Ito Lemma Dan Ekspansi Taylor Series

Agak mirip. Yang satu untuk stochastic differential equation (Ito), yang lain untuk Ordinary Differential Equation.

Why so much fuss about Taylor Series Expansion?
Team latte
Oct 20, 2005

Taylor Series expansion is one of the most, if not the most, applied mathematical concept in financial engineering. It is a simple expansion series of any function around a small increment .

Where are higher order terms (greater than the power of three). If we ignore
the higher order terms as well as the third order term then we can write:

And in the limiting case when we have

If you carefully look at the above equation then we will see a striking resemblance
to the famous Ito’s stochastic equation, which is the basis of Black-Scholes option
pricing partial differential equation (PDE). The only difference is that in Ito’s
equation we have the term  instead of   , that is in an Ito process we have

The above equivalence has a great intuitive meaning with respect of geometric Brownian motion and asset price movement, but that is not the purpose of this article. What we wanted to show was that even without any knowledge of stochastic calculus and the corresponding Ito process one can approach the Black-Scholes option pricing PDE if one were to make the accommodation shown above. It also shows that any asset price movement, such as the movement of a bond, can be approximated by the Taylor series expansion and regardless of whether we know the exact closed form solution for the pricing of that asset, it is possible,

using first principles of differential calculus to approximate the movement of the asset over small increments of state space and time.

In keeping with our arguments above let us show you that the Taylor series expansion can be used to calculate the value of any variable with a great degree of precision and speed, which may not be possible numerically.

Say, you wanted to calculate the value of , that is eleven raised to the power of eight. Doing this manually could take you forever. Of course, you can do it in less than a second using a scientific calculator or an Excel spreadsheet. But suppose you don’t have an Excel spreadsheet (or a calculator near you) and you wanted a quick approximation of this number.

You can say that and that a small increment of   is one and therefore,
and . First we calculate the value of which is very
straightforward and simple: 100,000,000. We need to use the Taylor series to
calculate the change in value of this number around the point . (Remember
by making x = 10, we have made all our calculations very simple and that is where
the usefulness of Taylor series lies)

Now using Taylor series for calculation we get:

Thus the change in the value of the function f which is given by is:

And the value of the function is approximated as:

This is quite good an approximation given the fact that if you were to use your Excel spreadsheet then for 11^8 (eleven raised to the power of eight) you would get 214,358,881.

Barrier Option

Pernah dengar barrier option? Agak beda dengan option yang konvensional. Valuasinya juga tentunya beda dari sekedar menggunakan formula Black-Scholes. Berikut artikel mengenai produk ini.

A Probability Conundrum: First Passage time of a Brownian Motion
Rahul Bhattacharya
Jan 25, 2005

Recently we encountered a bit of a problem with the use of first passage time in our forecasting models. The problem appeared more fundamental in terms of the mathematical formulation. I asked a trader who trades currency options in a large bank in Hong Kong as to what is the probability that an asset, say a currency or a stock, which is currently trading at $100 with a historical volatility of 15% and a mean historical return of 10% will hit a barrier at $95. It took him ten seconds to plug it in his option pricing calculator the values and he gave me an answer of 57.3% (assuming the implied vol for 3 months is 15%).

I am sure this is correct given the fact that he had calculated the probability of hitting the barrier by using the pricing model of a one touch digital option. However, when we use a different formula for the probability of the first passage time, as the problem is known, in asset forecasting models we get a different result for the probability of the asset hitting the barrier.

First passage time is a central concept in the analysis of Brownian motions with absorbing barriers. Consider a Brownian particle undergoing a one-dimensional random walk within a domain between x=0 and x=L, the boundaries being such that the particle will be reflected at x=0 and absorbed at x=L. If the particle starts from some position within the domain when will it hit the absorbing boundary for the first time? This first-passage problem in stochastic processes, in various spatial dimensions and with a variety of geometries and boundary conditions, is of longstanding interest and has ramifications and applications in diverse areas.

In simple English, First Passage Time refers to the probability of a random particle hitting and being absorbed by a barrier or a boundary. This barrier or a boundary could be the barrier of an asset price as in a barrier option, or a predetermined band such as a government determined peg rate of a currency.

Mathematically, the probability that an asset following a geometric Brownian motion will hit a barrier is given by:

Probability that an asset following a geometric Brownian motion will hit a barrier

Let us assume that an asset, for example a stock or a stock index, follows a geometric Brownian motion. The stock is currently at $100, the annualized volatility of the stock is 15% and the mean historical return of the stock is 10%. Then the probability that in three months time the stock will hit a barrier at $95 will be given by the above formula and is simply equal to 38.46%.

This could explain the difference – fairly large – between our result and the result that my friend got from his digital option pricing formula. Later on he used Monte Carlo simulation and yet he came up with a probability number much greater than 40%.

We believe that the formula mentioned above gives the correct probability of a stock hitting a barrier provided it follows a geometric Brownian motion with a drift equal to the mean return. This could be viewed as the real probability of hitting the barrier as opposed to a risk neutral probability measure.

They Tried to Outsmart Wall Street

Emanuel Derman expected to feel a letdown when he left particle physics for a job on Wall Street in 1985.

Nicole Bengiveno/The New York Times

“Nobody ever took these models to be playing chess with God.” — Emanuel Derman

After all, for almost 20 years, as a graduate student at Columbia and a postdoctoral fellow at institutions like Oxford and the University of Colorado, he had been a spear carrier in the quest to unify the forces of nature and establish the elusive and Einsteinian “theory of everything,” hobnobbing with Nobel laureates and other distinguished thinkers. How could managing money compare?

But the letdown never happened. Instead he fell in love with a corner of finance that dealt with stock options.

“Options theory is kind of deep in some way. It was very elegant; it had the quality of physics,” Dr. Derman explained recently with a tinge of wistfulness, sitting in his office at Columbia, where he is now a professor of finance and a risk management consultant with Prisma Capital Partners. Continue reading

October Pain Was ‘Black Swan’ Gain

For most of October, it seemed nearly everything that could go wrong with the markets did. But the rout turned into a jackpot for author and investor Nassim Nicholas Taleb.

Mr. Taleb last year published “The Black Swan,” a best-selling book about the impact of extreme events on the world and the financial markets. He also helped start a hedge fund, Universa Investments L.P., which bases many of its strategies on themes in the book, including how to reap big rewards in a sharp market downturn. Like October’s.

[Nassim Nicholas Taleb]

Nassim Nicholas Taleb

Separate funds in Universa’s so-called Black Swan Protection Protocol were up by a range of 65% to 115% in October, according to a person close to the fund. “We’re discovering the fragility of the financial system,” said Mr. Taleb, who says he expects market volatility to continue as more hedge funds run into trouble.

A professor of mathematical finance at New York University, Mr. Taleb believes investors often ignore the risk of extreme moves in the market, especially when times are good and volatility is low, as it was for several years leading up to the current turmoil. “Black swan” alludes to the belief, once widespread, that all swans are white — a notion that was proven false when European explorers discovered black swans in Australia. A black-swan event is something that is highly unexpected.

Assets under management at Universa have neared $2 billion since the fund launched early last year with $300 million under management. While Mr. Taleb frequently consults with Universa’s traders, the Santa Monica, Calif., fund is owned and managed by Mark Spitznagel, who worked for several years in the 1990s as a pit trader on the Chicago Board of Trade.

[Mark Spitznagel]

Mark Spitznagel Continue reading

Strategi Opsi (Options) Saat VIX – Implied Volatility Diatas 70%

Akhir2 ini Dow Jones, S&P ataupun Nasdaq bergerak sangat fluktuatif. Akibatnya, implied volatility (IV) naik drastis ke lebih dari 70%. Biasanya, disituasi yang demikian, jika anda baca buku2 options, kebanyakan akan merekomendasikan strategi selling options: positive theta, negative gamma atau negative vega. untuk memanfaatkan IV yang tinggi tersebut. Masalahnya, walapun IV tinggi, pergerakkan harganya pun cukup dahsyat. Akibatnya banyak yang psikologinya akan goncang akibat adanya serangan bertubi-tubi dari gamma dan delta. Kalau anda kuat, bagus. Kalau tidak, akan sering cut-loss disaat yang tidak tepat.

Disituasi yang sulit ini, kalau anda kurang kuat mental tradingnya, sebaiknya menggunakan strategi yang long atau short delta dengan vega, gamma ataupun theta yang kecil. Atau dengan kata lain, gunakan options dengan resiko terbatas. Tentunya, semakin kecil resikonya, semakin kecil pula expected returnnya.

There is no free lunch.

Just my 2 cents.


Harga Minyak, APBN dan Hedging

Baca buku-buku mengenai ‘Peak Oil Theory’ cukup meyakinkan saya bahwa harga minyak tinggi tampaknya tidak dapat dihindarkan.  Usd 150 sampai usd 200 pr barrel bukan lagi angka yang muluk. Oleh karena itu, ada baiknya pemerintah untuk mulai mempertimbangkan untuk hedging atas kemungkin melambung terusnya harga minyak di pasar dunia.  Cara terbaik adalah dengan menggunakan oil futures market.

Beberapa perusahaan penerbangan dunia sudah mulai melakukan hedging harga minyak sejak beberapa tahun yang lalu. Hasilnya, untuk 2 – 3 tahun kedepan, rata-rata harga minyak mereka hanya sekitar USD 55 per barrel! 

Memang benar untuk melakukan hedging perlu biaya yang tidak sedikit. Akan tetapi, biaya ini tidak akan seberapa dibandingkan potensi biaya yang akan timbul jika harga minyak mencapai USD 200 per barell, baik itu biaya ekonomi maupun sosial (economic and social unrest).

Tanpa hedging, negara kita akan terus terombang-ambing oleh gejolak di pasar minyak. Jika harga minyak terus naik, APBN kita akan tambah berdarah-darah. Tidak ada salahnya bagi kita untuk mengurangi pendarahan ini. Cara tercepat dan terbaik untuk memberhentikan pendarahan ini adalah dengan melakukan hedging. Baru, setelah itu, fokus dapat dialihkan untuk mencari solusi jangka panjang untuk mengatasi masalah permintaan (penggunaan) BBM yang relatif tinggi ini.

Tapi, sekali lagi, dalam jangka pendek ini, we have to stop the bleeding now. Karena kalau tidak, seperti yang Keynes katakan: In the long run, we are all dead.

Harga minyak memang masih ada kemungkinan turun lagi. Akan tetapi, menurut peak oil theory, penurunannya tidak akan seberapa. Potensi kenaikkan harga masih jauh lebih mungkin daripada potensi penurunan harga. Atau, dengan kata lain, potensi upside vs downside tidak sebanding. Jadi, hedge!


Just a thought.


Delta Neutral Strategi Yang Jitu?

Di dunia trading options, tidak ada strategi yang paling jitu. Yang ada adalah strategi yang paling cocok dengan toleransi risiko atau temperamen anda. Setiap strategi options mengandung kekuatan maupun kelemahan.  Tidak ada strategi yang selalu untung di situasi pasar yang berubah. Kunci sukses di options trading adalah dengan menerapkan strategi yang cocok dengan situasi pasar. Masalahnya, situasi pasar berubah terus secara kontinyu. Akibatnya, banyak traders yang rugi karena menerapkan strategi yang salah. Tidak ada yang namanya “Instant Expert” di options trading.  Perlu pengalaman yang memadai untuk suskes.  Kalaupun anda untung banyak walaupun baru bermain sesaat di options market, faktor keberuntungan mungkin lebih berperan dalam hal ini. Tunggu beberapa bulan lagi, kemungkinan besar hasilnya akan berubah.

Rekan saya bingung, kenapa dia kalah terus. Strategi yang dia terapkan – “Delta Neutral” – seharusnya untung karena secara teori, kalau pasar bergerak, posisi delta neutral dia akan untung. Pasar sudah bergerak, tapi kenapa dia masih rugi?

Inilah contoh bagaimana “imperfect understanding” atau pengetahuan parsial (hanya sebagian) dari seorang trader bisa berakibat buruk. Rekan saya, yang sebenarnya cukup sophisticated, seorang fund manager, lupa bahwa trading options adalah trading di multi dimensi. Bukan trading di satu dimensi saja (harga), seperti trading saham. Di options, anda secara tidak langsung trading di : harga, waktu, volatilitas dan suku bunga. Posisi dia mungkin saja delta neutral (harga netral), akan tetapi bisa saya pastikan, posisi dia tidak netral secara volatilitas, waktu maupun suku bunga. Akibatnya, pergerakan harga yang dia harapkan akan menguntungkan posisinya (posisi dia delta neutral, long gamma), ter offset oleh time decay dan  volatilitas yang menurun. Jadinya, rugi! Bahasa teknisnya, posisi dia: delta neutral, lomg gamma, short theta dan long vega.

Moral of the story: hati-hati trading options.  Tidak semudah yang anda bayangkan seperti yang diberikan di seminar-seminar options yang kian menjamur. Kalau pengetahuan anda tanggung-tanggung saja, kemungkinan besar anda akan rugi dalam jangka panjangnya.

Just my 2 cents.