reduced comparing with that in the previous epoch}
		\State Save parameters of SDBS
		\EndIf

		\EndFor
	\end{algorithmic}
\end{algorithm}

The estimation procedure is similar to the training procedure.
There are three differences:
(1) in addition to the input of training procedure, the parameters of trained SDBS is loaded,
(2) the learning rate is fixed by $10^{-7}$ in line 12 of Algorithm 1,
(3) the line 13, 14, and 15 are replace by output Price $= \mathcal N(t,S(t) )$, $\Delta_i = \mathcal N_{S_i}(t,S(t) )$,
$\Gamma_i =  \mathcal N_{S_i S_i}(t,S(t) )$, $\Theta = \mathcal N_t(t,S(t) )$.

\subsection{Activation functions for differentiable SDBS}\label{sec:mathActi}

\begin{table}[t]
\centering
\begin{tabular}{|l|c|c|c|}
\hline
Name & Equation & Range & Order of continuity\\
\hline
sigmoid & $1/(1+e^{-x})$ & (0, 1) & $C^\infty$\\
tanh & $(e^x - e^{-x}) / (e^x+e^{-x})$ & (-1, 1) & $C^\infty$\\
sin & $\sin(\cdot)$ & (-1, 1)  & $C^\infty$\\
relu &
  $ \left\{
  \begin{array}{rl}
  0 & \text{if } x \le 0,\\
  x & \text{if } x > 0.
  \end{array} \right. $
 & $(0, \infty)$  & $C^0$\\
elu &
  $ \left\{
  \begin{array}{rl}
  \alpha (e^x - 1) & \text{if } x \le 0,\\
  x & \text{if } x > 0.
  \end{array} \right. $
 & $(-\alpha, \infty)$
 & $ \left\{
  \begin{array}{rl}
  C^1 & \text{if } \alpha = 1,\\
  C^0 & \text{otherwise} .
  \end{array} \right.$ \\
selu &
  $ \lambda \left\{
  \begin{array}{rl}
  \alpha (e^x - 1) & \text{if } x \le 0,\\
  x & \text{if } x > 0.
  \end{array} \right.$
 & $(-\lambda \alpha, \infty)$
 & $ C^0$ \\
softplus & $\ln (1+e^x)$ & $(0, \infty)$ & $C^\infty$\\
\hline
\end{tabular}
\caption{Classification of activation functions}
\label{tab:activation}
\end{table}

Activation functions determine the output of a node and the characteristics of NNs.
They have various shape, range, order of continuity, and magnitude of gradient.
Every function in Table ~\ref{tab:activation} has nonlinear shape
because nonlinearity of the activation functions allows NNs to be universal function approximators \cite{Cybenko, HORNIK1990}.
The range of function is finite for sigmoid, tanh, and sin while infinite for relu \cite{relu}, elu \cite{elu}, selu \cite{selu}, softplus \cite{softplus}.
The boundedness and continuity of partial derivatives of the activation functions up to order $m$ are required to approximate the functions with all partial derivatives up to $m$ are continuous and bounded \cite{HORNIK1990, GALLANT1992129}.
Hence, the order of continuity of activation functions critically affects the performance of the proposed method
in which the first and second derivatives of a neural network play a key role.
This consideration suggests that the following sufficiently smooth activation functions are suitable for the present work.
\begin{equation}
C^\infty \ \text{functions} = \{ \text{sigmoid, tanh, sin, softplus} \} \label{eq:smoothAct}.
\end{equation}
The magnitude of gradient of activation function should not vanish for the update of parameters of NNs during training.
When the sigmoid and tanh are either too high or too low, the gradient vanishing is observed.
The following relu-like functions have been widely used in many neural network applications
because they show high performance and their gradients are non-zero for all positive values.
\begin{equation}
\text{relu-like functions} = \{ \text{relu, elu, selu, softplus} \}, \label{eq:reluAct}
\end{equation}
where $\alpha=1$ is used for elu
and pre-defined constants $\alpha=1.67326324$ and $\lambda=1.05070098$ in Tensorflow are used for selu.

In section \ref{sec:activation}, we compare the performance of  activation functions in Table ~\ref{tab:activation}
in terms of estimation accuracy for value, the first and second derivatives of NNs.

\section{Numerical experiments}

In this section, numerical experiments are performed to demonstrate the effectiveness of the proposed method for pricing multidimensional options such as exchange and basket options.

\subsection{Estimation of price and Greeks of exchange options }\label{sec:exNu}

We can compare the performance of numerical methods using an exchange option
because this option has an exact solution.
\cite{Margrabe1978} provided the formula for the exchange option price in Eq.\eqref{exCall}.
By differentiating the Margrabe formula, we obtained the Greeks in Eq.\eqref{exdelta}-\eqref{extheta}.
In the same notation in section \ref{sec:exchange},
initial stock prices $S_1 (0) = 60$ and $S_2(0) = 60$,
time to maturity $T=1$,
volatility of first asset $\sigma_1 = 0.4$,
volatility of second asset $\sigma_2 = 0.2$,
interest rate $r=0.1$,
and correlation coefficient $\rho_{12}=0.4$ are parameters of the first experiment.
We define the error between exact solution and the estimate by relative error
\begin{equation}
\text{ rError} = \left | \frac{exact - estimate}{exact} \right |.\label{eq:error}
\end{equation}

\begin{table}
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
 & Exact & FDM1 & FDM2 & MC1 & MC2 &MC3  \\
\hline
Price & 8.777591 & 8.765359 &  8.776234 & 8.784203 &8.776402 &8.777109  \\
rError & 0 & 1.39e-03  &  1.55e-04  & 7.53e-04 & 1.35e-04  & \textbf{5.49e-05} \\
\hline
$\Delta$ & 0.573140 &0.572740    & 0.573102  & 0.573611 & 0.573094 & 0.57313 \\
rError & 0 & 7.09e-04 &7.80e-05 & 8.10e-04 &9.24e-05 & \textbf{2.97e-05} \\
\hline
$\Gamma$ &0.017726& 0.017728 & 0.017726&0.017738&0.017724 & 0.017725\\
rError & 0 & 1.15e-04 &\textbf{1.07e-05} &6.82e-04&9.47e-05 & 4.06e-05 \\
\hline
$\Theta$ & -4.339281 &-4.344155&-4.339812&-4.343678&-4.338946  & -4.339065\\
rError &0 & 1.12e-03 &1.22e-04 & 1.01e-03 &7.72e-05 & \textbf{4.99e-05} \\
\hline
\end{tabular}
\caption{Numerical estimate of the price and Greeks of the exchange option. The best results are highlighted in bold face.}
\label{tab:exRes}
\end{table}

\begin{table}
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
nEpoch & Exact & 25,000 & 50,000 & 100,000 &200,000 & 400,000\\
\hline
Price & 8.777591 &8.777736 & 8.777681 &8.777660 &8.777587 & 8.777516  \\
rError & 0 & 1.65e-05 &  1.02e-05 & 7.85e-06 & \textbf{4.85e-07} & 8.51e-06 \\
\hline
$\Delta$ & 0.573140 &0.573155  &0.573158   &0.573152  & 0.573139 & 0.573152 \\
rError & 0 & 1.40e-05 &1.84e-05 & 9.44e-06 &1.45e-05 & \textbf{9.37e-06} \\
\hline
$\Gamma$ &0.017726&0.017692  &0.017653 &0.017699 &0.017708 &0.017701 \\
rError & 0 & 1.90e-03 &4.12e-03&1.52e-03&\textbf{1.01e-03} & 1.38e-03 \\
\hline
$\Theta$ & -4.339281 &-4.323961 &-4.311028 &-4.330462&-4.330742 &-4.331632  \\
rError & 0 & 3.53e-03  &6.51e-03 & 2.03e-03 &1.97e-03 & \textbf{1.76e-03} \\
\hline
\end{tabular}
\caption{SDBS estimate of the price and Greeks of the exchange option using several training epochs. The best results are highlighted in bold face.}
\label{tab:exSDBS}
\end{table}

In Table ~\ref{tab:exRes}, FDMs discretize the domain of stock price $[0, 300]$ with 100 uniform intervals for FDM1 and 300 for FDM2,
and the time domain $[0, T]$ with 5,000 uniform intervals  for FDM1 and 50,000 for FDM2.
MC simulations calculate the price and Greeks of options given by
Eq. \eqref{eq:mcPrice} and Table \ref{tab:Greeks}.
MC1 uses $10^7$ simulated paths, $10^{8}$ for MC2,  and $10^{9}$ for MC3.
As can be seen in Table ~\ref{tab:exRes}, FDMs and MC methods give accurate estimation.
Furthermore, the finer discretization makes FDMs more accurate; MC methods estimate more accurately with more simulated paths.

Five independent SDBS models are trained for several nEpochs using the Algorithm 1.
The batch size 10,000 of simulated stock price $S(t)$ was used for training and estimation procedure.
The price and Greeks are generated 1000 times for each trained NN.
The average of 5,000 (5 models x 1,000 times) values is used as the final estimated values.
Comparing with the results in Table ~\ref{tab:exRes}, the SDBS in Table \ref{tab:exSDBS} gives comparable performance to the classical FDM and MC methods. In particular, the estimation of price for nEpoch 200,000 is significantly accurate.
These experiments show that the differential neural network based SDBS performs accurately
which estimates Price $= \mathcal N(t_0,S(t_0) )$, $\Delta_i = \mathcal N_{S_i}(t_0,S(t_0) )$,
$\Gamma_i =  \mathcal N_{S_i S_i}(t_0,S(t_0) )$, and $\Theta = \mathcal N_t(t_0,S(t_0) )$.

\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
  \multicolumn{2}{|c|}{$S_1(0), S_2(0)$} & 20,60 & 40,60 & 60,60 & 60,40 & 60,20 \\
\hline
\multirow{3}*{ Price} &FDM2&5.64e-02 &5.55e-04 & 1.55e-04 & \textbf{4.08e-07} &\textbf{1.81e-08}     \\
& MC2 & \textbf{3.13e-03} & \textbf{2.36e-05} & 1.35e-04 &9.42e-05 & 6.75e-05   \\
& SDBS & 1.25e-02 &6.22e-05 & \textbf{7.85e-06} &1.31e-06 &  2.90e-07 \\
\hline
\multirow{3}*{$\Delta$ } &FDM2  &4.67e-02 &2.15e-04 & 7.80e-05&1.03e-04 &3.28e-06 \\
& MC2 &\textbf{1.52e-03} & 2.91e-04& 9.24e-05&1.20e-04 &4.21e-05\\
& SDBS &2.22e-03 & \textbf{6.63e-05} & \textbf{9.44e-06} & \textbf{1.78e-05} & \textbf{2.25e-06} \\
\hline
\multirow{3}*{$\Gamma$} & FDM2  &4.06e-02 & \textbf{1.70e-04} & \textbf{1.07e-05} & 2.48e-04 & \textbf{2.07e-04}\\
& MC2  &1.74e-03  & 2.55e-04& 9.47e-05& \textbf{5.14e-05} & 2.06e-02\\
& SDBS &\textbf{1.36e-03} & 2.11e-03 &1.52e-03 & 4.14e-03 &8.42e-02 \\
\hline
\multirow{3}*{$\Theta$} & FDM2 &8.95e-02 & \textbf{6.51e-06} & 1.22e-04& \textbf{2.82e-05} & \textbf{2.07e-04}\\
& MC2 &1.15e-03 &1.55e-04 & \textbf{7.72e-05} & 2.71e-04&2.39e-02\\
& SDBS & \textbf{8.80e-04} & 3.73e-03 &2.03e-03 &6.84e-03 &1.09e-01 \\
\hline
\end{tabular}
\caption{rErrors of numerical methods for the exchange options with several initial stock prices. The best results are highlighted in bold face. }
\label{tab:exSDBSs1s2}
\end{table}

Table \ref{tab:exSDBSs1s2} shows the estimates of price and Greeks of exchange options with  different initial stock price pairs ($S_1 (0)$, $S_2(0)$), where the SDBS uses the same experimental configuration as in Table \ref{tab:exSDBS} with 100,000 nEpoch.
In the estimation of Price and $\Delta$, the SDBS shows good performance; five bests and five second bests out of total ten combinations of initial stock price pairs ($S_1 (0)$, $S_2(0)$);
but, the performance of SDBS is degraded for the estimation of $\Gamma$ and $\Theta$.

From these experiments, we find that
the differential neural network, SDBS, accurately estimates the value, the first and second derivatives
of solution function.
But, the SDBS consumes large memory for storing many simulated paths and model parameters of NNs;
our simulation configuration needs 8GB memory of NVIDIA GTX 1080.
The SDBS takes about 4.5 hours for 10,000 training nEpoch.
Thus, the current SDBS cannot be a practical substitute of MC simulations for the pricing of
multidimensional European options.

\subsection{Estimation of price and Greeks of basket call options }

We consider a call option on a basket with four independent stocks.
The parameters are as in \cite{KORN} ; $T=0.5, r=0.06$, $(S_1, S_2, S_3, S_4)$ $= (40, 50, 60, 70)$, $(w_1, w_2, w_3, w_4)$ $= (0.25, 0.25, 0.25, 0.25)$, and various volatilities.
Since there is no exact solution for the basket option,
we need a reliable numerical method to give us the benchmark value.
In Table \ref{tab:ba}, we used MC simulation based benchmark values in \cite{KORN} where the number of MC simulations paths is $10^6$.
We compare our method, SDBS used in Table ~\ref{tab:exSDBSs1s2}, with LN (the log-normal approximation of \cite{LEVY}), RG (the reciprocal gamma approximation of \cite{Milevsky}),
SLN (the shifted log-normal approximation of \cite{KORN}),
JU (the Taylor expansion approximation of \cite{Ju2002}),
and MC2 which is the MC method with $10^8$ simulated paths used in Table ~\ref{tab:exRes},
We rounded up to four decimals to be consistent with previous results in \cite{KORN}.

\begin{table}[]
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
K  & MC & LN & RG & SLN & JU & MC2 & SDBS \\ \hline

\multicolumn{8}{c}{$(\sigma_1, \sigma_2, \sigma_3, \sigma_4)=(0.2, 0.2, 0.2, 0.2)$} \\ \hline
\multirow{2}*{50} & 6.5355  & 6.5412 & 6.5340 & 6.5653 & 6.5404 	& 6.5407 & 6.5404 \\
  &  0 & 8.72e-04	& \textbf{2.30e-04}	& 4.56e-03	& 7.50e-04	& 7.96e-04	 & 7.50e-04 \\ \hline
  \multirow{2}*{55} & 2.5063 & 2.5104 & 2.5010 &	2.5343 &	2.5092 &	2.5094 &	2.5092 \\
  &  0 & 1.64e-03&	2.11e-03&	1.12e-02&	\textbf{1.16e-03} &	1.24e-03&	\textbf{1.16e-03} \\ \hline
  \multirow{2}*{60} & 0.5041 & 0.5037 &	0.5133 &	0.4719 &	0.5049 &	0.5049 &	0.5049\\
  &  0 & \textbf{7.93e-04} &	1.83e-02&	6.39e-02&	1.59e-03&	1.59e-03&	1.59e-03  \\ \hline

\multicolumn{8}{c}{$(\sigma_1, \sigma_2, \sigma_3, \sigma_4)=(0.5, 0.5, 0.5, 0.5)$} \\ \hline
\multirow{2}*{55} & 4.8324 &4.8499 &	4.7920& 	4.9492& 	4.8384& 	4.8382& 	4.8377\\
  &  0 & 3.62e-03&	8.36e-03&	2.42e-02&	1.24e-03&	1.20e-03&	\textbf{1.10e-03}  \\ \hline
  \multirow{2}*{60} & 2.7402 & 2.7463 &	2.7444 &	2.6729 &	2.7450 &	2.7441& 	2.7436 \\
  &  0 & 2.23e-03	&1.53e-03	&2.46e-02	&1.75e-03	&1.42e-03	& \textbf{1.25e-03} \\ \hline
  \multirow{2}*{65} & 1.4468 & 1.4413 &	1.4831 &	1.2550 &	1.4488 &	1.4479 &	1.4476 \\
  &  0 &3.80e-03&	2.51e-02	&1.33e-01&	1.38e-03&	7.60e-04&	\textbf{5.48e-04} \\ \hline

\multicolumn{8}{c}{$(\sigma_1, \sigma_2, \sigma_3, \sigma_4)=(0.8, 0.8, 0.8, 0.8)$} \\ \hline
\multirow{2}*{60} & 5.3401 & 5.3897 &5.2725 &	5.3819 &	5.3563 &	5.3468 &	5.3457 \\
 &  0 &9.29e-03&	1.27e-02&	7.83e-03&	3.03e-03	&1.25e-03&	\textbf{1.05e-03} \\ \hline
  \multirow{2}*{65} & 3.8179 & 3.8418 &	3.8123& 	3.5776& 	3.8336& 	3.8230 &	3.8215 \\
  &  0 & 6.26e-03	&1.47e-03	&6.29e-02	&4.11e-03	&1.34e-03&	\textbf{9.45e-04} \\ \hline
  \multirow{2}*{70} & 2.7011 &2.7003 &	2.7430 &	2.2590 &	2.7135& 	2.7025& 	2.7019 \\
  &  0 & \textbf{2.96e-04} &	1.55e-02&	1.64e-01&	4.59e-03&	5.18e-04&	2.98e-04 \\ \hline

\multicolumn{8}{c}{$(\sigma_1, \sigma_2, \sigma_3, \sigma_4)=(0.6, 1.2, 0.3, 0.9)$} \\ \hline
\multirow{2}*{60} & 5.5569 & 5.9128 &	5.7558& 	5.9371 &	5.5922& 	5.5635& 	5.5619 \\
  &  0 & 6.40e-02&	3.58e-02&	6.84e-02&	6.35e-03&	1.19e-03&	\textbf{8.98e-04} \\ \hline
  \multirow{2}*{65} & 4.1555 & 4.3459 &	4.2836 &	4.0874 &	4.1973 &	4.1604 &	4.1588 \\
  &  0 & 4.58e-02	&3.08e-02	&1.64e-02	&1.01e-02	&1.18e-03&	\textbf{8.03e-04} \\ \hline
  \multirow{2}*{70} & 3.1196 &3.1607 &	3.1798 &	2.6941 &	3.1710 &	3.1222 &	3.1207 \\
  &  0 & 1.32e-02	&1.93e-02	&1.36e-01	&1.65e-02	&8.33e-04	& \textbf{3.62e-04} \\ \hline

\end{tabular}
\caption{Basket call option prices and rErrors (the upper and lower values in each cell). The best results are highlighted in bold face.}
\label{tab:ba}
\end{table}

Table \ref{tab:ba} shows that SDBS gives the best accurate estimation.
Out of 12 combinations of $K$ and $(\sigma_1, \sigma_2, \sigma_3, \sigma_4)$,
SDBS gives 8 bests and 1 equal best, LN 2 bests, RG 1 best, and JU 1 equal best performances.
SDBS gives good performance for all levels of strike prices;
it gives more accurate price of options with increasing volatilities and
non-uniform volatilities.
The simulation based methods such as MC2 and SDBS can effectively process the options with large and non-uniform volatilities accurately, while the analytic approximation methods like LN, RG, SLN, and JU are degraded with such volatilities.
This trend is consistent with the results in \cite{KORN}.

\begin{table}[]
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
  \multicolumn{2}{|c|}{ $(\sigma_1, \sigma_2, \sigma_3, \sigma_4)$} & \multicolumn{3}{|c|}{ (0.5, 0.5, 0.5, 0.5) }  &  \multicolumn{3}{|c|}{ (0.6, 1.2, 0.3, 0.9)}\\ \hline
  \multicolumn{2}{|c|}{$K$} & 55 &  60 & 65 & 60 & 65 & 70 \\
\hline
\multirow{3}*{$\Delta_1$ } &JU  &2.15e-04 & \textbf{1.17e-04} & 2.33e-04&5.73e-03 &5.63e-03 & 4.44e-03\\
& MC2  & \textbf{4.01e-05} &1.26e-04 & \textbf{1.08e-04}& \textbf{1.58e-04} & \textbf{8.17e-06} &1.69e-04\\
& SDBS  & 5.31e-04& 2.86e-04& 4.49e-04&5.19e-04   & 7.49e-05& \textbf{7.32e-05} \\ \hline

\multirow{3}*{$\Delta_2$ } &JU  &9.75e-05 & \textbf{1.88e-05} &2.13e-04 &8.27e-03 &7.92e-03 &  3.32e-03\\
& MC2  &6.57e-05 &1.51e-04 &2.11e-04 & 1.36e-04& 9.60e-05&1.98e-04\\
& SDBS  &\textbf{4.51e-05} & 1.29e-04 & \textbf{8.88e-05} & \textbf{1.19e-04}  & \textbf{2.03e-05} & \textbf{6.44e-05}\\ \hline

\multirow{3}*{$\Delta_3$ } &JU  & 1.28e-04& \textbf{2.31e-05}&4.09e-04 &2.52e-03 & 8.49e-04&3.05e-03\\
& MC2  & \textbf{1.99e-06}&9.24e-05 & \textbf{1.08e-06} & \textbf{9.81e-05} & \textbf{5.51e-05} &\textbf{1.27e-04}\\
& SDBS  & 5.57e-05&9.23e-05 & 2.46e-04& 5.15e-04&  6.89e-04& 7.48e-04\\ \hline

\multirow{3}*{$\Delta_4$ } &JU  &5.34e-04 &2.27e-04 & 7.55e-04&1.09e-03 &9.20e-03 &1.87e-02 \\
& MC2  & \textbf{8.03e-05} &2.37e-04 & 2.93e-04& 2.49e-04& \textbf{2.17e-04}& \textbf{3.11e-04} \\
& SDBS  &1.10e-04 & \textbf{2.42e-05} & \textbf{1.73e-04} & \textbf{6.00e-05} & \textbf{2.17e-04}&3.85e-04\\ \hline

\multirow{3}*{$\Gamma_1$} & JU   &2.47e-03 & 1.84e-03& 1.49e-03& 5.22e-03&8.22e-03 &1.43e-02\\
& MC2   & \textbf{3.34e-04}&\textbf{1.19e-04} &\textbf{1.12e-04} & \textbf{2.89e-05}&\textbf{1.13e-03} &\textbf{7.64e-04}\\
& SDBS  &7.99e-03 &2.98e-03 &7.61e-03 & 2.29e-03 &9.91e-03 &6.70e-03 \\ \hline

\multirow{3}*{$\Gamma_2$} & JU   &2.96e-03 & 3.81e-03& 4.10e-03&1.38e-01 &1.50e-01 &1.54e-01\\
& MC2   &2.01e-04 &\textbf{2.32e-04} &\textbf{1.25e-04} & \textbf{3.43e-04}&3.58e-04 & \textbf{1.96e-04}\\
& SDBS  & \textbf{9.01e-05}& 1.38e-03& 1.11e-03&2.20e-03 & \textbf{2.95e-04}&3.91e-04 \\ \hline

\multirow{3}*{$\Gamma_3$} & JU   & 1.25e-04& 8.40e-04& 1.70e-03&1.96e-02 & 2.24e-02&1.28e-02\\
& MC2   & \textbf{1.86e-05}& \textbf{5.42e-05}& \textbf{4.13e-04}& \textbf{1.01e-04}& \textbf{9.00e-04}&\textbf{5.93e-04}\\
& SDBS  & 1.23e-03& 8.23e-04&2.08e-03 &8.70e-04 & 1.03e-02&6.38e-03 \\ \hline

\multirow{3}*{$\Gamma_4$} & JU   & 1.16e-03& \textbf{1.01e-04}&8.68e-04 & 6.16e-03& 3.64e-03&6.54e-03\\
& MC2   & \textbf{2.02e-04}& 5.04e-04&\textbf{6.97e-04} &\textbf{1.94e-04} &\textbf{2.16e-04} &\textbf{2.89e-04}\\
& SDBS  & 3.81e-04& 9.63e-04& 2.49e-03&8.83e-04 & 8.10e-04&1.89e-03 \\ \hline

\multirow{3}*{$\Theta$} & JU  & 1.44e+00&8.76e-01 &5.12e-01 &1.40e-01 & 5.32e-02&1.91e-01\\
& MC2  &\textbf{1.35e-04} &\textbf{1.79e-04} & \textbf{1.85e-04}&\textbf{3.20e-04} & \textbf{3.26e-04}&3.71e-04\\
& SDBS  &1.59e-03 &1.95e-04 &1.95e-03 & 4.16e-04& 1.14e-03&\textbf{4.98e-05} \\ \hline
\end{tabular}
\caption{rErrors of Greeks estimation of basket options. The best results are highlighted in bold face. }
\label{tab:baGreeks}
\end{table}

Table \ref{tab:baGreeks} shows the rErrors of Greeks estimation obtained with the methods JU, MC2 and SDBS used in Table ~\ref{tab:ba}.
Since benchmark values are required for the calculation of the relative errors of Greeks estimations, large number  of MC simulated paths(=$10^{11}$) were generated
and Greeks were estimated with the formulas shown in Table ~\ref{tab:Greeks}.
In $\Delta$ estimation experiments, MC2 gives 11 best and 1 equal best, and SDBS 10 best and 1 equal best, JU 2 best  performances in 24 combinations of  $K$ and $(\sigma_1, \sigma_2, \sigma_3, \sigma_4)$.
In estimation experiments of $\Gamma$ and $\Theta$,
MC2 gives 26 best, SDBS 3 best, and  JU 1 best performances in 30 combinations of  $K$ and $(\sigma_1, \sigma_2, \sigma_3, \sigma_4)$.
The estimation performance of SDBS is high for $\Delta$, but low for $\Gamma$ and $\Theta$;
this trend is consistent with the results on exchange options in Table ~\ref{tab:exSDBSs1s2}.

As can be seen in Table ~\ref{tab:ba} and \ref{tab:baGreeks},
the SDBS estimates accurately for price and $\Delta$ of basket option with four assets;
for the estimation of $\Gamma$ and $\Theta$, the SDBS is worse than MC simulation methods
but better than analytic approximation methods.

\subsection{Effect of the loss function of the Black-Scholes equation}

We expect that the loss $L_{BS}$ in Eq.~\eqref{eq:lossD} constrains the SDBS to satisfy the Black-Scholes partial differential equation; the derivatives of SDBS, Greeks, are estimated more accurately with $L_{BS}$ than without it.
To validate the effectiveness of this constraint,
we test several weights ($w=10^{-3}, 10^{-2}$, $10^{-1}, 1, 10^1$, $10^2, 10^3$) in weighted loss $L_w$
\begin{equation}
L_w = L_{SDE} + w * L_{BS} + L_T\label{eq:Lw},
\end{equation}
where $w = 1$ corresponds to the loss in Eq.~\eqref{eq:loss};
small $w$ means small contribution of PDE constraint and relatively large contribution of the SDE,
and large $w$ means large contribution of PDE constraint and relatively small contribution of the SDE.
We perform the same experiments as in Table \ref{tab:exSDBSs1s2} with $(S_1, S_2 )=(20, 60), (60,60)$, and $(60,20)$
except that this experiment uses the weighted loss $L_w$ instead of the original loss in Eq.~\eqref{eq:loss}.

\begin{table}[t]
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$w$  & $10^{-3}$ &  $10^{-2}$ &  $10^{-1}$ & 1 &  $10^{1}$ &  $10^2 $ &  $10^3 $ \\
\hline

\multicolumn{8}{c}{$(S_1, S_2 )=(20, 60)$} \\ \hline
Price          &4.49e-02  &3.35e-02 &  1.57e-02 &  \textbf{1.25e-02} & 2.67e-02  &5.55e-02 &1.81e-01  \\ \hline
$\Delta$     &1.92e-02  &7.05e-03 &5.91e-03 & \textbf{2.22e-03} & 1.45e-02 &1.07e-01 & 2.96e-01  \\ \hline
$\Gamma$ & 5.65e-02 & 1.12e-02& 5.22e-03 &  \textbf{1.36e-03} & 7.56e-03  &4.02e-01&6.03e-01   \\ \hline
$\Theta$    &5.82e-02  &2.78e-02 & 3.70e-03 &   \textbf{8.80e-04} & 9.64e-03  & 3.73e-01&5.74e-01 \\ \hline

\multicolumn{8}{c}{$(S_1, S_2 )=(60, 60)$} \\ \hline
Price   & 2.84e-05 &5.56e-05 & \textbf{2.97e-06} & 7.85e-06 & 1.19e-05 &3.97e-04 &2.50e-04 \\ \hline
$\Delta$  &5.29e-04 & 3.90e-05  &4.99e-05 & \textbf{9.44e-06}  &4.56e-05 &5.46e-04 &2.11e-03\\ \hline
$\Gamma$ &1.25e-01 &1.61e-02  &3.55e-03 & 1.52e-03 & \textbf{1.12e-03}  &2.90e-03 &1.02e-02\\ \hline
$\Theta$   & 1.05e-01 &1.63e-02  &5.26e-03 &  \textbf{2.03e-03}  & 2.73e-03&4.72e-03 &4.42e-03 \\ \hline

\multicolumn{8}{c}{$(S_1, S_2 )=(60, 20)$} \\ \hline
Price          & 9.28e-06 &\textbf{4.15e-08} &  1.57e-07 & 2.90e-07&7.17e-07   &3.32e-06 & 8.15e-06 \\ \hline
$\Delta$     &1.14e-03  &1.13e-04 &7.86e-06  &  \textbf{2.25e-06}&  3.29e-06 &2.95e-05 & 1.06e-04 \\ \hline
$\Gamma$ &  6.86e+00&1.03e+00 & 2.15e-01 & 8.42e-02 & 5.11e-02  & \textbf{9.77e-03} &  1.31e-01\\ \hline
$\Theta$    &3.66e+00 & 9.36e-01& 2.66e-01 & 1.09e-01 &  5.81e-02 &  \textbf{1.83e-02} & 1.65e-01 \\ \hline

\end{tabular}
\caption{rErrors of weighted loss $L_w$ for the exchange option. The best results are highlighted in bold face.}
\label{tab:LD}
\end{table}

Table~\ref{tab:LD} shows that this version of SDBS performs accurately for wide range of $w$, but it degrades when the $w$ deviates from unity.
This means that suitable constraint of Black-Scholes equation enhances the performance of option pricing.
Price estimation performs better with $w \leq 1$
while $\Gamma$ and $\Theta$ estimation perform better with $w \geq 1$.
This is because weighting more to the loss of differential equation $L_{BS}$ gives accurate estimation for derivatives
while weighting more to the loss of value $L_{SDE}$ gives accurate estimation for values.
When $L_{BS}$ is removed completely, no convergent value is obtained.
These observations reveal that the loss $L_{BS}$ is essential for the accurate estimation of gradient and Hessian of NN, and both loss $L_{BS}$ and $L_{SDE}$ play a key role in making the SDBS suitable for option pricing problems.

\subsection{Effect of the smoothness of activation functions}\label{sec:activation}

In section \ref{sec:mathActi}, we described the mathematical conditions of activation functions for the approximation of  the functions with all partial derivatives;
the boundedness and continuity of all its partial derivatives up to order $m$ are required to approximate the functions with all partial derivatives up to $m$ are continuous and bounded \cite{HORNIK1990, GALLANT1992129}.
The activation functions such as sigmoid, tanh, and sin satisfy these conditions.
But, the practical performance can differ from one activation function to another.
We compare the performance of several activation functions in terms of
their suitability for option pricing problems.

\begin{table}[t]
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
  &  sigmoid & tanh & sin & relu & elu & selu & softplus \\
\hline

\multicolumn{8}{c}{$(S_1, S_2 )=(20, 60)$} \\ \hline
Price          & 8.78e-01 & 1.76e-02 &  3.93e-01  &  1.01e+00 &  1.39e-02 &  5.55e-01 &  \textbf{1.25e-02} \\ \hline
$\Delta$     & 5.06e-01 &  5.29e-03  &  9.93e-01  &  9.94e-01 &  2.01e-01 &  7.59e-01 &  \textbf{2.22e-03}\\ \hline
$\Gamma$ & 5.83e-01 &  1.90e-02  &  9.92e-01  &  1.00e+00 &  1.39e-01 &  9.51e-01 &  \textbf{1.36e-03}\\ \hline
$\Theta$    & 5.55e-01 &  3.40e-02  &  1.00e+00  &  1.00e+00 &  2.59e-01 &  9.07e-01 &  \textbf{8.80e-04}\\ \hline

\multicolumn{8}{c}{$(S_1, S_2 )=(60, 60)$} \\ \hline
Price   & \textbf{2.83e-06} & 1.43e-04 & 3.79e-04  & 8.69e-01 & 6.593-05 &1.88e-01 & 7.85e-06\\ \hline
$\Delta$  & 4.57e-05 & 3.07e-04 &  1.14e-03 & 1.02e-01 &2.973-04 & 1.42e-01 & \textbf{9.44e-06}\\ \hline
$\Gamma$ & 2.99e-03 & 3.65e-03 & 5.49e-03 & 1.00e+00 &1.813-03& 5.04e-03 & \textbf{1.52e-03}\\ \hline
$\Theta$   & 8.14e-03 & \textbf{1.90e-03} &  5.84e-03 & 9.96e-01 & 3.763-03 & 2.74e-02 & 2.03e-03\\ \hline

\multicolumn{8}{c}{$(S_1, S_2 )=(60, 20)$} \\ \hline
Price          & 6.10e-06 & 5.09e-05 &  6.08e-05  &  1.48e-04 &  1.66e-06 &  3.75e-03 &  \textbf{2.90e-07} \\ \hline
$\Delta$     & 8.50e-05 &  3.37e-04  & 4.49e-04  &  6.84e-05&  1.68e-04 &  5.05e-03 &  \textbf{2.25e-06}\\ \hline
$\Gamma$ & \textbf{8.41e-02} &  4.82e-01  &  6.48e-01  &  1.00e+00 &  2.63e-01 &  3.75e+00 & 8.42 e-02\\ \hline
$\Theta$    & 1.57e-01 &  5.39e-01  &  7.42e-01  &  1.06e+00 &  4.46e-01 &  4.04e+00 &  \textbf{1.09e-01}\\ \hline

\end{tabular}
\caption{rErrors of activation functions for exchange option. The best results are highlighted in bold face.}
\label{tab:actEx}
\end{table}

Using the same experimental setting used for SDBS in Table~\ref{tab:exSDBSs1s2} with nEpoch 100,000,
we examine the rErrors of estimation with several activation functions.
As can be seen in Table~\ref{tab:actEx}, the accuracy differs substantially from activation function to  activation function.
Sigmoid and softplus show significantly better accuracy than other methods for estimating price and $\Delta$;
but, sigmoid degraded for $(S_1, S_2 )=(20, 60)$.
The relu-like function such as relu, elu, selu, softplus
have given good performance in most applications of NNs.
But, the insufficient smoothness of relu, elu, and selu
degrade the estimation accuracy of the $\Gamma$ and $\Theta$.
These accuracies are exactly proportional to the smoothness of the relu-like functions.
Non-differentiable $C^0$ functions such as relu and selu give the lowest performance,
especially large errors( $\sim$ e-01 and e+00).
Another relu-like $C^1$ function elu gives intermediate performance,
while relu-like infinitely differentiable function softplus shows high performance.
This is because the smoothness of the neural network is determined by the smoothness of the component functions,
and sufficient smoothness of activation functions is an essential factor for training the SDBS which is strongly based on differentiability of NNs.
The accuracy of price estimation by relu-like functions in Eq.~\eqref{eq:reluAct}  is comparable to
that by $C^\infty$ functions in Eq.~\eqref{eq:smoothAct};
for example, elu shows good performance for estimating price comparing with $C^\infty$ functions.
This is because the vanishing gradient problem is reduced for relu-like functions.

The above considerations suggest that relu-like functions are required for
the estimation of accurate price
and $C^\infty$ functions are required for accurate Greeks estimation.
Hence, the intersection of relu-like functions in Eq.~\eqref{eq:reluAct} and  $C^\infty$ functions in Eq.~\eqref{eq:smoothAct}, softplus, gives the best performance for the present work.

\section{Conclusions}
We improve the approximation capability of NNs by making NNs have the differential relationship of the problem we wish to solve. Following this approach, the proposed method learns the Black-Scholes equation of option price.
The sufficient smoothness of activation functions is an essential factor
for the proposed method which relies heavily on the differentiability of NNs;
The softplus activation function is suitable for our method.

The SDBS is trained using more realistic asset price paths generated by SDEs.
Hence, the SDBS can accurately model option price  and
does not suffer from the curse of dimensionality associated with high dimensional FDMs.
Since the SDBS utilizes the exact differential relationship along a single simulation path,
it can easily use parallel and distributed computing.
We can also use this single-path dependency for backward recursion in American option pricing which will be discussed in a separate paper.

Further improvements can be made using sample paths
generated by advanced MC simulation methods such as variance reduction techniques and low-discrepancy sequences.
Although the proposed method was applied to a typical Black-Scholes model of option pricing in this work, it can be modified to suit different models and option types.
In addition, differential neural networks can be applied to solve problems related to partial differential equations such as inverse problems, the Navier-Stokes equation, and the Schr{\"o}dinger equation.

\section*{Appendix}

In this section we derive the calculation formulas of Greeks in Table~\ref{tab:Greeks}.
Although the exchange option has the exact solution, we prepare the MC approximation of Greeks for comparing with other numerical methods. Pathwise derivative estimate (PW) method first differentiates  $e^{-r(T-t)}V(T) $ in Eq.\eqref{eq:mcPrice}
and then expectation is performed \cite{glasserman}, i.e., differentiation and expectation is interchanged.
For the delta of exchange option, $e^{-r(T-t)} [ S_1(T) - S_2(T) ]\mathbbm{1} \{S_1(T) > S_2(T) \}$ at $t=0$ is differentiated with respect to $S_1(0)$,
\begin{align}
\frac{d}{d S_1(0)} e^{-rT} V(T)   &= e^{-rT}  \frac{dS_1(T)}{d S_1(0)} \frac{dV(T)}{d S_1(T)}   \nonumber \\
&=  e^{-rT} \frac{S_1(T)}{S_1(0)}  \mathbbm{1} \{S_1(T) > S_2(T) \} ,\label{eq:deltaEx}
\end{align}
where $S_1 (T)$ is presented in Eq.~\eqref{eq:stock} and  $\mathbbm{1}$ denotes a indicator function.

Similarly to the calculation of delta, the theta of  exchange option is obtained by the expectation of the following,
\begin{align}
& \frac{\partial e^{-r(T-t)} V(T-t)}{\partial t} |_{t=0} =
    \frac{\partial e^{-r(T-t)} }{\partial t} |_{t=0} V(T) + e^{-rT}\frac{\partial V(T-t)}{\partial t} |_{t=0} \nonumber \\
& = r  e^{-r T} V(T) +   e^{-r T}  \frac{\partial [ S_1 (T-t) - S_2 (T-t) ] \mathbbm{1} \{S_1(T-t) > S_2(T-t) \}  }{\partial t} |_{t=0} \nonumber \\
& = e^{-rT} \big( r [S_1(T) - S_2(T) ] \nonumber \\
& \quad - [ S_1(T) ( r - \frac{\sigma_1^2}{2} + \frac{\sigma_1 L_1 Z}{2 \sqrt{T} } ) - S_2(T) ( r - \frac{\sigma_2^2}{2} + \frac{\sigma_2 L_2 Z}{2 \sqrt{T} } ) ]  \big) \mathbbm{1} \{S_1(T) > S_2(T) \} \nonumber \\
& = e^{-rT} [ S_1(T) ( \frac{\sigma_1^2}{2} - \frac{\sigma_1 L_1 Z}{2 \sqrt{T} } )
- S_2(T) ( \frac{\sigma_2^2}{2} - \frac{\sigma_2 L_2 Z}{2 \sqrt{T} } ) ] \mathbbm{1} \{S_1(T) > S_2(T) \}
\label{eq:thetaEx}
\end{align}
where $L_i$ and $Z$ are described in ~\eqref{eq:stock}.

We also use the PW method for pricing basket call option.
The delta of basket option is calculated with $V(T)$ in Eq.\eqref{eq:BasCall}.
The delta is given by the expectation of
\begin{equation}
\frac{d e^{-rT} V(T)}{d S_i (0)} = e^{-rT} \frac{S_i (T)}{S_i (0)}  w_i \mathbbm{1} \{ \sum_{i=1}^{n} w_i S_i(T) > K \}
\end{equation}

The theta of  basket call option is given by the expectation of
\begin{align}
& \frac{\partial e^{-r(T-t)} V(T-t)}{\partial t} |_{t=0} = \nonumber \\
& e^{-rT} [  -rK + \sum_{i=1}^{n} w_i S_i(T) ( \frac{\sigma_i^2}{2} - \frac{\sigma_i L_i Z}{2 \sqrt{T} } ) ]
 \mathbbm{1} \{ \sum_{i=1}^{n} w_i S_i(T) > K \}
\end{align}

For the calculation of the gamma using PW method,  the first derivatives of payoff functions are required to be differentiable functions of the parameter $S_i(0)$.
But, the first derivatives, the delta, are not differentiable for exchange and basket options and
PW cannot be applied for these options.
Likelihood ratio method (LR), an alternate method, assumes
that the assets $S_i$ has a probability density $g_\alpha$
and that $\alpha$ denotes a parameter of this density.
To emphasize that the expectation is computed with respect to $g_\alpha$,
$E_\alpha$ is used.
LR interchange the order of differentiation and integration to derive a Greeks
\begin{align}
& \frac{d}{d\alpha}  E_\alpha [V(T)] = \int_{R^n} V(T) \frac{d}{d\alpha} g_\alpha (x) dx  \nonumber \\
& = \int_{R^n} V(T) \frac{\dot g_\alpha(x)}{g_\alpha(x)} g_\alpha (x) dx = E_\alpha [V(T)\frac{\dot g_\alpha(x)}{g_\alpha(x)}]
\end{align}

In the present work, Eq.\eqref{eq:stock} suggests that  $S_i(T)$ for $i=1,\ldots,n$, has the distribution $\exp(Y_i)$.
The vector $Y=(Y_1, Y_2, \ldots, Y_n)^\prime $ follows the normal distribution
$N(\mu (\alpha), \tilde \Sigma (\alpha))$,
where $\mu (\alpha;\cdot)$ denotes a vector with elements
$\mu_i = \ln(S_i(0)) +(r-\frac{1}{2} \sigma_i^2)T$,
$\tilde \Sigma (\alpha) = T A\Sigma A^\prime$
($A=diag(\sigma_1, \ldots, \sigma_n)$), and $\alpha = (S_1(0), \ldots, S_n(0))$.
The probability density function of $Y$
could be written by
\begin{equation}
g_\alpha (Y) = \frac{1}{\sqrt{(2\pi)^n | \tilde \Sigma (\alpha) | }}
\exp [ - \frac{1}{2 } (y-\mu (\alpha))^{\prime} \tilde \Sigma (\alpha)^{-1} (y-\mu (\alpha))].
\label{eq:multiN}
\end{equation}
We generate the sample paths $Y$ by
$\mu + \sqrt{T}ALZ$, where $LL^\prime=\Sigma$, and $Z$ denotes a standard normal distribution.
Hence,
\begin{align}
\frac{\dot g_{S_i(0)}}{g_{S_i(0)}} &= \frac{d}{d S_i(0)} \log(g_{S_i(0)}(Y) )
= (Y-\mu)^{\prime} \tilde \Sigma^{-1} \frac{d\mu}{dS_i(0)}
= (\sqrt{T} A L Z)^{\prime}   \tilde\Sigma^{-1}  \frac{d\mu}{dS_i(0)} \nonumber \\
&= (\sqrt{T} A L Z)^{\prime}   (T A\Sigma A^\prime)^{-1}  (0, \cdots, \frac{1}{S_i(0)}, \cdots, 0)^\prime
= \frac{ ( Z^{\prime} L^{-1} A^{-1} )_i }{\sqrt{T} S_i(0)}
\end{align}
where $ ( Z^{\prime} L^{-1} A^{-1} )_i$ denotes the $i$th component of the row vector
 $Z^{\prime} L^{-1} A^{-1}$.

For calculating gamma in the present study, LR method is first applied to the delta calculation
and this delta is differentiated and averaged by the PW method.
This LR-PW method for gamma calculation gives superior performance
compared with the pure LR method in \cite{glasserman}.

For the exchange option, the gamma is given by the expectation of the following expression
\begin{align}
\frac{d }{dS_1(0) }
& \big( e^{-rT} \max [ S_1(T) - S_2(T), 0 ] \frac{ ( Z^{\prime} L^{-1} A^{-1} )_1 }{\sqrt{T} S_1(0)} \big) \nonumber \\
&=e^{-rT} \big( \frac{d \max [ S_1(T) - S_2(T), 0 ] }{dS_1(0) } \big) \frac{ ( Z^{\prime} L^{-1} A^{-1} )_1 }{\sqrt{T} S_1(0)}
\nonumber \\
& \quad + e^{-rT} \max [ S_1(T) - S_2(T), 0 ] \frac{d  }{dS_1(0) } \big( \frac{ ( Z^{\prime} L^{-1} A^{-1} )_1 }{\sqrt{T} S_1(0)}  \big) \nonumber \\
&=e^{-rT}  \mathbbm{1} \{(S_1(T) > S_2(T) \frac{ ( Z^{\prime} L^{-1} A^{-1} )_1 }{\sqrt{T} S_1(0)}
\nonumber \\
& \quad + e^{-rT} \max [ S_1(T) - S_2(T), 0 ]   \frac{ ( Z^{\prime} L^{-1} A^{-1} )_1 }{-\sqrt{T} S_1^2(0)}  \nonumber \\
&=  e^{-rT}  \frac{ ( Z^{\prime} L^{-1} A^{-1} )_1 }{\sqrt{T} }  \frac{ S_2 (T) }{ S_1^2(0)}
 \mathbbm{1} \{(S_1(T) > S_2(T) \}.\label{eq:gammaEx}
\end{align}

For the basket call option, the gamma is given by the expectation of the following expression
\begin{align}
&\frac{d }{dS_i(0) } [ e^{-rT} \max( \sum_{i=1}^{n} w_i S_i(T) - K,0)
\frac{ ( Z^{\prime} L^{-1} A^{-1} )_i }{\sqrt{T} S_i(0)} ] \nonumber \\
&= e^{-rT}  \frac{d }{dS_i(0) } [ \max( \sum_{i=1}^{n} w_i S_i(T) - K,0) ]
\frac{ ( Z^{\prime} L^{-1} A^{-1} )_i }{\sqrt{T} S_i(0)} \nonumber \\
&\quad +  e^{-rT} \max( \sum_{i=1}^{n} w_i S_i(T) - K,0)
\frac{d }{dS_i(0) } [ \frac{ ( Z^{\prime} L^{-1} A^{-1} )_i }{\sqrt{T} S_i(0)} ] \nonumber \\
&= e^{-rT} w_i  \mathbbm{1} \{ \sum_{i=1}^{n} w_i S_i(T) > K  \}
\frac{ ( Z^{\prime} L^{-1} A^{-1} )_i }{\sqrt{T} S_i(0)} \nonumber \\
&\quad +  e^{-rT} \max( \sum_{i=1}^{n} w_i S_i(T) - K,0)
 [ \frac{ ( Z^{\prime} L^{-1} A^{-1} )_i }{ -\sqrt{T} S_i^2(0)} ] \nonumber \\
&=  e^{-rT}  \frac{ ( Z^{\prime} L^{-1} A^{-1} )_i }{\sqrt{T} S_i^2(0) }
\{ w_i S_i(T) - \sum_{i=1}^{n} w_i S_i(T) + K \}
 \mathbbm{1} \{ \sum_{i=1}^{n} w_i S_i(T) > K \}.\label{eq:gammaBa}
\end{align}

redpajama_set_name: RedPajamaArXiv


Thousands of North East children are risking their lives every night by charging their phone next to their bed, according to research by Electrical Safety First.
The organisation's study has found three in four children in our region have a potentially fatal fire hazard in their bedroom, whether it be a charging mobile or fairy lights.
In our region, it also discovered more than half of young people admit to charging their phone next to their bed, meanwhile one in three leave it under their pillow.
The findings also show children have on average 10 electrical items in their bedroom.
The study also shows parents are even more likely than their children to take risks with electrics with over eight in 10 admitting they take risks.
For example, 41% of parents have used or purchased a cheap unbranded charger.


text:
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PRE-EST
The magnetic cradles of solar tornadoes
What are solar tornadoes? Read the new post in our #TheScienceofEST series by Dr. Robertus Erdelyi, from the University of Sheffield (UK).
Vortex flows in the atmosphere of the Sun, often referred to as solar tornadoes, have attracted the imagination of scientists in the past few decades. These massive energetic phenomena are observed with a unique combination of ground-based telescopes, like the European Solar Telescope in the near future, and space-borne facilities, for example NASA's Solar Dynamics Observatory (NASA Sun Science).
These rapidly swirling magnetic structures are similar to the bathtub effect but, here, solar plasma locked to the magnetic field lines goes upwards from the solar surface into higher regions of the atmosphere. The rotating speeds could be even a few 100 km/s. Their size could be the size of the entire Earth or more. The related braiding and twisting of the magnetic field lines contribute to the heating of the solar chromosphere and corona up to a few million degrees. There, energy is carried in the form of magnetic waves that are often referred to as Alfvén waves. For their prediction, a Noble prize was awarded in the 1970s.
For more details, see https://www.sheffield.ac.uk/…/super-space-tornadoes-magneti… and http://www.solartornado.info
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text:
Tag Archives: inuksuit
The Essential John Luther Adams
Posted on March 29, 2018 by Michael Schell
by Michael Schell
Did you miss Second Inversion's John Luther Adams Marathon on March 28? Are you interested in exploring the music of America's most famous ecologist-composer by sampling a few key pieces? If so, check out this selection of JLA's most indispensable albums to date.
Earth and the Great Weather
If you're ever remanded to a desert island where you can take along a single John Luther Adams album, this is the one to pick. Subtitled A Sonic Geography of the Arctic, this ten-movement composition from 1993 was Adams' breakout piece. It's both an ecological oratorio of the far North and a compendium of the techniques that Adams would hone over the next 25 years: haunting drones and trills, ritualistic taiko-like drumming, and overtone-based textures inspired by his teacher James Tenney (compare the latter's Shimmer to this album's track Pointed Mountains Scattered All Around). It even has some things you don't find in other Adams pieces, such as Alaska nature recordings and texts from Native Alaskan languages
This is another fine sampler album from 1993 that features three medium-length pieces for large ensemble. Dream in White on White is a plaintive work for strings and harp reminiscent of Stravinsky's Orpheus. The early choral composition Night Peace openly displays its debt to Feldman's Rothko Chapel. The Far Country of Sleep begins with a solo trumpet motif that's almost identical to Strauss' Also sprach Zarathustra, but as this orchestral piece progresses, it makes clear that its philosophical affinity is with Rachel Carson rather than Nietzsche.
Inuksuit
This outdoor piece for multiple percussionists has been performed all over North America (including here in Seattle in 2015). Adams considers this recording, three years in the making and captured on location in rural Vermont, to be a definitive representation.
Become Ocean
And here it is: Seattle Symphony's Grammy Award-winning recording of Adams' Pulitzer Prize-winning piece. Released in 2014, it's the first recording of Adams' music by a major orchestra. Although the sound world of Become Ocean isn't all that far from Ravel's daybreak scene in Daphnis et Chloé, Adams' instinct as an ecologist is to let his textural soundscape unfold on its own terms and at its own pace, with a minimum of intervention. Indeed, this work is so well proportioned that it seems much shorter than its 42-minute duration. Become Ocean is both a fulfillment of the trajectory of Adams' work since Earth and the Great Weather and a searchlight illuminating the wonders yet to come from this imaginative composer.
The Seattle Symphony presents the world premiere of John Luther Adams' Become Desert on Thursday, March 29 and Saturday, March 31. For tickets and additional information, please click here.
A Spotify version of our Essential JLA playlist is available below:
Posted in Announcements, Concerts, Reviews, West Coast | Tagged Become Desert, become ocean, contemporary classical, Earth and the Great Weather, inuksuit, john luther adams, new music, seattle symphony, The Far Country, world premiere | Leave a reply
Music in the American Wild: Looking to the Future
Posted on August 24, 2016 by maestrobeats
by Seth Tompkins
This is the final installment in a series covering Music in the American Wild. Our earlier posts include a series preview and concert review.
Music in the American Wild at Hurricane Ridge Visitors Center. Photo Credit: Geoff Sheil
Following the completion of the Music in the American Wild tour in celebration of the centennial of the National Park Service, I took some time to consider the state of the interaction between music, the parks, and wild places in general. Inspiringly, there are many projects happening now that explore this terrain. The interaction of wild spaces and music is a topic which many people both in the United States and around the world are eager to explore.
In addition to Music in the American Wild, another group recently completed an entirely separate tour that brought new music to the national parks in celebration of the centennial. The Grand Valley State University New Music Ensemble's tour visited Badlands, Wind Cave, Grand Teton, and Yellowstone National Parks. This tour concluded on July 9, 2016. They also commissioned new music for the centennial, presenting eight new works alongside three previously commissioned works from a 2014 tour of national parks of the Southwest.
David Biedenbender's Red Vesper, a commission from the 2014 tour
Also like Music in the American Wild, the GVSU tour was supported by a grant from the National Endowment for the Arts. The fact that there were at least two separate new music ensembles touring the national parks and celebrating the centennial with newly commissioned works is outstanding! However, beyond recognizing projects like these, we must address the deeper meaning of what they are trying to accomplish. The effect of music in wild places is defined by two axes: the interaction of music and venue, and the content of the music itself.
When we consider music in the context of wild places, there is a spectrum of ways in which music and location interact. At the most basic, some projects are little more than outdoor concerts, with music (usually) written for indoor spaces presented outdoors. Moving further along the spectrum, some outdoor concerts include music that was written with the outdoors in mind, inspired by the outdoors, or even specifically written to be performed outdoors. Some projects go a bit further and curate music for the specific space in which it will be performed, whether written with outdoor performance in mind or not. Further along are projects that include music written specifically for the outdoors, sometimes combined with aforementioned curation of music for specific venues. Finally, there are other projects written for exactly the specific outdoor venue (and sometimes, time) at which they are performed.
Heart o' the Hills Campground. Photo Credit: Geoff Sheil
Music in the American Wild falls squarely in the middle of this spectrum, with a robust, if not completely essential, connection to their specific performance locations. The music that they presented was certainly written with the outdoors in mind and was designed to be performed outdoors (indoors, too, I suspect). However, not all of their pieces were designed for or inspired by specific places, and not all of their pieces took an apparent interest in interacting with the location in which they were to be performed. The group did, however, take the care to specially curate the programs of each of their performances so that the music they offered complemented each different performance space. The project was successful, to be sure, but the interaction of music with wild spaces shouldn't be limited to the form it took in Music in the American Wild.
Music in the American Wild in the Hoh Rain Forest. Photo Credit: Geoff Sheil
Consider a piece like John Luther Adams' Inuksuit, in which the music intentionally acts like an assistive device, helping the audience experience the performance location in a new way. The music is not written for any specific outdoor space, but is intended to be performed outdoors and to deepen the experience of the space. This overtly intentional interaction between music and outdoor performance spaces illustrates one way concepts from Music in the American Wild and similar projects could be extended.
Inuksuit was performed in Seattle in September, 2015. Photo Credit: Melanie Voytovich
Another piece that illustrates how music and location can be even more deeply connected is Michael Gordon's collaborative piece Natural History, which was premiered on July 29, 2016 at Crater Lake in Oregon and commissioned by the Britt Music and Arts Festival. This piece was written specifically to be performed at Crater Lake, making both location and music essential to the project. Further, Natural History's collaborative nature extends its connection to the performance location; the piece involved local musicians, especially focusing on performers from the Klamath Tribes, for whom Crater Lake has always been a special spiritual place.
Crater Lake. Photo Credit: Britt Festivals
The totality of connections to place in Natural History invites consideration of the role of content in projects that include the interaction of music and wild spaces. Natural History was conceived with a keen awareness of Native American issues and culture. Other pieces, like many of John Luther Adams' works, are centered on the issues of climate change and ecology. Pieces like these, which openly confront and explore the serious issues facing our wild spaces, are leading the way as musicians become more interested in and adept at exploring the intersection of music and nature.
On this front, one piece from Music in the American Wild's set list deserves special recognition. Aaron Travers' piece Sanctuary, inspired by the armed occupation of the Malheur National Wildlife Refuge earlier this year, takes a clear-eyed look at a difficult issue that is relevant to modern audiences.
Perhaps, as time moves on, more projects like Music in the American Wild will delve even deeper in search of connections to place. Maybe they will attempt to explore some of the modern challenges surrounding our national parks and wild spaces. The issues of conservation, Native American history, and land use are as relevant now as they have ever been. Exploring these sometimes-unpleasant facets of our National Parks and wild places is good for everyone involved; musicians get to participate in relevant modern conversation on topics that truly matter, and the public gets a new set of tools with which to connect with wilderness and consider the issues.
American Camp at San Juan National Historical Park. Photo Credit: Geoff Sheil
I have immensely enjoyed covering the Music in the American Wild ensemble this year. I would like to thank them for all of their hard work and their openness in presenting these enjoyable concerts, and for visiting the beautiful (and sometimes overlooked) National Parks of Washington. They deserve praise for contributing to the movement to connect music with our national parks and wild places. Projects like this not only bring music to new and exciting places, they deepen the public's awareness of wild places and can foster much-needed conversations on the issues surrounding them. I sincerely hope that the project continues into the future and grows even broader in scope. Until next time, I suggest we all go "take a hike!"
Seth Tompkins, mid-hike.
Posted in Concerts, On Demand Clips, Reviews | Tagged aaron travers, climate change, crater lake, david biedenbender, ecology, grand valley state university, gvsu, gvsu new music ensemble, hurricane ridge, inuksuit, john luther adams, maw, michael gordon, music and nature, music in the american wild, national park service, national parks, native american history, natural history, new music ensemble, nps, olympic national park, sanctuary, site specific music, take a hike, washington national parks | Leave a reply
CONCERT REVIEW: Inuksuit at Seward Park Amphitheater
Posted on September 24, 2015 by maestrobeats
by Roger Downey
Inuksuit was performed on Saturday, September 19 at the Seward Park Amphitheater, organized by Melanie Voytovich. Photo credit: Seth Tompkins.
John Luther Adams says he wrote his Inuksuit for "9 to 99 percussionists." That sounds as if it's supposed to be epic in scale, and it is, in ambition. When you hear the commercial recording, with just 30-odd players, that feeling is confirmed. But Adams is thinking in terms of multiples of musicians, and 99 players would probably need more than nine times the footprint of just nine.
So the main shock of encountering Inuksuit live, on a sunny-cloudy, coolish-warmish Saturday afternoon in Seattle was the discovery that the piece is chamber music; that at its thundering climax about a half hour in iit's just as calm and transparent as at its amplified breathing opening and its shimmering-triangles conclusion.
Every review of earlier performances round the world mentions how Adams' slowly shifting soundscape captures the random sounds of its surroundings. That bare statement doesn't convey the way it bends those sounds and enfolds them, makes them seem as appropriate to their moments as the sounds of the written score.
Other composers are surely studying this piece, learning from it how to tear down the material concert hall built into their preconceptions and replace it with a virtual cathedral, where the winds are the walls and the sky Shelly's dome of multicolored glass, "staining the white radiance of Eternity."
Posted in Reviews | Tagged free concerts, inuksuit, john luther adams, Melanie Voytovich, seward park amphitheatre | Leave a reply


The Department of Community Development will be moving its offices to the Senior Center lcoated at 6 Crawfords Corner Road. Community Development includes: Construction/Building, Zoning, Fire Prevention, Housing, Code Enforcement and Planning.
Mailing addresses shall remain 4 Crawfords Corner Road.
Employee phone #s and email addresses will remain the same.
Thank you for your cooperation while we make efforts to improve our service to the community.
Our goal is to attain compliance with Township ordinances regarding Property Maintenance, enforcement and abatement to improve the Quality of Life for the entire Township.
Potential violations are discovered through inspector's observations, residential complaints and referrals from other agencies. Once a violation is confirmed to exist a notice of violation will be sent to the property owner given a time frame to abate said violation.