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You don't mention if the puts in question are exotic or vanilla, but assuming they are vanilla, you should read this paper by Chen and Joshi. In it, they find optimal performance by using smoothed, truncated Tian-parameter binomial lattices with Richardson extrapolation -- where the idea is to run one extra low-cost (long $\Delta T$) tree in order to ...

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Not all binomial trees take $u=e^{\sigma\sqrt{\Delta t}}$. Thinking of the binomial tree as a discrete approximation (on a grid) to a continuous process, it makes sense that a variety of choices for where to place grid points will work. For a listing of a few different choices of $u$, see the Tian Tree settings and others. From this Sitmo page you can ...

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The model here is the binomial option pricing model, so the second term in the brackets represents the expected future value of the option (under riskneutral probabilities). The aim of the option holder is always to maximize the value of his option. He can at any point sell the option at the fair market price $E(V_{n+1})$ or exercise it to get $G_n$. So if ...

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In binomial tree models, there is no such a thing as a path. The binomial tree represents information about the distribution of the zero-curve at a given time and preserve enough information between different times to let you compute conditional expectations. Generally, you can not price path-dependant instruments in a model based on trees—because there is ...

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The condition $$ud=1\text{, or equivalently }u=1/d$$ is necessary to ensure convergence of the Binomial tree's mean $\mu$ and standard deviation $\sigma$ to nonfinite values when $n$ (number of steps) goes to infinity. Cox-Rubinstein-Ross showed in their famous paper, that to achieve this, we must have: u=e^{\sigma\sqrt{t/n}}\text{, ... 4 For a martingale dX=a(X,t)\,dt+b(X,t) dW(t) where a and b are not constant, your tree will not recombine in general . This is the main issue. See for instance: Florescu, I. and F. G. Viens (2008, March). Stochastic volatility: Option pricing using a multinomial recombining tree. Applied Mathematical Finance 15 (2), 151-181. It deals with the case ... 4 You know that the Ho-Lee model is represented by the stochastic differential equations \begin{align} dr_t=\lambda_t\,dt+\sigma\,dW_t \end{align} In order to Implementation our binomial tree, we use the Euler discretization. \begin{align} r_t=r_{t-\Delta t}+\lambda_{t-\Delta t}\,\Delta t+\sigma\,\sqrt {\Delta t} \,Z \end{align} where Z is a standard normal ... 3 There is a good quick well-known approximation for at-the-money options:\textrm{Call,Put} = 0.4 S \sigma \sqrt{T}.$$See further discussion at What are some useful approximations to the Black-Scholes formula?. 3 you don't need ud=1. In fact, there are now about 30 binomial trees which converge to Black--Scholes in the large step limit. Most of them do not have ud=1. All you need is$$ d < e^{r \Delta t} < u $$The tree recombines provided u and d don't change from step to step. See my book More Mathematical Finance for a comprehensive review and ... 3 You can calibrate the model by discretizing in time, and using a forward induction method as originally proposed by Jamishidian in 1991: F.Jamshidian, Forward Induction and Construction of Yield Curve Diffusion Models, J.Fixed Income 6, 62-74 (1991). Although he formulated this induction in the language of the binomial tree, the method is more general, and ... 3 All you need is to use the discretization to implement the MC approach. The following links should get you started: http://www.lcy.net/files/BDT_Seminar_Paper.pdf http://www-2.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote23.pdf ... 3 A condition for correct calibration of the short rate model is that it exactly reproduce the present values of fixed (option-free) cashflows - that is, that it give the same answer as ordinary discounting at the spot rate. If it doesn't, you've done something wrong - sort of like using a model that violates put-call parity. (Actually, it's exactly like ... 3 You have to look at the terms and conditions on your individual bond. The way the specifications usually work is that a call will result in accrued interest being paid, effectively making up for the lost coupon. Sometimes there's even an extra penalty. A put will result in a loss of coupon in almost all cases, and so is almost always done just after a ... 3 if you let \delta t be small enough, this won't happen. So the solution is to take more steps. The CRR tree is very out dated in any case. 2 I would create separate estimates for volume and choice of debt instrument. There are tools to estimate these simultaneously but I do not see a compelling advantage here. I assume the volume is conditional on the choice of debt issuance so you might start by predicting choice of debt issuance and use this as an input to the volume model. The volume model ... 2 Let's start with question (2). If you are not obtaining S=1.5295e+009 after backwardation, then you have a bug in your binomial tree code. You may wish to find and eliminate that before proceeding. One simple check is to make all the terminal nodes have value 1.0. You should obtain that the initial node has value e^{-rT}. This assumes, of course, ... 2 Under the typical Black-Scholes model, you "cannot" do it, because the assumption is that each of the securities in the portfolio has a lognormal terminal distribution, and the sum of lognormally distributed variables it not itself lognormally distributed. In theory one needs an N-dimensional tree (or grid) to treat an N-element portfolio. I write "cannot" ... 2 I see your porblem - Hull unfortunately does not explain the reasoning behind the approach. The hint the books gives is correct. Using Taylor series e^x can be written as e^x = 1 + x + \frac{x^2}{2}+.... Hull also incoporates a dividend rate q but we can disregard it here. p is given by p=\frac{e^{r\Delta }-d}{u-d}. We also have u=\frac{1}{d}. ... 2 In R you can use fOptions package to draw Binomial Tree graphs. Here is a simple code snippet #Install the package and load it install.packages('fOptions') library(fOptions) #Calculate the value of the option and plot optionVals<-BinomialTreeOption(TypeFlag="ce",S=100,X=100,Time=3,r=0.05,b=0,sigma=0.2,n=3,title="example binomial tree") ... 1 I think that you can find the answer to this question here: http://people.stern.nyu.edu/wsilber/chuang-silber%20approx%20option%20value.pdf 1 I read the question as follows: You have one stock S_0 and after one period it either goes up to S^+ where the option takes the value f^+ or it goes down to S^- where the option takes the value f^-. The bond grows from B_0 to B_1 = B_0 \exp(r). Then you need to solve$$ a S^+ + b B_1 = f^+ \\ a S^- + b B_1 = f^- $$for a,b which are 2 ... 1 S_0: The stock price today. p: The probability of a price rise. u:The factor by which the price rises. d: The factor by which the price falls. Three equations are required to be able to uniquely specify values for the three parameters of the binomial model. Two of these equations arise from the expectation that over a small period of time the ... 1 You could solve this by constructing a binomial tree with the stock price ex-dividend. Also keep in mind that you have to adjust your volatility by muliplying with S/(S-PV(D)). 1 You can view the arbitrage-free statement as being about infinitely liquid trading. If one has to trade x by paying a half-spread \nu, and you have a trivial payoff \Phi(\cdot) \equiv 1+R then there is no solution. You would be trying to solve$$ (1+R)x - \nu + suy = (1+R)x = (1+R)x - \nu + sdy $$1 If the stock you'd like to hedge with is the same as the option's underlying obviously just find the net delta and hedge with that amount of stock. If you have different types of stocks and would like to hedge with an index you can multiply the delta with the beta of each stock versus the index. Beta is analogous to delta in a way. With delta we describe ... 1 If there is no chance of default, and you have an extremely simple set of terms and conditions (T&C) on the bond, then the two are equivalent. In the real world T&C are complex for all bonds currently traded, and default is important. Therefore something closer to the binomial model, which allows the embedded option to disappear in the event of ... 1 R denotes the implied risk-neutral cumulative growth which is different for each note. You get them through: 1 + cumulative risk-neutral return through the tree to a particular node. It should be in the paper, though, for example "a 20% growth in the underlying gives R = 1.20" 1 I'll take a stab. In short rate modelling we start by postulating dynamics under pricing measure Q and then calibrating our model to market prices. General short rate dynamics under Q can be described as dr(t)=b(t_0+t,r(t))dt+\sigma(t_0+t,r(t))dW(t) , where W is Q Brownian motion. Note that the short rate process itself is not a Q-martingale ... 1 Binomial model is just a model, and a rather simplistic one. Think about it, anywhere you put the strikes of the call spread, provided that, as you did, they are between su and sd, you get the same value! Also, to be fair to the binomial model, the delta hedge in the binomial model is not defined as the derivative of the value with respect to s, but rather ... 1 One way to start thinking about this is to work out a couple of Discrete versions of Ito's lemma Øksendal (6th edition) Example 3.1.9: almost surely,$$ B_t^2 - t = \int_0^t 2B_s dB_s $$This has a discrete version which holds everywhere: let X_n=\pm 1 and S_n=\sum_{i=1}^n X_i, then$$ S^2_n-n = 2\sum_{i=0}^{n-1} S_i X_{i+1}  To verify ...

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