Pricing Convertible Bonds-An Example

In the previous post, we outlined the main steps in pricing a convertible bond using the Binomial Tree approach. In this follow-up post, we provide a hypothetical example of a straight convertible bond.

The specifics of the hypothetical convertible bond are as follows,

Stock price 100
Volatility 0.25
Risk Free Rate 0.02
Risky Rate (risk free+credit spread) 0.08
Coupon 0.06
Maturity (in years) 5
Conversion ratio 10

Using the formula provided in the previous post, we calculated the up and down moves, and probability of the up move. The results are

dt 1.00
u 1.28
d 0.78
Probability Up 0.48

Once the tree parameters are calculated, we next build the tree, and then work backward from the end  nodes in order to obtain the convertible bond’s price at time zero. As the final result, the bond price is $1319 (per $1000 notional)

To see the details of the calculations of the tree, click on the link below to download an Excel spreadsheet.


Note that this is a simplified example. In real life, convertible bonds are usually more complex. They often include features such as call, put, contingency conversion options. The call and put options can be implemented using the formula given in the previous post.

I hope that this and previous posts demystified a little bit the complexities of convertible bonds. Let us know if you have any questions.

Pricing Convertible Bonds and Preferred Shares

A convertible bond (or preferred share) is a hybrid security, part debt and part equity. Its valuation is derived from both the level of interest rates and the price of the underlying equity. Several modeling approaches are available to value these complex hybrid securities such as Binomial Tree, Partial Differential Equation and Monte Carlo simulation. One of the earliest pricing convertible bond approaches was the Binomial Tree model originally developed by Goldman Sachs [1,2] and this model allows for an efficient implementation with high accuracy. The Binomial Tree model is flexible enough to support the implementation of bespoke exotic features such as redemption and conversion by the issuer, lockout periods, conversion and retraction by the share owner etc.

In this post, we will summarize the key steps in pricing convertible bond method using the Binomial Tree approach. Detailed description of the method and examples are provided in references [1,2].

Generally, the value of a convertible bond with embedded features depends on:

  • The underlying common stock price
  • Volatility of the common stock
  • Dividend yield on the common stock
  • The risk free interest rate
  • The credit worthiness of the preferred share issuer

Within the binomial tree framework, the common stock price at each node is described as

pricing convertible bond

where S0 is the stock price at the valuation date; u and d are the up and down jump magnitudes. The superscript j refers to the time step and i to the jump. The up and down moves are calculated as


where is the stock volatility, and  is the time step.

The risk neutral probability of the up move, u, is

and the probability the down move is 1-p

After building a binomial tree for the common stock price, the convertible bond price is then determined by starting at the end of the stock price tree where the payoff is known with certainty and going backward until the time zero (valuation date). At each node, Pj,i  the value of the convertible is

where m denotes the conversion ratio.

If the bond is callable, the payoff at each node is

The payoff of a putable bond is

Here C and P are the call and put values respectively; r denotes the risk-free rate.

The above equations are the key algorithms in the binomial tree approach. However, there are several considerations that should be addressed due to the complexities of the derivative features

  • Credit spreads (credit risk) of the issuers which usually are not constant.
  • Interest rates can be stochastic.
  • Discount rate ri,j depends on the conversion probability at each node. This is due to the fact that when the common share price is well below the strike, the preferred share behaves like a corporate bond and hence we need to discount with a risky curve. If the share is well above the strike then the preferred behaves like a common stock and the riskless curve need to be used.
  • The notice period: the issuer tends to call the bond if the stock price is far enough above the conversion price such that a move below it is unlikely during the notice period. For most accurate results, the valuation would require a call adjustment factor. This factor is empirical and its value could be determined by calibration to stock historical data.

This approach in pricing convertible bond can be implemented in scripting languages such as VBA and Matlab. In the next installment, we will provide a concrete example of pricing a convertible bond. If you have a convertible bond that you want us to use as example, send it to us.


[1] Valuing Convertible Bonds as Derivatives, Quantitative Strategies Research Notes, Goldman Sachs, November 1994.

[2] Pricing Convertible Bonds, Kevin B. Connolly, Wiley, 1998.



Brazil Beats Germany

The World Cup is over and Germany won the much coveted trophy. This World Cup will be remembered for its beautiful attacking-style games as well as one of the most crushing defeats in football history: the Host lost 1-7 to Germany in the semi final.

However, the Brazilians do not have to wait for another 4 years in order to have a chance to revenge against Germany. Their stock market has already beaten the German’s one. The chart below shows the ratio of the Brazil ETF (EWZ) v.s. Germany ETF (EWG) and we can see that the ratio is in an uptrend.

Statistically, the Brazilian market has underperformed over the last 3 years. It lost -29%, while the German market gained 45%. However, EWZ started bouncing at the beginning of this year from an oversold condition.

The bounce is also supported by fundamental factors: Brazil is an exporter of commodities and the uptrend in base metals lends support to its market recovery. Additionally, the upcoming Summer Olympics in Rio in 2016 will give a boost to the local economy.

Germany, on the other hand, is suffering from problems in Europe:   deflation threat, a weak euro, negative interest rates, geopolitical tension in Ukraine and Russia, and a possible bank failure in Portugal, just to name a few. This is a good opportunity for pair traders who want to take advantage of the divergence.

Volatility of Gold Pairs

It is commonly believed that commodity pairs are relatively easy to trade because their underlying stocks are pegged to a certain commodity market. Sometimes, however, this is not the case.

Gold stock pairs have been difficult to trade lately. One of the economic reasons is that as the gold spot declines, it approaches the production cost of around $1200 per ounce, and a small change in the spot would induce a big change (in percentage terms) in the profit margin of the producer. In other words, a small change in the spot would make a larger impact on the company’s profit and loss, thus causing a bigger fluctuation in the stock price. The big fluctuation magnifies the fundamental discrepancies inherent in the stocks of the pair. Consequently, deviations from the norm are likely due to more fundamental than statistical reasons. For quantitative traders who rely solely on statistics to make decision, it has been more difficult trading gold pairs profitably.

We can look at this problem from the options pricing theoretic point of view.  The average production cost of $1200 per ounce can be considered a put option strike. If gold spot deeps below $1200, then the stock is considered in the money. Since late 2012 the “options” are near at the money, and gold stocks behave more or less like ATM options that have greater gamma risks.

In the chart below, the solid black line shows the ratio of two gold stocks, AngloGold Ashanti (AU) v.s. Harmony Gold Mining (HMY). As it can be seen, starting August 2012 (marked with the blue vertical line), the declining in the stock prices started accelerating, and the oscillation in the pair ratio started increasing accordingly. Put it differently, the gamma has caused a greater oscillation in the pair ratio.

In the future, if gold spot trends up, the “options” will get out of the money, and the gamma risk will decrease. In this case gold stock pairs are expected to behave more regularly, thus providing better trade opportunities for statistical arbitrage traders.

Brazil, Emerging Markets and the World Cup

There are now less than 2 weeks until the start of the 2014 FIFA World Cup of Soccer, which is the biggest sport event in the world. The event is being organized in Brazil. From an economic point of view, Brazil is one of the BRIC countries; it has underperformed the overall emerging market during the last 4-5 years.  The chart below shows the relative strength of Brazil ETF (EWZ) with respect to the emerging market ETF (EEM).

The ratio has been in a down trend for more than 4 years. We can observe, however, a rebound taking place in early 2014. Some analysts said that this rebound is supported in part by the preparation of the 2014 World Cup of Soccer and the 2016 summer Olympics.

Interestingly, Brazil is the favorite for winning the World Cup this year. It has the highest chance of winning the World Cup, followed by Argentina, Germany and Spain.

The implied probability of winning calculated from the various bookmakers odds is in the range of 20%-25%. A World Cup win can boost consumer confidence and hence the local economy in general. (We saw a similar situation before in 1998 when France won its first ever World Cup at home).

To play a potential recovery in Brazil, one can go long EWZ and hedge the downside with EMM.  If you worry about the negative impact of the host nation’s not winning the World Cup, you can hedge by laying against Brazil on a sport exchange.

Good luck and enjoy the Game!

Why Do Arbitrage Opportunities Still Exist?

A question arbitrageurs are frequently asked is “why aren’t the pricing inefficiencies arbitraged away?”  This is a very legitimate question.

I believe that in some areas of trading and investment, the number of arbitrage opportunity is diminishing. Take, for example, statistical arbitrage; its profitability is decreasing due to the increasing popularity of the method, competition among traders and advancement in information technology. In other areas of trading, opportunities still exist and persist. For example, in option trading, the volatility risk premium seems to persist despite the fact that it has become widely known.  Here are some possible explanations for the persistence of the volatility risk premium:

  • Due to regulatory pressures, banks have to meet Value-at-Risk requirements and prevent shortfalls. Therefore, they buy out-of-money puts, or OTC variance swaps to hedge the tail risks.
  • Asset management firms that want to guarantee a minimum performance and maintain a good Sharpe ratio must buy protective puts.
  • The favorite long-shot bias plays a role in inflating the prices of the puts.
  • There might be some utility effects that the traditional option pricing models are not capable of taking into account.
  • There are difficulties in implementing and executing an investment strategy that exploits the volatility risk premium and that is at the same time within the limits of margin requirements and drawdown tolerance.

We believe, however, that with a good understanding of the sources of cheapness and expensiveness of volatility, a sensible trading plan can be worked out to exploit the volatility risk premium within reasonable risk limits. We love to hear your suggestion.

Current Implied Volatility is Cheap

The phenomenon of the favorite long-shot bias (or volatility risk premium) can be exploited in order to construct a profitable options trading strategy. Basically, such a strategy would consist of selling overpriced options and hedging the risks using cheaper (or more precisely less overpriced) options. As can be seen from the graph below, the implied volatility (IV, yellow line) is generally higher than the historical volatility (HV, blue line). However, as of late, IV deepened below HV, which is rather unusual. A premium selling strategy in this kind of volatility environment might still profitable. However, the risks will likely outweigh the rewards.

The cheapness of IV is also confirmed by the volatility cone. As can be seen below, at the money IV, (depicted in white diamonds) is below the current HV (depicted by the white line).

A long volatility trade will have a higher probability of success in this kind of market.

The Favorite Long-shot Bias in Option Markets

The favorite long-shot bias is a phenomenon that has been studied extensively in gambling markets. A question has arisen naturally: does this bias exist in financial markets?

In a paper entitled “The Favorite /Long-shot Bias in S&P 500 and FTSE 100 Index Futures Options: The Return to Bets and the Cost of Insurance”, Hodges, Tompkins and Ziemba examined whether the favorite/long-shot bias that has been found in gambling markets (particularly in horse racing markets) applies to options markets. The motivation came from the fact that puts and calls on stock index futures represent leveraged short or long positions on the index and their behavior might have similar features to racetrack bets.

The authors found that both call and put options on the S&P 500 (and FTSE 100 to some degree) with one and three months to expiration display a relationship between probabilities of finishing in the money and expected returns that is very similar to the favorite long-shot bias in horse racing markets. In another word, they found evidence that was consistent with the hypothesis that investors tend to overpay for put options as the expected cost of insurance and call options as low-cost, high-payoff gambles.

This finding is consistent with results published by other researchers under the general framework of the volatility risk premium which refers to the fact that implied volatility in equity index options often tends to overstate realized volatility. In fact, the favorite/long-shot bias provides an alternative explanation for the existence of volatility risk premium.

For practitioners, a question arose: how to structure a position that takes advantage of the favorite long-shot bias?

Another BRIC in the Wall

A good reward/risk trade is a one where fundamentals and technicals are aligned. We have seen two fundamentally similar countries (Canada/Australia) but they did not make a good pair for short-term trading. We have also seen two seemingly different economies (Australia/Indonesia) but that made a good pair.

There exist, however, some pairs that have good technical and fundamental relationships. India (INDL) and Russia (RUSL) are two emerging markets; they are part of the BRIC countries. The ratio chart (upper panel, below) exhibits a regular oscillating pattern, albeit somewhat volatile. The backtested equity line (lower panel) is, however, orderly upward.

Backtest results showed a winning percentage of 88% and a profit factor of 1.86, so this is a good pair to trade. As a bonus, these stocks are leveraged ETFs, hence the pair is also suitable for intra-day scalping.


Quantitative Poker Player

If there were an equivalent of a quantitative trader in the poker world, then Chris Ferguson would be a perfect example. Chris earned a PhD in computer science in the late nineties and went on to become a professional poker player and gambling and lottery industry consultant.  He has won five World Series of Poker events, including the 2000 WSOP Main Event, and the 2008 NBC National Heads-Up Poker Championship. His playing style is highly mathematical as he uses a strong knowledge of game theory and has developed computer simulations to improve his understanding of the game.  Last but not least, Chris also tried his hands at day trading.

I watched a video of him being interviewed by Michael Covel, who is known as one of the original Turtles. Although the video is short, it successfully brings across the following important points:

  • Play with the best players and learn from them.
  • Spend more time away from the tables doing analysis, thus developing an analytical thought process.
  • Do not focus on last hands’ results. Winning or losing does not matter in the short run. Focus on the process and long-term results.
  • Put faith in mathematics. Try to make decisions less emotional as possible and mathematical as much as possible.
  • Study the game hard. Analyze the game from mathematical stand points.
  • Practice and practice and practice.

Although Chris discussed these points in order to educate aspiring professional poker players, they are definitely important lessons for quantitative traders as well.