The Price Theory of Bond Stock Swaps

The Price Theory of Bond Stock Swaps

1. The use of the credit default option and the stock option
(1) The points of the precondition
　As a way to derive a price theory of bond stock swap, a credit default option dealings and a stock option dealings are applied. A precondition for these two dealings, are set first and compared..
　The points of the precondition are as follows:
　(1) When an object company defaults, a similar cash flow occurs.
　(2) When an object company does not default, cash flow does not occur.
　(3) Both values are made equal by satisfying two conditions above.
Furthermore, it is assumed that this precondition does not include stock dividends for simplification.

(2) Setting of a precondition
　For simplification, let me set a concrete business example as a precondition.
Business examples（① and ② are the same company.）
①The credit default option
　　Notional： 20,000,000,000 yen
　　Period： One year
　　Default time recovery： The amount of 40% (8,000,000,000 yen)
　　Default time settlement:12,000,000,000yen (20,000,000,000 yen -8,000,000,000yen)
　　Premium： 600,000,000 yen

②The stock option
Type： An American-type put option
　　Stock prices： 120 yen
　　Number of put stocks： 100,000,000 stocks 　
　　Period： One year　
　　Strike price： 120 yen (At The Money)
　　Knock in price： 1 yen (Assuming default time)
　　Default time settlement ： 12,000,000,000 yen
　　　　　　　　　　　　　　　（11,900,000,000 yen to be exact）
　　 Premium： 600,000,000 yen
When a default occurred within one year, for business examples ① and ②, the amount of settlement (cash flow) became almost the same amount. (an error of less than 1%) Since the business example ② is an American-type, the option is exercised when the knock in price becomes 1 yen. (it is default time). In addition, cash flow does not occur when a default does not occur. Therefore, the premium that is the business value becomes the same amount theoretically.

(3) Equilibrium point of default probability
　Default probability is examined here.
　For example, in the case of selling an option, the amount of payment becomes 12,000,000,000 yen at default, and since the premium of 600,000,000 yen is received in return, the default probability is as follows.

The default probability =600,000,000 yen /12,000,000,000 yen =5%（If the default payment is adjusted to its present value, the default probability rises.）

Because the object company of both examples is the same, naturally default probability of both becomes equal. The next step is to derive a formula for a price theory by finding an equilibrium point of default probability of both from this precondition.

2. A relational expression of the credit default option and the stock option
(1) Setting of a precondition
　The business example of a precondition is generalized to set a precondition.
Business example（① and ② are the same company.）
①The credit default option 　
　　Notional： D
　　Period：Td
　　Default time recovery： The amount of R %　
　　Default time settlement： D ( 1 - R %)
　Premium： Pd

②The stock option
Type： An American-type put option
　　Stock prices： E
　　Number of the put stocks： X
　　Period： Te
　　Strike price： E (At The Money)
　　Knock in price： 1 yen (I assume default time)
　　Default time settlement ： EX
　　Premium： Pe
　
(2) Deriving the equation
The equation for both cases are derived by finding an equilibrium point of default probability.
　From the above default probability, following equality holds, if Td = Te.

　Default probability = Pd / D (１- R %)= Pe / EX
　EX = D (１- R%) Pe / Pd

The amount of the stock (EX) becomes equal to the amount of the default option notional except that the recovery at default time theoretically multiplied by the premium ratio (Pe / Pd). Since the market price gets closer to the theoretical price due to arbitrage, if the market price deviates apart from the theoretical price, E, Pd, Pe will vary to get closer to the theoretical price while affecting each other.

(3) Consideration about a premium
　In the case of a credit derivative, credit default swap is generally performed. Credit default option Pd is found by using a credit default swap. The total swap interests are discounted to their present worth, and this present worth is Pd. Generally the credit risk premium is added to the market interest rate to discount the present worth.　The theoretical value of the premium Pe of an American-type stock option with knock in price, can be calculated using the binomial model. Like Black Sholes model, it is calculated from 6 parameters, strike price, period, stock price, stock price interest rate, short-term interest rate, and volatility. A major variable factor of premium Pe is volatility which is the only unknown quantity to calculate the premium. Volatility is calculated by the average rate of change for the daily stock prices. That is standard deviation of statistics. And, future expected volatility is called implied volatility. By guessing from historical volatility, which was calculated by the average of a rate of change for the daily stock prices of the past, implied volatility market is formed and option transaction is performed.

3. The price theory of bond stock swap
(1)The credit default option notional = The amount of a bond
As for the credit default option notional, an fictitious bond of the price of 100 yen (par) is assumed, and an optional price based on the supposition is shown. One of the settlement methods at default time is "Physical Settlement", and the bond goods settlement of the same default option notional are performed. （To be precise, unpaid accrued interest is also added.）Therefore, the credit default option notional is replaced with the amount of a bond.
Actually, the price of both sometimes does not agree completely. There is an article of the proof analysis that pays attention to the price difference of both. As its reason, I can suppose that there are some factors such as the supply and demand relations of a market or a default article which are peculiar to a credit derivative. However, the price difference of both is a narrow margin particularly in the recent market, so I think it is not an important matter in this article.

(2)The equation of a switch from a bond to a stock
　The equation which switches a bond to a stock is derived.

　EX / D = (１- R% )× Pe / Pd
　Remarks ① D is the amount of a bond
　　　　 ② Each variable shall satisfy the above-mentioned precondition. 　

　When market price differs apart from a theoretical price, the equation above will not be formed, and arbitrage is enabled. Major variable factors, E, Pd, and Pe will vary towards an equilibrium point and will converge to a theoretical price.
　In this equation, the only unknown quantity is recovery rate R%, and all the other variables can be specified. Because there are various models about a recovery, and there are many proof analyses, it is necessary to specify the most suitable model, and it is also necessary to choose a strong model for an error when we specify a recovery.

A relational expression of the Nikkei Stock Average and a bond price (interest rate)

(1)A theoretical value to exchange a bond value with the Nikkei Stock Average value is calculated by applying the price theory of Bond Stock Swap.
A specific calculation method based on a business example is described in the following.

Business example
The Nikkei average option　 Type： American-type put option
Stock price： E
Number of the put stocks： X
Period： Te (year)
Strike price： E (At The Money)
Knock in price： 10 yen (default point)
Default settlement amount： (E-10)*X
Premium： Pe

Regarding the business example, we assume such business transactions as Nikkei 225 price index of stocks options listed on the Osaka Securities Exchange. Although there is no actual business transaction with knock in feature, it is assumed to be newly implemented. In the business example, we assume that a fictitious stock of Nikkei average stock defaults at the point when the stock price falls down to 10 yen. Since the business example is an American-type, the option is exercised when the knock in price becomes 10 yen (it is default time). In the mean time, the default point may be set freely at one’s estimation.
Now, we examine the default probability within the period of Te years. For example, in the case of selling a put option, the amount of payment becomes (E-10)*X at default, and since the premium of Pe is received in return, the default probability is as follows, Default probability = Pe / (E -10 )× X.
(If the default payment is adjusted to its present value, the default probability rises.）
The theoretical value of the premium Pe of an American-type stock option with knock in price can be calculated using the binomial model like in the case of individual stock option business.

(2)Let us assume a fictitious bond, Nikkei average bond, next.
Although the interest rate of the bond represents the default probability, it is necessary to consider the default loan recovery rate. A business example is shown and explained as follows;

Business example
The Nikkei average bond
Price： D
Period：Td (year)
Default recovery rate： R %
Default payment： D ( 1 - R %)
Premium： Pd　

In the above example, the premium Pd is used for descriptive purposes. The premium Pd should be interpreted as the total bond interest discounted to the present worth. Generally the credit risk premium is added to the market interest rate to discount the present worth.
In this case, since the amount of default payment is D(1-R%) and the premium of Pd is received in return, the default probability is as follows.
　Default probability = Pd / D(1 - R %）
From the above default probability, following equality holds, if Td = Te, .
　Pe / (E -10 )× X = Pd / D(1 - R %）.

　Regarding the recovery rate R%, it is necessary to specify the most suitable model considering the Nikkei 225 index.
　Regarding the premium Pd, because there is no such bond as the Nikkei average, bond interest and Pd do not exist either. So, Pd, which is the only unknown quantity is calculated from above equation.

(3)Finally, because there is no such bond as the Nikkei average on the market, we derive the spread between the Nikkei average and some existing commodity such as the Japanese government bond future for actual utilization.

Business example
The Japanese government bond future　 Amount：D
　　　　　　　　　　　　 Price　：Y

The Japanese government bond future is a fictitious bond with 6% interest and a period of ten years. A premium is the difference between the present worth of this fictitious bond and the current price Y. Assuming a premium PS,
　PS+ α= Pd(α is the spread between the two）
　Pe / (E -10 )× X = (PS+ α) / D(１ - R %）.

The Nikkei average and the Japanese government bond future market can be connected by finding the premium Pd of a fictitious Nikkei average bond and creating a market to deal the spreadα of the futures.
As the last remarks, it is obvious that swap business described above is also applicable to foreign markets such as the New York Dow average and U.S. government bond future, although only the example between the Nikkei average and the Japanese government bond future was discussed

The relational expression between the FX (foreign exchange) rate and the bond price (interest rate).

We calculate a theoretical value to exchange the FX rate and the bond price by applying a similar price theory as the Bond Stock Swap.
A specific calculation method based on a business example is given below.

Business example
US dollar / Japanese yen
Type： American-type US dollar put/ Japanese yen call currency option
FX rate： E yen/US dollar
Amount： X US dollar
Period： Te (year)
Strike price： E yen (At The Money)
Knock in price： 10 yen (Default)
Default settlement ： (E-10)*X yen
Premium： Pe yen

In the business example, the US default point is assumed at the point where the FX rate of Japanese yen / US dollar falls down to 10 yen. Since the business example is an American-type, the option is exercised when the knock in price becomes 10 yen (it is default time). In addition, the default point can be set freely by one’s estimation.
Now, we examine the default probability within the period of Te years. For example, in the case of selling an option, the amount of default payment becomes (E-10)*X yen. Since the premium of Pe is received in return, the default probability is as follows,
Default probability = Pe / (E -10 )× X.
(If the default settlement is revised to its present worth, default probability rises.)
This is a model to calculate the country risk from its default probability.
The theoretical value of the premium Pe of FX option which is an American-type with knock in price, can be calculated using the binomial model.

The default probability assuming US government bond is calculated next.
Although the interest rate of a bond represents its default probability, default recovery rate has to be considered. A business example is given and explained below.

Business example
US government bond
Amount： D US dollar
Period：Td (year)
Default recovery rate： The amount of R %
Default settlement： D ( 1 - R %)
Premium： Pd　

In the above example, the premium Pd is used for descriptive purposes. The premium Pd should be interpreted as the total bond interest discounted to the present worth. Generally the credit risk premium is added to the market interest rate to discount the present worth.
In this case, since the amount of default payment is D(1-R%) and the premium of Pd is received in return, the default probability is as follows,
　Default probability = Pd / D(1 - R %）.
From the above default probability, following equality holds, if Td = Te.
　Pe / (E -10 )× X = Pd / D(1 - R %）

In this equation, the only unknown quantity is recovery rate R%, it is necessary to choose the most suitable model for US government.

In this way, the relational expression between the US government bond and the dollar-yen FX rate is derived. Theoretically speaking, US dollar-yen FX market and US government bond market can be connected.
As the last remarks, it is obvious that the relational expression between the Japanese bond and dollar-yen FX rate can also be derived based on the Japanese default probability, although only the US default probability was used in this example. In addition, similar relational expression can be derived for all currencies and countries with a FX market and a bond market.