Which one is least likely correct? Difference 1: The Monte Carlo approach does not require calibration, whereas the binomial tree approach does. Difference 2: The Monte Carlo approach is typically used when cash flows are path dependent, whereas the binomial tree approach only allows one expected cash flow per node, regardless of the path of interest rates. Difference 3: The Monte Carlo approach randomly simulates a fixed number of interest rate paths and values the security only across those paths, whereas the binomial tree approach values the security across all possible interest rate paths on the tree. Donât get this at all, anyone can help?
Poorly worded questions, but it should be number 2, because you can change the expected cash flows at a node depending on the interest rates like in the case of an option.
MC does not require calibration, because it takes the average of thousands of simulations.
Number 3 should be incorrect, as it is not common to use a ârandomâ approach to simulating interest rates, but they are random within a set of given inputs.
Not sure what Monte Carlo calibration means but pretty sure Difference 2 is a correct statement and it is NOT the correct answer.
Path dependence IS an issue in binomial trees.
The LEAST likey correct answer is 3. So that one you should choose. The statement put both definitions reverted, monte carlo uses all possible ways (like 10,000 paths for example), it is an exhaustive method. On the other hand, binomial trees use a fixed number of paths: 2^(n-1) where ânâ is the number of periods looking forward.
Statement 1 is correct, monte carlo is exhaustive, so calibration is implied inside in all incredibly amount of paths. You get at the real average by fatigue. However, binomial trees has very limited paths, so you need to calibrate the tree by force.
Statement 2 is also correct, as krokodilizm stated. Monte carlo is capable to value at dependent paths, thats why you use it on mortgage-backed securities valuation for example (and hence not binomial trees). Binomial trees values flows that are path independent.
Hope this helps.
Fixed that for you.
OMG I stole S2000magician line!
Binomial trees assume path independence, so its use is wrong for securties that describe cash flows with path dependency like Mortgage-backed portfolios or securities. Therefore you must use MonteCarlo for those ones.
LOL. I still think I put it correctly. What I mean is that path dependence âdoes not apply in binomial treesâ. It is a problem/issue.
So what is calibration again? Drawing out branches?
Like I said, poorly worded questions.
The correct answer depends (no pun) on how you interpret them.
Re-adjusting the inputs to match a desired output.
Sorry, interpreted it all the way around. Your statement is correct.
Calibration is what MrSmart said, iterate inputs to make the output arrive at a desired amount or value. In a bimonial tree, spot rates, forecasted forward rates and interest rate volatility are inputs, and the price of the security analysed is the output.
Difference 2 is false: a binomial interest rate tree can have path-dependent cash flows (e.g., when valuing a floating-rate bond).
I have an example in the article I wrote on valuing floating-rate bonds: http://financialexamhelp123.com/valuing-floating-rate-bonds/
Difference 3 is true. Suppose that you have a binomial interest rate tree for valuing MBS: 360 periods. The number of paths through the tree is incomprehensible: 2.35 Ă 10^108. MBS models will randomly generate paths through that tree â 500, 1,000, 10,000, whatever â and value the bonds only along those paths. The randomness is in generating the paths, not in the interest rates along those paths.
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Hi @S2000magician , in this case, is Difference 1 true? Monte Carlo approach does not require calibration, whereas the binomial tree approach does?
Thanks in advance.
This seems to be taken from a mock exam and if you google, you can find the official answer, which seems to be that the least likely to be correct is 1 (calibration):
Monte Carlo forward rate simulation randomly generates a large number of interest rate paths that
will correctly value benchmark bonds only by chance.
A fixed amount, known as a drift term, is added to every forward interest rate on every simulated path
to calibrate the simulation so that the values estimated for benchmark bonds equal their market
prices.
Some of these statements seem to be intended to confuse, such as
Difference 3: The Monte Carlo approach randomly simulates a fixed number of interest rate paths
and values the security only across those paths, whereas the binomial tree approach values the
security across all possible interest rate paths on the tree.
Each of the two parts of Difference 3 is true, but I would make the point that a binomial tree has a finite number of possible paths and you value the security on all of those paths, while a Monte Carlo Method has an infinite number of possible paths and you only value the security on a finite number of them.
Added: what I understand by Monte Carlo is as follows:
Itâs simplest for equity securities, the stock price S obeys a log-normal random walk described by the stochastic differential equation dS=\mu dt + \sigma S dX where dX is a random variable
the option value = e^{-r(T-t)}E[\textrm{payoff}(S)]
You use the equation for dS to find S between t=0 (with S(0) given) and t=T.
One way of doing this is the Euler method, where the equation for dS is replaced by an equation for \delta S,
\delta S =rS\delta t +\sigma S\sqrt{\delta t}\phi (where the drift \mu has been replaced by the interest rate r and \phi is drawn from a standardized Normal distribution).
You do this N times say, with each realization being a different path, and at each time step on each path, \phi will be randomly generated.
For interest rate securities, the random walk is a little more complicated and there are several different models,
e.g. CIR (Cox, Ingersoll, Ross) model
the random walk for the interest rate r is
dr=(\eta-\gamma r)dt+\sqrt{\alpha r}dX
where \eta, \gamma, and \alpha are constants, and part of the calibration would be finding values of these constants to match any values given in the question (eg T-bill rates at certain times)
Difference 2: The Monte Carlo approach is typically used when cash flows are path dependent,
whereas the binomial tree approach only allows one expected cash flow per node,
regardless of the path of interest rates.
This is true.
With a floating-rate bond, the interest rate payment at a given node will be the same, no matter what path on the tree you took to get there, whether it was up-down or down-up. Thatâs not path-dependent in the sense they meant.
An example of path-dependency would be MBS (mortgage-backed securities) because people are
more likely to prepay when rates are high, and once youâve made a prepayment, the remaining repayments will be different. If you go up-down that may trigger a prepayment while going down-up does not.
Binomial trees use backward-induction, which means when you value a security, you start at maturity and work backwards in time using the tree to the present time.
Monte Carlo works forward along each path from the present time until maturity.
With Monte Carlo, as youâre going forward in time, you know what happened at earlier times.
With Binomial Trees, as youâre going backwards in time, you know what happened at later times but not at earlier times.