Another Magic Formula

I am psyched! After months of trying to understand the logic, derivations and theorems that lead to the result today we finally derived the famous Black-Scholes option pricing formula. Weeks ago, I already introduced a one-period derivative pricing model. Well, since then the topic has become a lot more complex and in this post, I will tell about the logic of anther magic formula.

As we did already in the one period case we will start this analysis by assuming that there is no arbitrage in the market. Thus, every contingent claim X(t) with a specific payoff can be replicated. Our goal is to price this contingent claim.

In this case the contingent claim is a European call option with strike price K. The payoff of this option at time is equal to X(T) = max(S(T) – K, 0) as we have seen before already. Additionally, we already know that there are always two ways to price a contingent claim when markets are complete:

  1. Find a strategy that replicates the payoff. The price of the option then equals the cost of setting up the following strategy, where Sx(t) is the price of the contingent claim at time t, θ0(T)B(t) is the position in the risk-free security and θ1(t)S(t) represents the position in the risky security.unbenannt
  2. The expectation at time t under the risk neutral probability of the discounted price at time T equals equals the price at time t as can be seen below, where EQ means expectation under the risk-neutral probability Q, eδ(T-t) is the discount factor between T and t in the continuous time and the information that is known is the information at time t, ft.


With the help of advanced integrals, we can model the development of the stock price as


This formula tells us that the stock price at time t equals the stock price at time 0 times an exponential. The exponential depends on: the instantaneous risk-free rate of interest δ, the volatility σ and the standard Brownian motion W*(t) which corresponds to the part of the return that is totally random and unpredictable. However, W*(t) is distributed according to a normal distribution with mean 0 and variance t (N(0,t)).

So, what do we have so far? We have the equation for the payoff of our call option, we have the condition for the price under the risk neutral probability (we choose this approach to value the option) and we have the development of the stock price. Let’s put all this together. The price of the call option at time 0 (c(0)) can be calculated as follows:



For convenience, we start our valuation at time 0 which makes the equation easier to work with. Next we standardize the standard Brownian motion so that W*(t) becomes sqrt(T)Z where Z is normally distributed with zero mean and variance of 1 (N(0,1)).

Thanks to various Russian and Japanese mathematicians the expression above can be expressed as an integral. Thus, we get


where the last part is the density function of the normal distribution.

Now we are at a point where we should ask us which development actually is of relevance for the price of the call option. Obviously, we would only pay for the option if we think it to be possible that the option will be in the money. Consequently, we only consider the case where S(t) => K. What does this depend on? On Z! The stock price develops exponentially according to different values that Z takes on. However, K is fixed and thus there is a unique value of Z that equalizes K and the stock price as can be seen below.


Thus, the system below can be solved for a unique value of Z.


This is a big step since it enables us to get rid of the max function in the integral. We can re-write the equation


This consists of two parts:

  • There average price of the stock if we exercise the option
  • The average cost of exercising the option (the present value of the strike price times the probability of being in the money)

The difference between these two is the cost of the option. I will not bother you anymore with further derivations but finally the option can be priced as follows:

c(0) = S(0)N(d1) – e-δTKN(d2)


d1 = (ln(S(0)/K) + (δ+0.5σ2)T)/(σT)

d1 = (ln(S(0)/K) + (δ-0.5σ2)T)/(σT)

This is it. This is the famous (magic) formula that allows us to price options.

One side note. I did some research about the normal distribution assumption implied by the formula and found out that indeed it is possible to introduce bimodal models or jumps in the distribution. Thus, what I have writing in my entry about Deutsche Bank regarding possible option pricing inefficiencies might not be true after all.

Wow, we got a lot done here. Is your head exploding as well? But how cool that we just derived one of the most famous and fundamental formulas in Finance!




Demystifying Buffet

We can be as successful as Warren Buffet! This sounds too good to be true. The world´s most famous investor has achieved continuous incredible returns for decades. Both he and his fellow value investors from Graham and Doddsville are often cited when trying to prove the efficient market hypothesis wrong.

But how could Buffet achieve these incredible returns over decades and compound invested capital like this hundreds of times? Is this related to skills, sheer luck or is he just remunerated for exposure to market factors that other investors have not considered yet?

In order to answer this let´s talk about market models, especially multi-factor models. Probably the most famous model to determine security and portfolio returns is the Capital Asset Pricing Model (CAPM). The CAPM relies on the assumption that markets are efficient, that idosyncratic risk of shares can be diversified away and that thus only systematic risk should be remunerated. The proposed relationship is as follows.

E(Ri) – rf = x +ßi(E(Rm) – rf) + ei

where E(Ri) – rf and E(Rm) – rf are the excess returns over the risk free rate (secure U.S government bond for example) of security i and a diversified market portfolio (S&P500 for instance) respectively. The term ß is the crucial component since it measures how exposed the security is to the market movement. It  measures whether market and security are highly, less or even negatively correlated. Accordingly, securities with higher ßs should achieve higher expected excess returns than low ß securities since they pose a higher systematic risk. X finally is the intercept of the regression equation and since markets are supposed to be efficient it is assumed to be 0. We do not consider the error term ei for this analysis. While the CAPM has been proven to be wrong many many times it still is a helpful tool in understanding how return is driven by factors.

This single factor model (with the market being the only model) can further be extended to a multi market model. The probably most famous multi-factor model is the Fama-French multi-factor model which looks as follows:

 E(Ri) – rf = x +ß(E(Rm) – rf) +bs*SMB + bvHML + ei

This three factor models extends the CAPM by the two factors SMB and HML and the respective security´s sensitivity to these factors bs and bv. SMB is a portfolio that consists of a long position in small stocks (measured by market capitalization) and a short position in large stocks. This is included to incorporate the fact that for many decades small stocks did outperform large stocks. However, since the 80s this effect has disappeared either due to arbitrageurs  or due to the fact that  had never existed and was subject to data mining. The HML factor on the other hand measures the performance of value stocks (do you see where Buffet might fit in here?). It is composed by a long position in stocks with high book-to-market equity ratios and a short position in securities with the opposite characteristic. Historically and empirically value stocks outperform growth stocks and that is what HML is to indicate.

Indeed the Fama-French three factor model has been verified empirically.But what about Buffet? Let´s see. In a empirical test a portfolio replicating buffet´s strategy and performance has been created to demystify the great investor´s return. (I am not able here to provide the name of study and data set since the study never has been published.

The first thing that we have to consider is Buffet´s high leverage. Buffet and leverage? The conservative value investor would never use leverage. Everybody knows that leverage is “the fastest way to the poorhouse”. But this is wrong. Buffet indeed has employed considerable leverage over his investment life time. Leverage has been as high as 1.4. This is made possible by Berkshire Hathaway´s consistent income from its insurance business. Generally, the annual cost for the insurance float was 2.2% and thus almost 3% below the average t-bill cost of financing. Consequently, Buffet was able to finance himself cheaply. But doing so he might have incurred risks as well that should be accounted for.

But let´s test the multi factor model for Buffet. The above multi factor model is applied including three additional factors. These are UMD, BAB and QMJ. UMD (up minus down) represents a portfolio with a long position in momentum stocks and a short position in loosing stocks. The rationale behind this is that mementum stocks indeed tend to continue to win while loosing stocks keep on loosing. BAB (betting against beta) measures an anomaly that empirically has been observed. This anomaly refers to the fact that large beta stocks return less than expected under CAPM while low beta stocks obtain higher returns than expected. Finally QMJ (Quality minus Junk) is another value portfolio that is long equity with low price to book-equity  ratios and short stocks with high ratios. Applying this model provides us with  the following results.

rt – rf = 0.063  +0.79MKTt -0.15SMBt +0.46HMLt – 0.05UMDt + 0.29BABt +0.43QMJt + et

I do not include the t-statistics of these estimates but it turns out that the intercept is statistically insignificant. This would indicate that Buffet has no special skill but just was able to invest in the right risk factors of securities and invest in these long before other people found out about them. Regarding the betas only the ones for MTK, HML, BAB and QMJ are statistically significant and positive. This makes rational sense. Buffet invests in value securities and has a strong exposure to the market. Below we can plot the possible return of this portfolio applying Buffet like leverage.


The obtained Buffet like portfolio even performs better than Buffet´s Berskhire Hathaway stock. This seems remarkable.

Of course, one should not make rapid conclusions. Firstly, I for my part strongly believe in Buffet´s investing abilities. Secondly, the model presented above might be subject to data mining issues. For example the QMJ factor seems to be an integral part of the HML factor according to its accounting definition. On the other hand the included factors make sense and at least some truth seems to lie inside this model.

What we can take away from this analysis in my opinion is that understanding factor analysis might be greatly beneficial for investment decisions. Maybe one day we can even be as successful as Warren Buffet.



Donald Makes Shipping Great Again!

Donald makes freight shipping great again! No one knows really why, but the market for the shares of companies in the shipping services industry is crazy at the moment. Even the fact, that the market seems to anticipate future inflationary policy, possibly leading to economic growth (which would strengthen the shipping industry) can´t explain this. The list of recent supernovas is long: DRYS, GLBS, SINO, DCIX, SHIP, ESEA, TOPS, GNK, DAC. These are all companies that have increased their values by 100´s-or even 1000´s of percents.The undisputed king of this irrational exuberance is DRYS which went from less than $10 to over $110 just in a matter of days. Some short sellers got seriously squeezed in this play.  But have a look yourself.



Ouch, not again!

Ouch, it happened again! Well, I recently declared to be completely transparent. I guess then I have to share the trade which I did today (unfortunately). This was another penny stocks day-trade – I just keep on losing on this kind of trades.

So what happened?

I recently have condensed the list of stock promoters that I consider of high quality. Do not get me wrong. These are all very sketchy companies but I have been following the promotion activity for quite some time and some of the promoters seem to generate great returns (I will come back to this). Yesterday, I received the alert that one of these promoters,, was promoting the technology company TWER (Towerstream Corporation ). Since recently had produced incredible results, I decided to put it on my watchlist. This morning I realized that two more promoters, that had performed superbly in recent times, DamnGoodPennyPicks and SmallCapleader, jumped on the wagon and had announced TWER to be their pick. This made it highly likely that much volatility and profit potential would result (very subjective probability indeed). My plan was to buy the stock before the market opened and then sell into a possible morning spike. Seems quite easy, doesn´t it?.

Below you can see the TWER´s chart from 14:00 until 18:00 central European time. The 1.5 hours of pre-market trading took place in the grey part of the chart. What you can see is that the stock had closed at 78 cents on Friday. What happened in the first minutes of trading was a surge up to a high of $1.06 cents before retracing. Just before the market opened at 15:30 the stock was trading at around 98 cents and I decided to buy stock.

The following factors are to consider:

  1. I expected a strong morning spike, possibly breaking through the pre-market high of $1.06 due to retail traders buying stock when the market opened. My plan was to sell within the first 20 minutes into this morning spike.
  2. I entered my position at 98 cents with my full account size. This was necessary due to trading fees. However, fortunately this account is very small and the percentage loss exaggerates the true loss of the trade.


When the market opened there actually was a short battle at the $1 mark before sellers came in and TWER started a continuous downtrend. Well, this only took around 15minutes but seemed to me like 4 hours. At the end of this downtrend the share was trading at 90 cents and held this support steadily. I decided to sell my position if this level would break. Of course all retail traders had put their stops at this round number of 90 cents as well and when this level gave away the stock plunged to 86 cents before rebounding. I ended up getting out of my position at 98 cents.

The facts.

  1. I lost around 9% of my capital in this trade. Including transaction costs the loss is more than 12% (seems incredible doesn´t it).
  2. The stock currently trades at around 92 cents. A strong rally did not occur so far for this promotion.

This was the first time I have traded this account and it showed me that its account size is way too small to trade with realistic expectations. The fees just eat you alive. On the other hand, I am happy that I traded this volatile setup with a small account because obviously there is so much I have to learn about this way of trading.

Some more observations

  1. I keep on loosing on day-trading: Maybe at some point I will have to admit to myself that I should focus on the trading style that is the most profitable for myself. If the Pareto principle holds for trading and 80% of my winners are generated by 20% of my trades then I might better focus on these and cut my loosing strategies (CPLA for example still is rocking). However, I am still learning and it seems to be too early to make this conclusion.
  2. Selection bias: I was so eager to enter the trade because three (!!!) very successful promoters announced TWER to be the pick. Given their recent performance what happened today seems like unbelievable bad luck. Maybe it was not my fault, but I was just unlucky! While it feels good to say something like this it might be appropriate to investigate another thought. Maybe the incredible profit potential of these promoters only has looked so good in the past because I wanted it to look good. I wanted to believe that this trading opportunity exists. When I looked at the promotions I assumed that I would have entered (and got filled) early enough to ride the up move and that I would have sold at the very top (maybe not that realistic).
  3. I need a clear plan: While I did have an approximate plan on how to trade this play it was not clear enough. I had not decided when and under which exact conditions to enter the market and I ended up buying because I did not want to be left out of a possible up move.
  4. This trade humbled me: This trade humbled me incredibly.Just this weekend I thought that this strategy would work wonders and that I could easily multiply my capital within days. I actually wanted to increase my account size in  order to compound it faster. Now I am back to reality.





How to be a Stock Market Genius



You can be a Stock market Genius! Yes, it is true! Well, at least that´s the title of the first book written by Joel Greenblatt, the author of “The Little Book That Beats The Market” – very promising titles indeed. I recently have finished reading this book, loved its style and Greenblatt´s humor and learned many relevant lessons.

The book is build on the premise that you should buy what other people are given but do not want. At the same time you should not try to compete with the world´s financial elite picking stocks. Very, very few investors can successfully do this. But there are special situations that allow exceptional returns and that generally are not followed closely on Wall Street. The performance extract of Greenblatt´s fund Gotham Capital, investing in these kinds of special situations, backs this up. The fund was closed in 1995 due to asset size impeding the fund´s performance.


A 50% average yearly return over a decade with the lowest return being almost positive 30%. That seems very impressive.

So let us look at the investments enabling such astounding returns.

Opportunity Nr. 1: Spin-offs

A spin-off takes place when a division or subsidiary of a company is made independent. Usually, the result is the emergence of a new entity. The motives for spin-offs are manifold: Separation of very diverse or qualitative differing businesses, tax purposes, legal issues and shareholder creation are all factors to be considered. More important however, is the fact that the shares of the spun-off entity and even of the parent-company subsequently tend to outperform the market.

Why do spin-offs tend to do so? When the spin-off occurs shareholders of the parent company usually receive some sort of compensation such as shares of the new entity. It follows however, that the new owners did not actively seek to own the company. Rather they are passively handed the shares. This often results in selling pressure on the newly issued asset because:

  • Owner´s don´t want the stock.
  • Institutional investors don´t want the stock.
  • Even if institutional investors wanted the stock in many cases they would not be allowed to hold it due considerably low market capitalization of the spun-off entity.

These factors create selling pressure in the first several months. Thus, the shares often are severely undervalued before the market discovers the true value.

Additional positive situations to look for in a spin-off are:

  • Insiders do want the stock of the new company. This might occur for example if executives´ compensation depends on stock performance.
  • A unique investment opportunity is created. For instance the spun-off  company is loaded with debt, creating a highly leveraged investment that might multiply its initial valu  manifold.

Opportunity Nr. 2: Merger Securities

Empirically mergers tend to create and destroy value at the same time. They destroy value for the shareholders of the acquiring corporation while increased value for the shareholders of the acquiree results. The condition for this however, is that the deal is successful. There are in general two risks:

  • The risk of the deal not going through. For example a government authority might block the merger.
  • Timing risk.

That´s why the shares of companies to be acquired usually trade below the announced acquisition price. As of today (09.11.2016) the shares of Time Warner trade at $86.5 and thus $21 below the announced acquisition price of $107.5 per share. Funds specialized in merger-arbitrage try to take advantage of these “inefficiencies”, shorting the stock of the acquiring company while buying the stock of the acquiree. But the risks remain and this activity is compared to picking up pennies in front of a steam roller. This is what Greenblatt concludes as well.

There are exceptions however. These exceptions are called merger securities. In the case of merger securities the merger is achieved by  reimbursing shareholders not with cash but completely or partly with merger-securities such as bonds. What follows is the same as for spin-offs. The owner´s do not want to own the securities and institutional investors might not even be allowed to do so (as in the case of low-grade bonds). The resulting selling pressure may considerably undervalue the securities and give you a great investment opportunity.

Opportunity Nr. 3: Bankruptcy and restructuring

A similar argumentation holds for restructurings of bankrupt companies. In this case the companies´ debt holders may be paid with new common stocks or bonds while the old shareholders generally are wiped out. As before the new owners of these securities generally did not intend to be the owners which results in the securities being sold off. Still, investing in bankrupt companies is a risky endeavor. Thus it is important to screen for fundamentally “good companies”. This of course is hard to do. One way is to look for bankrupt leveraged buyouts that have failed due to special market conditions. In leveraged buyouts the executives generally are heavily invested into the company creating a positive incentive structure as Greenblatt points out.

Opportunity Nr. 5: Recapitalization and Stub Stocks and 

A recapitalization is another opportunity we should have on our radar. In this case a company purchases its own stock for bonds, cash or other stock resulting in a highly leveraged entity. Do the owners of these new securities want to be invested in such a heavily leveraged company? Probably not and as you know by now the resulting sell-off can leave you with superb opportunities to take. Stub stocks are the stocks of the newly leveraged company. There are two reasons you should have a look at them. Firstly, if they are partly distributed to the old shareholders the stub stocks may be undervalued due to known reasons. Secondly, buying these shares essentially is like investing in a leveraged buyout. You invest in a highly leveraged corporation with the benefits (and of course risks) associated to leverage. Additionally in this case management tends to be highly incentivized to make the company perform successfully.

These are the main investment opportunities that can make you a stock market genius. Of course, this sounds too easy to be true and in fact successful investing requires disciplined and often hard work. As always successful people distinguish themselves from unsuccessful people by doing the stuff that other people don´t want to do. In this case this includes reading 100´s of pages of legal work in order to understand the often complicated structure of these deals. Additionally, Greenblatt mentions that since you do not want to compete with the big players it is always beneficial to watch for small companies to invest in. While risky, small companies are covered by less analysts what makes their shares more inefficiently priced.

And inefficiency is something we are always looking for.







Today´s my lucky day

Following some traders on Twitter one just gets an incredible perception of the winning percentages of these traders. Some of them actually seem to win in as many as 140% of their trades (Yeah – not really possible. But you can get the impression). This of course stems from the fact that people love to tell us about their successes but kindly do not inform us about their losing trades. Exactly that is what I will do now. As I already indicated, recently pretty much all of my trades have been loosing trades. But instead of letting you guys know about these trades I will talk about my one profitable one – seems  fair doesn´t it? But at least I commit to post short notes about all the trades I will be doing, winning or loosing ones, from now on.

The most recent investment opportunity that I have found is Cappella Education (CPLA) which provides online services in the education area. The company was on my watchlist due to its strong fundamentals. According to the “magic formula” Cappella Education´s fundamentals convinced with both relatively cheap price and a high return on capital. Additionally, earnings developed in a strong way for the last quarters and years as you can see below.


When I first looked at the chart however, the 21 day moving average was below the 50 day moving average indicating a downtrend. Additionally, earnings were to be reported on October 25 and I did not want expose myself to the volatility of the earnings season too early. So, I kept the stock on my radar.

When earnings expectations for the fiscal year of 2016 were reported in the $3.50 to $3.55 range versus expectations of $3.30 to $3.40 this caused the stock to soar from around $62.5 to more than $67.5 in one day. At the same time the 21 days moving average (green) crossed over the 50 day moving average (red) giving me my entry signal. Additionally, the stock broke out of resistance at around $63. Even though in danger of chasing the stock I decided to enter my position at $66.22, accepting a reversal of the crossover as sell signal.


Since then the stock has moved further and is currently trading at around $74.5 (02.11.2016). Even though the spike seems overextended to me and cashing in some profits seems like the right choice I will follow my plan. My sell signal will be the 21 day moving average crossing below the 50 day moving average. Even though this might make me give back my profits I want to give the stock the opportunity to continue its trend.

Probably my winning trade (so far) in this case is just due to luck (after all how likely is it that I lose all the time). However, in this case various indicators seemed to be lining up  perfectly.

These are:

  • Strong fundamentals – cheap stock price with high return on equity
  • A positive trigger in form of favorable earnings report
  • Buying signal triggered by the crossover of 21 and 50 day moving average
  • Breakout of resistance level at around 63$

These could be just made up reasons in order to confirm me being right but we will see how the trade turns out.

Until then…