interest rate volatility & binomial model

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pepp

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Hello, I think i am able to solve a few problems correctly that asks to compute the price of the bond. But I don’t understand how the interest rate volatility is coming to play. In none of the questions I’ve done so far, interest rate volatility is specified.
a) What if its specified, how are we supposed to use it to calculate the arbitrage free price?
b) also on a side question, this is completely ridiculous i am asking, but what is VOLATILITY??? the fact that interest rates can move up or down is called VOLATILITY?
c) really confused between spot rates, forward rates, interest rates volatility, and yield to maturity, coupon.
d) is a 7.5 YTM equivalent to saying 7.5% coupon?
 
a- you will need to be given at the very least all the rates on the bottom sections of every node. most of the time you’re just given the tree. the formula to calculate the upper nodes from lower is something like 1F1H = 1F1L*e^(2sd)
b- yes, there is an entire reading about interest rate volatility
d- yes assuming the bond was issued at par
 
a. not sure but the effect of volatility on stock prices is measured by duration & convexity, on option prices it measured by Rho
b. yes
c. sport rate: exchane rate today
foward rate= sport(1.Rf)^t
interst rate voliatility same as b, the rest check d
d. no, coupon is used to calculate cash inflows during the holding period. you multiply it with the par value
YTM is used for bond valuation i.e. 1000/1+y. if the bond is selling at par YTM and coupon will be the same.
 
expanding on sharpshooter std dev = SQR[sum(Xt - Xbar)^2/n-1
the important thing is that Xt = 100ln( Y(t) / Y(t-1) )
forward spot is a whole book… called derivatives and also a whole chapter in econ.
simply put forward is a spot rate in the future.
 
pepp, in these problems, volatility means a 0.50 probability of going up by 10% or down by 10% (or whatever percentage they give you). Later on, they change the up/down percentage, but the probability is usually kept at 0.50.
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Spot rate is the price of the bond assuming no coupons. So, a 1-year spot rate is today’s bond-equivalent yield for a 1-yr T-Bill. The 2-year spot rate is the annualized rate (i.e., bond-equivalent yield) you get for holding on to a 2-year zero-coupon bond, etc.
The spot rate for a 1-year T-Bill 4, years from now, is called the the forward rate of the T-bill, 4 years from now, written as 1f4, or in some other places, written as 4f1.
So, if you look at a 5-year term structure, like 0, 1, 2, 3, 4, 5, with zero being today, you can list the spot rates for every year, but be careful with terminology. If I put 4% under year 3, this could mean two completely different spot rates:
(1) It is the 1-year spot rate for year 3 (meaning if you are at the end of year 2, 4% is the rate you get if you buy a 1-year bond, i.e., it is the 1-year forward rate, 2 years from now.
(2) It is the spot rate for buying a 3-year bond today, a zero coupon bond.
If they don’t say it is the 1-yr rate, then you assume it is an m-year spot rate, like (2) above.
If you want more just ask.
 
AudreyMwala Wrote:
——————————————————-
> a. not sure but the effect of volatility on stock
> prices is measured by duration & convexity, on
> option prices it measured by Rho
Isn’t volatility measured by Vega and Rho measures interest rate?
 
b) Don’t worry about volatility in the binomial trees. The calculation has already been performed for you. But if you really have to know, volatility comes into the calculation of the interest rates. It’s a measure of uncertainty or variance.
If you are sitting at a node where the short rate is 5%, and you want to forecast an up and down move in the next period, you’d need to know the volatility of interest rates. To be completely technical, if volatility were 1%, you forecast the up interest rate change “u” as exp(0.01*sqrt(time interval). If time interval is 1 year, then you’d get exp(.01) = 1.01, and the new r would be r*this - depending on what interest rate model you are using of course - or 5.05%. The down move “d” is exp(-.01) or 1/u, of 4.95%. So your interest rate tree would look like:
5.05%
5%
4.95%
and so on. But really, the curriculum doesn’t go this deep.
d) YTM is NOT the same as the coupon unless the bond sells at par. YTM is a mathematical concept that equates the PV of the bond cash flows to the price. Let me give an example.
Suppose you have a two year bond paying a coupon of 6% annually. The price is 100. What is the YTM? Write out the bond value like this:
PV = CF(year 1)/(1+YTM)^1 + CF(year 2)/(1+YTM)^2
Set this equal to the observed price of 100. The CF(year 1) is just the coupon * par, or $6 on a 100 par value. The CF at year two is the coupon plus par or $106. Then
100 = 6/(1+YTM)^1 + 106/(1+YTM)^2
YTM is unobserved and must be solved for. You will find however that in this case, YTM = 6%. The YTM always equals the coupon rate for a bond selling at par.
Try it again with observed price = 95. You’ll find that YTM is now the solution of 98= 6/(1+YTM)^1 + 106/(1+YTM)^2. Put it in Excel and use solver, you get YTM = 7.11%.
 
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