<p>Notation:</p>

<p>t - time; G(t) - zero-coupon yield curve; $r$, $r_d$, $r_u$ - interest rates.</p>

<p>The task is to find market price of a bond for today, while knowing the price of a number of other bonds. </p>

<p>Nelson-sigel model provides interest rate curve G(t). The next thing to do is to calibrate interest rate binomial tree using G(t). Also are known bond cash flow, par value and today market price.</p>

<p>The problem is how to do it properly. The start yield is given by G($t_1$), where $t_1$=0.25 (step for binomial model).</p>

<p>$100=\frac{1}{2}(\frac{G(t_2 )}{1+G(t_1)} +\frac{G(t_2 )+100}{(1+G(t_1))*(1+r_u )} + \frac{G(t_2 )}{1+G(t_1)} +\frac{G(t_2 )+100}{(1+G(t_1))*(1+r_d )})$</p>

<p>$r_u= r_d*exp⁡(2*σ)$</p>

<p>$t_2 = 2*t_1$</p>

<p>These conditions provide $r_d$, $r_u$ for the second step of i.r. tree.</p>

<p>Is the next set of conditions for $r_{dd},r_{du},r_{uu}$  correct?</p>

<p>$100= \frac{1}{4} (\frac{G(t_3)}{1+G(t_1)}+\frac{G(t_3)}{(1+G(t_1))(1+r_d)}+\frac{(G(t_3)+100)}{(1+G(t_1))(1+r_d)(1+r_{dd})}) + \frac{1}{4}(\frac{G(t_3)}{1+G(t_1)}+\frac{G(t_3)}{(1+G(t_1))(1+r_d)}+\frac{(G(t_3)+100)}{(1+G(t_1))(1+r_d)(1+r_{ud})}) + \frac{1}{4}(\frac{G(t_3)}{1+G(t_1)}+\frac{G(t_3)}{(1+G(t_1))(1+r_u)}+\frac{(G(t_3)+100)}{(1+G(t_1))(1+r_u)(1+r_{ud})}) + \frac{1}{4}(\frac{G(t_3)}{1+G(t_1)}+\frac{G(t_3)}{(1+G(t_1))(1+r_u)}+\frac{(G(t_3)+100)}{(1+G(t_1))(1+r_u)(1+r_{uu})})$</p>

<p>$r_{du}= exp⁡(2*σ) r_{dd}$</p>

<p>$r_{uu}= exp⁡(2*σ) r_{ud}=exp⁡(4*σ) r_{dd}$</p>

<p>$t_3=3*t_1$</p>

<p>Is there an easy way to define $r_{uuu}$, $r_{uuuu}$ and other interest rates (yields), so it would be possible to do it programmatically?</p>

<p>What are the ways to correctly determine σ?</p>

<p>I'm heavily relying on ideas used in this presentation:</p>

<p><a href="http://faculty.cbpa.drake.edu/root/Auvergne/DESS%20Analyst%20Binomial.ppt" rel="nofollow">http://faculty.cbpa.drake.edu/root/Auvergne/DESS%20Analyst%20Binomial.ppt</a></p>

CC BY-SA 3.0
2016-06-09T19:06:32.273Z
I think this would be more appropriate on the quant.SE ? :)
Alexis L.
CC BY-SA 3.0
2016-06-10T07:22:36.553Z
Too many questions at once, is usually a bad idea, and sharing thoughts, is, unfortunately not what Stack Exchange is for. I'd recommend to edit your question to focus on one question at once. Being specific and answerable.
clem steredenn
CC BY-SA 3.0
2016-06-10T09:10:19.577Z
Ok, thank you. I'll go on quant.SE and try to be more specific.
Binosaur
CC BY-SA 3.0
2016-06-14T02:15:31.813Z
I'm voting to close this question as off-topic because it doesn't belong on this SE
Jamzy
CC BY-SA 3.0
2016-06-14T10:29:54.517Z
I posted it on quantSE, but nobody answered neither here, nor there.
Binosaur
CC BY-SA 3.0
2016-06-14T10:32:33.217Z
Jamzy, could you please tell, why it doesn't belong here?
Maybe you know smbd who can answer it?
Binosaur
<interest-rate><bonds><asset-pricing>
Bonds with embedded options pricing via binomial model
Binosaur
# Bonds with embedded options pricing via binomial model

(Asked by: username_0 on 2016-06-09)

Notation:

t - time; G(t) - zero-coupon yield curve; $r$, $r\_d$, $r\_u$ - interest rates.

The task is to find market price of a bond for today, while knowing the price of a number of other bonds.

Nelson-sigel model provides interest rate curve G(t). The next thing to do is to calibrate interest rate binomial tree using G(t). Also are known bond cash flow, par value and today market price.

The problem is how to do it properly. The start yield is given by G($t\_1$), where $t\_1$=0.25 (step for binomial model).

$100=\frac{1}{2}(\frac{G(t\_2 )}{1+G(t\_1)} +\frac{G(t\_2 )+100}{(1+G(t\_1))\*(1+r\_u )} + \frac{G(t\_2 )}{1+G(t\_1)} +\frac{G(t\_2 )+100}{(1+G(t\_1))\*(1+r\_d )})$

$r\_u= r\_d\*exp⁡(2\*σ)$

$t\_2 = 2\*t\_1$

These conditions provide $r\_d$, $r\_u$ for the second step of i.r. tree.

Is the next set of conditions for $r\_{dd},r\_{du},r\_{uu}$ correct?

$100= \frac{1}{4} (\frac{G(t\_3)}{1+G(t\_1)}+\frac{G(t\_3)}{(1+G(t\_1))(1+r\_d)}+\frac{(G(t\_3)+100)}{(1+G(t\_1))(1+r\_d)(1+r\_{dd})}) + \frac{1}{4}(\frac{G(t\_3)}{1+G(t\_1)}+\frac{G(t\_3)}{(1+G(t\_1))(1+r\_d)}+\frac{(G(t\_3)+100)}{(1+G(t\_1))(1+r\_d)(1+r\_{ud})}) + \frac{1}{4}(\frac{G(t\_3)}{1+G(t\_1)}+\frac{G(t\_3)}{(1+G(t\_1))(1+r\_u)}+\frac{(G(t\_3)+100)}{(1+G(t\_1))(1+r\_u)(1+r\_{ud})}) + \frac{1}{4}(\frac{G(t\_3)}{1+G(t\_1)}+\frac{G(t\_3)}{(1+G(t\_1))(1+r\_u)}+\frac{(G(t\_3)+100)}{(1+G(t\_1))(1+r\_u)(1+r\_{uu})})$

$r\_{du}= exp⁡(2\*σ) r\_{dd}$

$r\_{uu}= exp⁡(2\*σ) r\_{ud}=exp⁡(4\*σ) r\_{dd}$

$t\_3=3\*t\_1$

Is there an easy way to define $r\_{uuu}$, $r\_{uuuu}$ and other interest rates (yields), so it would be possible to do it programmatically?

What are the ways to correctly determine σ?

I'm heavily relying on ideas used in this presentation:

<http://faculty.cbpa.drake.edu/root/Auvergne/DESS%20Analyst%20Binomial.ppt>

**Tags:** interest-rate, bonds, asset-pricing

> **Comments:**
> **username_1**: I think this would be more appropriate on the quant.SE ? :)  
> **username_2**: Too many questions at once, is usually a bad idea, and sharing thoughts, is, unfortunately not what Stack Exchange is for. I'd recommend to edit your question to focus on one question at once. Being specific and answerable.  
> **username_0**: Ok, thank you. I'll go on quant.SE and try to be more specific.  
> **username_3**: I'm voting to close this question as off-topic because it doesn't belong on this SE  
> **username_0**: I posted it on quantSE, but nobody answered neither here, nor there.  
> **username_0**: username_3, could you please tell, why it doesn't belong here? Maybe you know smbd who can answer it?  

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## 0 answers

