Binominal Tree - Which rates at the different Nodes?

[Thammer]

Hi,

Why does the Binomial Tree use forward rates? Why does it not consist of Spot rates? Is there anyway to explain this in an easy and simplified way?

Also, which rate do we have at each of the different nodes?

Time 0 = So

Time1

Node u: ?

Node d:?

Time 2 Node uu: ?

Node ud = forward rate (2.1) ?

Node dd: ?

Thanks for the help.

[S2000magician]

Probably the simplest way to explain it is that at each node you need to know the value of the bond so that you can (potentially) make a decision about how the cash flows might change: the bond may be callable, putable, or floating-rate. You could not do that if you had spot rates at each node.

[Thammer]

@S2000magician

Probably the simplest way to explain it is that at each node you need to know the value of the bond so that you can (potentially) make a decision about how the cash flows might change: the bond may be callable, putable, or floating-rate. You could not do that if you had spot rates at each node.

Thank you. However, I do not understand why we will have “different forward rates”? It makes sense to me at time 2, the middle one is f(2,1) and I understand that based on volatility we can find node uu and node dd. But how can we estimate the forward rate at time 1 (Since we don’t have a middle rate)?

[burberryjam]

Well…Spot rates and forwards are different packages of the same thing. Forwards are derived from spot rates.

I suppose you could use spot rates but you would have to add in a whole lot of extra calculations: If we say time y = time x + 1, the forward rate at time x+y is the spot rate at y divided by spot rate at x. You could use spot rates but you would have to discount the cashflows requiring another step in the process.

Does that make any sense, I realise I am probably not making it clear.

[S2000magician]

@Thammer

@S2000magician

Probably the simplest way to explain it is that at each node you need to know the value of the bond so that you can (potentially) make a decision about how the cash flows might change: the bond may be callable, putable, or floating-rate. You could not do that if you had spot rates at each node.

Thank you.

My pleasure.

@Thammer

However, I do not understand why we will have “different forward rates”?

I’m not sure I understand fully what you mean by ‘“different forward rates”.’

The n + 1 forward rates at time n are related by the volatility of interest rates for the tree: σ. Each forward rate at time n is e^2σ times the forward rate immediately below it.

[Thammer]

@S2000magician

@Thammer

@S2000magician

Probably the simplest way to explain it is that at each node you need to know the value of the bond so that you can (potentially) make a decision about how the cash flows might change: the bond may be callable, putable, or floating-rate. You could not do that if you had spot rates at each node.

Thank you.

My pleasure.

@Thammer

However, I do not understand why we will have “different forward rates”?

I’m not sure I understand fully what you mean by ‘“different forward rates”.’

The n + 1 forward rates at time n are related by the volatility of interest rates for the tree: σ. Each forward rate at time n is e^2σ times the forward rate immediately below it.

Thank you sir.

What I mean is that after 2 years I can calculate the forward rate in the middle based on 1+f(2,1) and go up and down based on volatility.

But in time 1 I don’t have a rate that is in the middle since I just have Node u and Node d. Meaning that from my understanding I cant calculate the rates based on 1+f(1,1). So how do I calculate the forward rates in the tree for those years that have an even number of nodes. Does that make sense?

[S2000magician]

At time t = 1, you can think of the upper rate as f(1,1) × e^σ and the lower rate as f(1,1) ÷ e^σ (= f(1,1) × e^−σ).

At time t = 2, you can think of the upper rate as f(2,1) × e^2σ, the middle rate as f(2,1), and the lower rate as f(2,1) × e^−2σ.

And so on.

Unfortunately, these relationships are only approximate; that’s why you have to calibrate the tree.

[Thammer]

@S2000magician

At time t = 1, you can think of the upper rate as f(1,1) × e^σ and the lower rate as f(1,1) ÷ e^σ (= f(1,1) × e^−σ).

At time t = 2, you can think of the upper rate as f(2,1) × e^2σ, the middle rate as f(2,1), and the lower rate as f(2,1) × e^−2σ.

And so on.

Unfortunately, these relationships are only approximate; that’s why you have to calibrate the tree.

Great. Thanks a lot. That was very helpful!

What do you mean by “Calibrate”?

[S2000magician]

@Thammer

@S2000magician

At time t = 1, you can think of the upper rate as f(1,1) × e^σ and the lower rate as f(1,1) ÷ e^σ (= f(1,1) × e^−σ).

At time t = 2, you can think of the upper rate as f(2,1) × e^2σ, the middle rate as f(2,1), and the lower rate as f(2,1) × e^−2σ.

And so on.

Unfortunately, these relationships are only approximate; that’s why you have to calibrate the tree.

Great. Thanks a lot.

You’re quite welcome.

@Thammer

That was very helpful!

I’m glad that it was.

@Thammer

What do you mean by “Calibrate”?

Aha!

That question explains a lot.

You need to read §3.4 in Reading 36: Constructing the Binomial Interest Rate Tree.

Or, you need to read the article I wrote on creating a binomial interest rate tree: http://financialexamhelp123.com/creating-a-binomial-interest-rate-tree/.

Calibration means determining what the interest rates are at each node so that the tree gives correct prices for par bonds at all maturities. The rate at node N₀ is easy: r₀ is the 1-period par rate which is the same as the 1-period spot rate which is the same as the 1-period forward rate starting today.

To get the rates at nodes N_L and N_H – r_1,L and r_1,H, respectively – you have to discount the payments on a 2-period par bond along both (equally weighted) paths to get today’s price, then adjust the rates (keeping in mind that r_1,H = r_1,L × e^2σ, and that r₀ is already known) until today’s price is par. This one can be solved analytically (i.e., using the quadratic formula), but in practice most people solve it numerically, using something like Excel’s Solver.

To get the rates at nodes N_LL, N_HL, and N_HH – r_2,LL, r_2,HL, and r_2,HH, respectively – you have to discount the payments on a 3-period par bond along all four (equally weighted) paths to get today’s price, then adjust the rates (keeping in mind that r_2,HL = r_2,LL × e^2σ, r_2,HH = r_2,HL × e^2σ, and that r₀, r_1,L, and r_1,H are already known) until today’s price is par. This one can be also be solved analytically (i.e., using the Ferrari’s formulae), but in practice most people solve it numerically, using something like Excel’s Solver.

And so on.

[Thammer]

@S2000magician

@Thammer

@S2000magician

At time t = 1, you can think of the upper rate as f(1,1) × e^σ and the lower rate as f(1,1) ÷ e^σ (= f(1,1) × e^−σ).

At time t = 2, you can think of the upper rate as f(2,1) × e^2σ, the middle rate as f(2,1), and the lower rate as f(2,1) × e^−2σ.

And so on.

Unfortunately, these relationships are only approximate; that’s why you have to calibrate the tree.

Great. Thanks a lot.

You’re quite welcome.

@Thammer

That was very helpful!

I’m glad that it was.

@Thammer

What do you mean by “Calibrate”?

Aha!

That question explains a lot.

You need to read §3.4 in Reading 36: Constructing the Binomial Interest Rate Tree.

Or, you need to read the article I wrote on creating a binomial interest rate tree: http://financialexamhelp123.com/creating-a-binomial-interest-rate-tree/.

Calibration means determining what the interest rates are at each node so that the tree gives correct prices for par bonds at all maturities. The rate at node N₀ is easy: r₀ is the 1-period par rate which is the same as the 1-period spot rate which is the same as the 1-period forward rate starting today.

To get the rates at nodes N_L and N_H – r_1,L and r_1,H, respectively – you have to discount the payments on a 2-period par bond along both (equally weighted) paths to get today’s price, then adjust the rates (keeping in mind that r_1,H = r_1,L × e^2σ, and that r₀ is already known) until today’s price is par. This one can be solved analytically (i.e., using the quadratic formula), but in practice most people solve it numerically, using something like Excel’s Solver.

To get the rates at nodes N_LL, N_HL, and N_HH – r_2,LL, r_2,HL, and r_2,HH, respectively – you have to discount the payments on a 3-period par bond along all four (equally weighted) paths to get today’s price, then adjust the rates (keeping in mind that r_2,HL = r_2,LL × e^2σ, r_2,HH = r_2,HL × e^2σ, and that r₀, r_1,L, and r_1,H are already known) until today’s price is par. This one can be also be solved analytically (i.e., using the Ferrari’s formulae), but in practice most people solve it numerically, using something like Excel’s Solver.

And so on.

Thanks a lot. Both for your response and pointing me in the right direction. I read the articles on your webpage, they were very helpful. Great quality and solid explanation.

I think the reason behind my confusion was not having a good enough knowledge on the different type of rates, it all makes more sense now. Thank you

[S2000magician]

My pleasure.