An Example of Snowball Contract

Given a contract of snowball note issued by Bank SinoPac which par value equals $ 10,000,000, and the contract could be redeemed with par value after the third year. The coupon is paid quarterly and the general form for i-th quarter coupon is ( )Coupon i =(Coupon i( -1)+Spread i( ) -Floating rate i( ))⁺

where the floating rate in this contract is fixing rate of 90 days CP and each coupon rate is illustrated in table 4.1.

Table 4.1 Coupons of Ten Years Snowball Bond.

Cn, i means that coupon rate at i quarter of n year. Notes that the floating interest rate (FR) is the fixing rate of 90 days CP; if i-1=0, Cn, i-1= Cn-1, 4 for n=1…10, i=1..4.

Year Coupon rate (Cn,i )

1 C1,i=3%, i=1,2,3,4

2 C2,i=C2,i-1+1.40%-FR2,i 3 C3,i=C3,i-1+1.65%-FR3,i 4 C4,i=C4,i-1+1.90%-FR4,i 5 C5,i=C5,i-1+2.15%-FR5,i 6 C6,i=C6,i-1+2.40%-FR6,i 7 C7,i=C7,i-1+2.65%-FR7,i 8 C8,i=C8,i-1+2.90%-FR8,i 9 C9,i=C9,i-1+3.15%-FR9,i 10 C10,i=C10,i-1+3.40%-FR10,i 4.2 Simulation and Analysis of Pricing

We will continuously analysis influence of the parameters of Hull-White model, spreads designed and zero rates on snowball price with contract in section 4.1. There are many figures in this section to particularly explain the associations between sensitivity of parameters and snowball price.

4.2.1 Sensitivity to mean reversion of Hull-White model

We will discuss how mean reversion parameter influence snowball price. In Figure 4.1, we can observe that if mean reversion increases, the price of snowball notes decreases as volatility equals 0.006, especially with non-redeemable contract.

Figure 4.1 Snowball Price vs. Mean Reversion.

The volatility is 0.006, par value is $1 and zero rate function is rate (t) =0.02363-0.007314*exp (-1.316*t). There is negative association of price and mean reversion. Because issuers of snowball

notes with non-redeemable contract can not call back the bond to hedge loss when its price move up,

the price with non-redeemable contract is more than with redeemable contract.

There is negative relationship between mean reversion and price in figure 4.1. At low mean reversion, that is to say, the higher and the lower interest rates would not quickly back to long-run average level, so it is possible to maintain low interest rates at low market short rates and get more profit because of inverse rate property on the coupons of snowball contract. Moreover, the discounted factor also rises at low interest rate, so the bond value would increase. Although it is possible to maintain high interest rates at low mean reversion, non-negative coupon contract would protect against price of snowball failing down violently. Only decrement of bond price results from low discounted factor at high interest rate. Hence, there is a negative association between mean reversion and price.

4.2.2 Sensitivity to volatility of short rate

Next, we discuss relationship between volatility of short rate and snowball price.

In figure 4.2, we can observe that if volatility increases as mean reversion equals 0.005, the price of snowball notes increases, especially with non-redeemable contract.

Figure 4.2 Snowball Price vs. Volatility of Short Rate.

The mean reversion is 0.005, par value is $1 and zero rate function is rate (t)

=0.02363-0.007314*exp (-1.316*t). There is positive association of price and volatility. Because

issuers of snowball notes with non-redeemable contract can not call back the bond to hedge loss when

its price move up, the price with non-redeemable contract is more than with redeemable contract.

There is positive relationship between volatilty and price in figure 4.2.At high volatility of Hull-White model, that is to say, the change of interest rate is violent, so it is possible to become high interest rates at low market short rates or low interest rates at high market short rates. Thus, non-negative coupon contract would protect against price of snowball failing down violently at previous high market rates and get profit of coupons at present low market rates because of high volatility. Figure 4.3

shows this phenomenon and could explain the detail clearly.

Figure 4.3 Snowball Price without Redemption Article.

The par value is $1 and zero rate function is rate (t) =0.02363-0.007314*exp (-1.316*t). In all different lines, they could obviously display the positive association between price and volatility.

Resulting from the negative relation between price and mean reversion, the line with bigger mean

reversion moves up slowly, oppositely, the line with smaller mean reversion moves up rapidly.

Combine the influence of mean reversion and volatility of short rate on non-redeemable snowball price in figure 4.3, the relation between these parameters and price consists with results in figure 4.1 and 4.2. With regard to price with redeemable snowball contract, most profit of coupons bond holder get is in the first three year because of issuer redeeming contract to protect the loss from more coupon payments on low market rates. Therefore, the price of redeemable Snowball notes would not move up rapidly than non-redeemable snowball price even volatility increasing and this phenomenon is showed in Figure 4.4.

Figure 4.4 Snowball Price with Redemption Article.

The par value is $1 and zero rate function is rate (t) =0.02363-0.007314*exp (-1.316*t).

Tendencies of all different lines are the same as Figure 4.3. If the price too higher, the issuer would

redeem the contract and this snowball note would be concealed. Therefore, price dose not reach $1 or

more.

Hence, there is a positive association between volatility and price without redemptive article but not obvious in redeemable snowball contract. Furthermore, in non-redeemable condition, if mean reversion is big enough, the negative association of mean reversion and price would eliminate some positive association of volatility and price.

4.2.3 Sensitivity to spread of snowball contract

Moreover, we would discuss how spreads influence snowball price in Figure 4.5.

Figure 4.5(a) Snowball Price with Redemptive Article vs. Spreads.

Figure 4.5(b) Snowball Price without Redemption Article vs. Spreads.

The par value is $1 and zero rate function is rate (t) =0.02363-0.007314*exp (-1.316*t). In graph (a), the price with twice spreads of snowball contract issued by Bank SinoPac in chapter 4.1 is larger than with one and half. However, it is not very distinct than graph (b) because redemptive article could make issuers to hedge loss. Moreover, in graph (b), even the price of non-redeemable snowball contract with half spreads is more than the price of redeemable snowball contract with origin spreads (shot dotted line).

Issuers may lose a lot for higher spreads of snowball contracts. However, the change of price in figure 4.5(a) is not more conspicuous than figure 4.5(b) even the price of non-redeemable snowball contract with half spreads of snowball contract from section 4.1 is more than the price of redemptive article with origin spreads. That is to say, the effective way to hedge snowball price is redemptive article, not how to contract spreads.

Figure 4.6 Snowball Price with Half Spreads vs. Mean Reversion.

The par value is $1 and zero rate function is rate (t) =0.02363-0.007314*exp (-1.316*t). With half spreads of snowball contract issued by Bank SinoPac, the redeemable price in situation of high

mean reversion is low enough for not redeeming the contract, thus non-redeemable price would

converges to redeemable price when mean reversion increases.

In figure 4.6, the negative relationship between price and mean reversion would influence on redeeming contract. In high mean reversion, the snowball price with redemption article may be low enough for not redeeming the contract. Thus, non-redeemable price would converges to redeemable price when mean reversion

increases. If mean reversion is large enough, the snowball price with redemption article maybe equal to non-redeemable price.

4.2.4 Sensitivity to interest rates of zero curves

In this section, we will compare the snowball contract in chapter 4.1 in situations of different zero rates. Figure 4.7 is showed the high zero rate and the low zero rate which we take different parameters in equation (2.27).

Figure 4.7 Zero Curves.

The two lines come from the same equation (2.27) which iszero rate t ( )=a e* ^−bt+c. The parameters of high rate curve (dotted line) is a=-0.05, b=0.18, c=0.08 and low rate curve (real line) is

a=-0.007314, b=1.316, c=0.02363.

As we known, the interest rates would take great effect on coupons because of inverse interest rate property of snowball notes. Figure 4.8 explains that under the same spreads, non-redeemable snowball price at low market rates is more than at high market rates. Moreover, even without redemptive article, price at high market rates is less than with redeemable contract at low rates.

Figure 4.8 Snowball Prices vs. Different Zero Rates.

The prices with different zero rate curves are according to the figure 4.7. Real lines are the price with non-redeemable snowball contract and dotted line is with redeemable snowball contract.

Figure 4.9 Snowball Prices with High Zero Rates vs. Volatility of Short Rate.

The prices in both lines are very low because in high zero rates, most coupons may be reset to zero and decrement of discounted factor makes bond value diminish. The prices don’t increase unless volatility of Hull-White model is big enough.

In Figure 4.9, because in high zero rates, most coupons may be reset to zero and decrement of discounted factor make bond value diminish, the redeemable and non-redeemable snowball price are very low and the price would increase unless volatility is big enough. To conclusion, if the issuers forecast the wrong tendency of interest rate and contract unsuitable spreads, they may be subjected to loss.

4.3 Estimation of parameters

There are two steps for estimating parameters: one is to find the coefficients of zero rate function; the other is to calibrate the mean reversion and volatility in Hull-White model.

4.3.1 Zero rate function

We take Hull-White zero rate equation (2.27) and use curve fitting tool of Matlab toolbox to find the coefficients of term structure function

* ^bt a e⁻ + c

where t means time of year. The observation of zero rates today is in table 4.2 and illustration figure 4.10 shows fitting coefficients.

Table 4.2 An Example of Zero Rates.

The zero rates for ten years from 2006/3/1 could be observed.

Maturity (year)

Zero rates

Maturity (year)

Zero rates

Maturity (year)

Zero rates

Maturity (year)

Zero rates

0.25 1.5160% 2.75 1.9420% 5.25 2.1466% 7.75 2.3572%

0.5 1.5900% 3 1.9678% 5.5 2.1691% 8 2.3744%

0.75 1.6505% 3.25 1.9898% 5.75 2.1918% 8.25 2.3917%

1 1.7115% 3.5 2.0118% 6 2.2145% 8.5 2.4090%

1.25 1.7497% 3.75 2.0339% 6.25 2.2372% 8.75 2.4264%

1.5 1.7880% 4 2.0561% 6.5 2.2601% 9 2.4439%

1.75 1.8264% 4.25 2.0730% 6.75 2.2830% 9.25 2.4614%

2 1.8649% 4.5 2.0900% 7 2.3059% 9.5 2.4790%

2.25 1.8905% 4.75 2.1070% 7.25 2.3230% 9.75 2.4966%

2.5 1.9162% 5 2.1241% 7.5 2.3401% 10 2.5143%

Figure 4.10 A Curve Fitting of Zero Rate Function.

The fitting coefficients are a=-0.01269, b=0.1475, c=0.02761 with 95% confidence bounds where SSE= 3.418e-006, R-square=0.9916.

In order to decrease errors, we use directly the observable zero rate from Table 4.2 to calibrate parameters of Hull-White model in section 4.3.2.

4.3.2 Calibration of mean reversion and volatility

In the first place, as zero rates known in Table 4.2, the forward rate could be calculated by

1 1 1 1

1 1

1

1

(0, ) ( , ) ( , ) (0, )

(0, ) (0, )

( , )

( , )

q q

iq q iq q q q

iq q iq q

q q

q q

r t i F i i t i i

r t i

r t i r t i

F i i

t i i

e e ^{− −} e ^{− −}

− −

−

−

⇒ = −

=

where ^r_{i q} is q-th quarter zero rate of i-th year, ^t(0,i_q⁾ is time from present to q-th quarter of i-th year, and F i(_q−1, )i_q is forward rate from q-1 to q quarter of i-th year.

In the second place, with variable volatilities of caplet on different strike rate observed in the market like table 4.3, we can compute each cap price in table 4.4 by Black’s formulas (2.22)

Table 4.3 Market caplet volatilities.

Each value is percentage of volatilities with different strike rates from 2006/3/1 to 2009/11/27.

Strike rate 1.5% 2.5% 3.5% 4.5%

2006/6/1 8 8 8 8

2006/8/31 8 8 8 8

2006/11/29 8 8 8 8

2007/3/5 8.538594 8.360483 8.321068 8.314355 2007/6/1 9.065219 8.712955 8.635001 8.621724 2007/8/30 9.603813 9.073438 8.956069 8.936078 2007/11/29 10.14839 9.437926 9.280704 9.253926 2008/3/4 11.1357 10.14639 9.837004 9.778085 2008/6/2 12.12301 10.85486 10.3933 10.30224 2008/8/29 13.08838 11.54759 10.93724 10.81475 2008/11/27 14.07569 12.25605 11.49354 11.33891 2009/3/4 14.81617 12.97623 12.06952 11.85515 2009/6/2 15.55665 13.69641 12.6455 12.37139 2009/8/31 16.29713 14.41659 13.22149 12.88762 2009/11/27 17.02116 15.12076 13.78467 13.39239

Table 4.4 (a) Caplet Price from 2006/6/1 to 2009/11/27.

The caplet price from Black’s formula is shown below.

Strike rate 1.5% 2.5% 3.5% 4.5%

2006/6/1 0.000416 0 0 0

2006/8/31 0.000683 3.58E-14 0 0

2006/11/29 0.000989 3.41E-09 0 0

2007/3/5 0.000993 6.95E-08 9.18E-18 0 2007/6/1 0.001176 1.53E-06 1.97E-13 0 2007/8/30 0.001364 1.04E-05 9.90E-11 5.62E-17 2007/11/29 0.001545 3.58E-05 5.69E-09 9.22E-14 2008/3/4 0.001441 4.27E-05 2.61E-08 3.05E-12 2008/6/2 0.00156 8.68E-05 2.53E-07 1.99E-10 2008/8/29 0.001687 0.000149 1.34E-06 4.30E-09 2008/11/27 0.001814 0.000228 4.78E-06 4.38E-08 2009/3/4 0.001802 0.000265 8.96E-06 1.55E-07 2009/6/2 0.001904 0.00035 1.96E-05 6.28E-07 2009/8/31 0.002014 0.000446 3.73E-05 1.97E-06 2009/11/27 0.002124 0.000549 6.33E-05 4.96E-06 Table 4.4 (b) Cap Price of maturities 1,2,3,4 years.

The cap price could be calculated by table 4.4(a); for example, the one year cap of strike rate 1.5% is summation of caplets from 2006/6/1 to 2006/11/29, namely, one year cap of strike rate 1.5%={0.000416+0.000683+0.000989}=0.0020879. All caps are computed by the same way.

Strike rate 1.5% 2.5% 3.5% 4.5%

1 year 0.0020879 3.41E-09 0 0

2 year 0.00716633 4.79E-05 5.79E-09 9.23E-14 3 year 0.0136686 0.000555042 6.41E-06 4.83E-08 4 year 0.0215128 0.00216609 0.000135513 7.76E-06

Nevertheless, we can sum up cap price of different maturity and compare with cap price which pricing caplet as a put option on a zero coupon bond (2.24) by Hull-White equation (2.20).

In order to find adapted mean reversion and volatility of Hull-White model, we Use summation of square error (SSE) method to find the coefficient

2

, ,

1

min min ( )

σ = σ

∑∑

₌^{n ki}− ^ki

a a

k i

SSE U V

{1.5%, 2.5%,3.5%, 4.5%}

k∈

where k is strike rate, n is maturity date, U is the market cap price from Black’s _ik formula and V is the price of cap given by the Hull-White model. _ik

The optimal parameters of mean reversion and volatility are 0.014485 and 0.004596. The cap price from Hull-White model with optimal parameters is showed in table 4.5.

Table 4.5 Cap Prices of maturities 1,2,3,4 years from Hull-White Model.

The caplet is calculated by Hull-White equation and the way to compute cap price is the same as table 4.4(b).

Strike rate 1.5% 2.5% 3.5% 4.5%

1 year 0.002331 3.20E-05 6.50E-09 4.96E-15 2 year 0.007852 0.000599 8.94E-06 1.79E-08 3 year 0.01489 0.002012 9.27E-05 1.24E-06 4 year 0.023215 0.00435 0.000373 1.32E-05

Moreover, the total fitting consequence of cap price is in figure 4.11. There are caps of 1, 2,3,4,5,7,10 years’ maturities and four kinds of strike rates.

Figure 4.11 Consequence of Calibration for Parameters of Hull-White Model.

The optimal estimated parameters are that mean reversion equals 0.014485 and volatility of Hull-White model equals 0.004596. The minimum summation of square error (SSE) is 7.80133e-005.

The dotted lines above are cap price of different strike rates calculated by Hull-White model with

optimal parameters and real lines are calculated by Black formula.

5. Conclusions and Future Work

We provide a numerical approach method to price sophisticated snowball notes.

Firstly, take Hull-White short rate model as a basic of term structure combined with trinomial tree. Secondly, construct the state variables of coupons and price in snowball notes. Finally, snowball price can be calculated by backward induction and linear interpolation method.

In the sensitivity analysis, we find that the parameters of Hull-White model have significant influence on snowball price. On the one hand, there is negative association between price and mean reversion of Hull-White model, and on the other hand, price and volatility of Hull-White model have positive relation. It is also important about contracting spreads of Snowball notes because of its positive relation to Snowball price. Moreover, the effective way to hedge snowball price is redemptive article which could protect issuers from losing a lot by using lower price to redeem contracts.

In the future, we maybe use different interest rate models to pricing snowball notes and compare with Hull-White tree model in this thesis. Moreover, we also could extend the algorithm of this thesis to price other sophisticated interest rate derivatives by the same term structure.

References

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Appendix A

Derive caplet as a put option of zero-coupon bond

At timet , Caplet value which is paid at time _k t_k+1 is following: option where the underlying is a zero-coupon bond which maturity is t_k+1.

Appendix B

Derive floorlet as a call option of zero-coupon bond

At timet , floorlet value which is paid at time _k t_k+1 is following: option where the underlying is a zero-coupon bond which maturity is t_k+1.

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