In this financial engineering research, we study the behaviour of an option premium of a call/put option which is embedded in a typical fixed coupon bond with finite maturity. The contribution of the research is the conclusion about the dynamics of premium changes; represented by direction and sensitivity; with respect to the changes in credit rating and also risk-free interest rate development. The aim of the research is also to clearly demonstrate this theoretically complicated topic to the financial practitioners using a practical example. We are about to consider a 3-dimensional process where the dimensions are: time, rating development process and risk-free interest rate development. We use Standard & Poor’s rating transition matrix to create rating tree and Hull-White model for modelling of risk-free interest rate development. We add embedded call/put option to the bond structure and assume the call/put option to be exercised in case of interest rates decline/rise or rating worsening/improvement. For valuation, we use the risk-neutral concept. Using a numerical solution on the 3-dimensional tree (implemented in MATLAB), we avoid problems that appear while analytical solving of partial differential equations.

A bond is probably the most traditional and important investment instrument with regard to the volume and liquidity of the transactions. Portfolios of banks and insurance companies contain a significant percentage of this instrument. In comparison to a stock, for example, it is not a simple product, moreover in case if it contains embedded options. Issuers frequently have the right to buy back a certain amount of the debt or to repay all the instrument on certain points of time before maturity (Bermuda Style Option in this case). Sometimes there is set a certain protection period for an investor, after the issuance. In this case, we speak about the call option (callable bond) and the issuer pays for the option – the bond is cheaper for an investor. If an analogical option is in the hands of a bondholder we speak about embedded put option. This option can be of considerable value. In some cases, the issuer might even have the right to pay back a bond at any time, i.e., the bond contains an American Style Prepayment Option.

The main contribution of the research is the conclusion about the dynamics of an option premium value changes; represented by the direction and sensitivity; with respect to the changes of credit rating and also risk-free interest rate development. We are about to consider a 3-dimensional process where the dimensions are: time, rating development process and interest rates development. The research contains an example of practical usage and appropriate 3-dimensional charts. For practical valuation, we use a fixed coupon bond with finite maturity, Standard & Poor’s rating transition matrix and Hull-White model for risk-free rate development. We add embedded call/put option to the bond structure and assume the call/put option to be exercised in case of interest rates decline/rise or rating worsening/improvement. All the presumptions have financial logic. Using a numerical solution on the 3-dimensional tree (implemented in MATLAB), we avoid problems that appear while analytical solving of partial differential equations.

When a bank or insurance company deal with the risks resulting from interest rate options (and also with other simple options), they usually calibrate an interest rate model to certain liquid instruments of the interest rate market, price the options using such model, and use the model output to manage the options’ risks (

Similar research of dynamics using a multifactorial approach, applied to credit default linked instruments, was done by

This research is quite generic and applicable to any type of bond. The default is modelled in a reduced form approach by explicitly modelling the credit rating of an issuer. A further advantage of this model is that it is based almost entirely on risk parameters that are already available in banks’ or insurance companies’ risk management systems and, therefore, hardly any model calibration is necessary. Modelling prepayment, optimality conditions are derived endogenously from the future level of interest rates and the debtor’s rating at prepayment times. Other embedded options like caps and floors of loans with floating interest rates or combinations of caps with prepayment rights can be included easily into this model. Finally, if a bank decides to hedge some of its exposure to embedded options with market instruments like European swaptions the model can be used for calculating the appropriate hedge ratios. However, as the model is not complete, these hedges will not be perfect. It will be shown how techniques from credit risk modelling can be used to quantify the risk of losses when hedging embedded options in loans on the portfolio level. We finally remark that a similar approach to modelling prepayment endogenously as in this article was suggested by

We have focused so far on modelling the stochastic structure of the default event by an intensity using rating transitions, for example,

With a numerical solution on the 3-dimensional tree, using computational finance methodology (implemented in MATLAB), we avoid problems to obtain an analytical solution of partial differential equations. Nowadays, computational finance methods allow demonstrating such solutions while analytical solutions do not exist in many cases as argue

By the way of example, we evaluate typical coupon bond with 30 years to maturity which is callable (case 1) and putable (case 2) and taking credit rating and risk-free rate development into the consideration. Coupon of the bond is the same as the initial risk-free rate, thus the initial price must be 100% at best rating (AAA). Price of the risk-free bond (rated AAA), with the coupon, equalled current risk-free rate, is in Figure

Price of risk-free bond with the coupon equalled current risk-free rate (source: own illustration)

We are about to consider a 3-dimensional process where the dimensions are:

1. dimension – time

2. dimension – risk-free interest rates development

3. dimension – rating development process

We use a combination of 2 trinomial trees. The first one is the tree of credit rating development and the second one is the tree of risk-free rate development.

The tree is more intuitive than a PDE solver. The trinomial tree (Figure

The trinomial tree (source: own illustration)

The nine probabilities sum to one. The risk-neutral probabilities and the short rate levels of the tree may be calibrated to the funding discount curve and a set of European swaptions.

This ensures that the pricing of basic market instruments is done correctly. For details on the calibration process, we refer to

We note that in practical applications one would rather use a PDE solver instead of the trinomial tree because of its superior convergence properties (

The sequence of random developments which is possible to observe is for the illustration in Figure

The sequence of random developments (simulation on the tree) (source: own illustration)

The initial price (required evaluation) is obtained using the calculations from right to left on the tree, or in other words, from the future to the purchase day (Figure

The principle of the calculation (source: own illustration)

The price

As we have stated we recognize 9 ways from each node (Figure

We use Standard & Poor’s rating transition matrix which is in figure 1 in the Appendix. We denote ratings “AAA – D” by numbers “1–18”. So we recognize 18 credit rating states.

Transition matrix could be displayed as in Figure

An example of a rating tree is shown in Figure _{0}; _{1}; … _{n}

Rating transition chart (source: own illustration)

Rating tree (source: own illustration)

The probability that a debtor in rating grade _{i}_{i}

Appropriate distribution based on Standard & Poor’s rating transition matrix is in Figure

Simulated path of credit rating development (source: own illustration)

Credit rating distribution (source: own illustration)

For the modelling of risk-free rate development, we assume Hull-White model using parameters given from the US market. Hull-White tree is constructed using MATLAB implementation which is also providing probability values of transition between rates (Figure

Hull-White trinomial tree for modelling of interest rates development (source: MathWorks)

We adopt certain presumptions regarding exercising of option which are reasonable from the point of view of a bondholder and also of a bond issuer.

The call option is exercised in case the interest rate falls below the initial interest rate. The issuer has the chance to issue a new bond at lower interest costs. The similar presumption is the case of rating improvement. The issuer may issue the bond at the lower rate. Analogically the put option may be used by a bondholder in the case of the increase of interest rate (the bond price decreases below strike which is 100%) and worsening of rating which also decreases the price.

Both the call and put options are of Bermuda style and may be used at the end of each year.

The final price of the option is calculated as the difference of the bond without and with the embedded option. It is also observable from tab 1 in the Appendix. In figure 10 a) there is the recovery rate set to 0%, in figure 10 b) the recovery rate is set to the empirical value of 35%.

Price of the bond, as an underlying asset, with respect to credit rating is in figure 10 a), b). Its value, of course, falls with worsening of credit rating. From figure 10 it is also clear that the sensitivity of the bond price with respect to credit rating changes generally increases with the rating worsening – the decreasing price/rating curve is steeper in the area of lower interest rate. This property could be well explained by higher price sensitivity in the area of lower interest rates while yield to maturity changes are caused by rating changes. For better imagination, we provide two figures.

Adequate YTM of bond with respect to credit rating is in Figure

Price of bond with respect to credit rating, a) recovery rate = 0%, b) recovery rate = 35% (source: own illustration)

YTM of bond with respect to credit rating (source: own illustration)

3d chart of a price of the embedded put option is in Figure

3d chart of the embedded call option is in Figure

Differences of option premium values with respect to rating changes are in the Figures

Price of embedded put option (source: own illustration)

Price of embedded call option (source: own illustration)

Call option premium sensitivity with respect to the development of credit risk (source: own illustration)

Put option premium sensitivity with respect to the development of credit risk (source: own illustration)

The main contribution of this financial engineering research are the conclusions about the dynamics of change (means the direction and value of change) of option premium of an option which is embedded in a typical fixed coupon bond with respect to the credit rating changes and risk-free rate changes. We have studied both callable and putable bonds.

From practical example, we have obtained 2 main findings in the research. Both findings we may intuitively feel, but in the research, we have demonstrated them by numerical calculation and quantification.

The first finding is that the value of option premium of embedded call/put option increases with the worsening of credit rating. It could be well explained by the higher volatility of the underlying asset price in the area of worse rating. Such property is very well observable in figure 10 a), b). This finding very well corresponds to the standard option theory, says that the price of call/put option increases with higher volatility of the underlying asset.

The second finding concludes into the assessment of the sensitivity of option premium value with respect to credit rating changes (or in other words: assessment of changes (differences) of option premium value with respect to the credit rating change). The sensitivity of option premium value with respect to credit rating changes depends on the current situation of a bond, given by credit rating and risk-free interest rate and it is demonstrated in the figures 14, 15. Based on the parameters of the rating transition matrix the sensitivity may not increase continuously; also, the surface (especially in figure 13) is not smooth because of parameters of the rating transition matrix.

This work was supported by the Czech Republic GA 16-21506S and VSE IG102027.

Prices (row) of the bond without option with respect to the credit rating and risk-free interest rate (source: own results)

Rating: | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 |

AAA | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 |

AA+ | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 |

AA | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 |

AA- | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 |

A+ | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 |

A | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 |

A- | 100.2 | 100.2 | 100.2 | 100.2 | 100.1 | 100.1 |

BBB+ | 100.1 | 100.1 | 100.1 | 100.1 | 100.1 | 100.1 |

BBB | 99.8 | 99.8 | 99.8 | 99.8 | 99.8 | 99.8 |

BBB- | 98.7 | 98.8 | 98.9 | 98.9 | 99.0 | 99.0 |

BB+ | 96.0 | 96.2 | 96.3 | 96.5 | 96.6 | 96.8 |

BB | 89.9 | 90.3 | 90.6 | 91.0 | 91.4 | 91.7 |

BB- | 79.2 | 80.0 | 80.7 | 81.4 | 82.0 | 82.6 |

B+ | 64.0 | 65.1 | 66.2 | 67.3 | 68.2 | 69.2 |

B | 46.4 | 47.7 | 49.0 | 50.2 | 51.4 | 52.6 |

B- | 30.0 | 31.2 | 32.4 | 33.6 | 34.7 | 35.8 |

CCC | 12.5 | 13.2 | 14.0 | 14.7 | 15.4 | 16.0 |

D | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Rating transition matrix (source: Standard & Poor’s)

Global Corporate |
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RATING | AAA | AA+ | AA | AA- | A+ | A | A- | BBB+ | BBB | BBB- | BB+ | BB | BB- | B+ | B | B- | CCC/C | D |

AAA | 87.91 | 4.72 | 2.68 | 0.68 | 0.16 | 0.24 | 0.14 | 0.00 | 0.05 | 0.00 | 0.03 | 0.05 | 0.00 | 0.00 | 0.03 | 0.00 | 0.05 | 0.00 |

AA+ | 2.62 | 76.06 | 11.67 | 3.93 | 0.89 | 0.66 | 0.30 | 0.12 | 0.12 | 0.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

AA | 0.47 | 1.32 | 80.64 | 8.01 | 2.89 | 1.41 | 0.43 | 0.42 | 0.14 | 0.09 | 0.05 | 0.04 | 0.02 | 0.00 | 0.00 | 0.02 | 0.05 | 0.02 |

AA- | 0.05 | 0.13 | 4.28 | 76.93 | 10.02 | 2.84 | 0.71 | 0.27 | 0.14 | 0.07 | 0.04 | 0.00 | 0.00 | 0.04 | 0.11 | 0.02 | 0.00 | 0.04 |

A+ | 0.00 | 0.11 | 0.58 | 4.46 | 77.42 | 8.80 | 2.57 | 0.71 | 0.40 | 0.09 | 0.09 | 0.12 | 0.01 | 0.09 | 0.04 | 0.01 | 0.00 | 0.07 |

A | 0.05 | 0.06 | 0.28 | 0.56 | 5.01 | 77.73 | 6.82 | 2.69 | 1.15 | 0.28 | 0.15 | 0.15 | 0.10 | 0.12 | 0.03 | 0.01 | 0.02 | 0.09 |

A- | 0.06 | 0.01 | 0.11 | 0.20 | 0.61 | 6.78 | 75.80 | 7.51 | 2.36 | 0.68 | 0.16 | 0.15 | 0.16 | 0.14 | 0.04 | 0.01 | 0.05 | 0.08 |

BBB+ | 0.00 | 0.01 | 0.07 | 0.09 | 0.31w | 1.05 | 6.93 | 73.19 | 8.85 | 2.01 | 0.47 | 0.40 | 0.17 | 0.26 | 0.15 | 0.02 | 0.10 | 0.16 |

BBB | 0.01 | 0.01 | 0.06 | 0.04 | 0.17 | 0.48 | 1.23 | 7.04 | 74.22 | 6.30 | 1.62 | 0.83 | 0.37 | 0.31 | 0.17 | 0.04 | 0.09 | 0.23 |

BBB- | 0.01 | 0.01 | 0.01 | 0.07 | 0.07 | 0.24 | 0.40 | 1.37 | 8.56 | 71.12 | 5.48 | 2.59 | 1.03 | 0.56 | 0.34 | 0.22 | 0.31 | 0.38 |

BB+ | 0.07 | 0.00 | 0.00 | 0.05 | 0.02 | 0.15 | 0.12 | 0.63 | 2.29 | 11.70 | 62.56 | 6.43 | 3.24 | 1.27 | 0.83 | 0.19 | 0.51 | 0.56 |

BB | 0.00 | 0.00 | 0.06 | 0.02 | 0.00 | 0.10 | 0.08 | 0.23 | 0.74 | 2.56 | 8.51 | 64.26 | 7.74 | 2.69 | 1.37 | 0.46 | 0.74 | 0.80 |

BB- | 0.00 | 0.00 | 0.00 | 0.01 | 0.01 | 0.01 | 0.07 | 0.13 | 0.30 | 0.48 | 2.06 | 8.23 | 63.76 | 8.43 | 3.06 | 0.97 | 0.91 | 1.31 |

B+ | 0.00 | 0.01 | 0.00 | 0.04 | 0.00 | 0.04 | 0.09 | 0.06 | 0.07 | 0.10 | 0.34 | 1.57 | 6.92 | 65.02 | 7.66 | 2.62 | 1.96 | 2.62 |

B | 0.00 | 0.00 | 0.02 | 0.02 | 0.00 | 0.09 | 0.07 | 0.04 | 0.11 | 0.04 | 0.23 | 0.39 | 1.69 | 8.39 | 57.67 | 7.95 | 5.42 | 5.90 |

B- | 0.00 | 0.00 | 0.00 | 0.00 | 0.04 | 0.07 | 0.00 | 0.14 | 0.07 | 0.14 | 0.18 | 0.21 | 0.61 | 3.13 | 10.22 | 51.30 | 10.82 | 9.15 |

CCC/C | 0.00 | 0.00 | 0.00 | 0.00 | 0.05 | 0.00 | 0.14 | 0.09 | 0.09 | 0.09 | 0.05 | 0.23 | 0.56 | 1.39 | 2.91 | 8.70 | 43.80 | 27.43 |