The n-period model converges in the limit to the Black-Scholes option pricing model. One advantage of the binomial approach is that not only does it accommodate a wide range of option pricing problems, but also it provides an interpretation of the Black-Scholes pricing model an n-period model of the ﬁnancial market where for each time interval of length ∆t the value of the stock may change by the factor u with probability q and by factor d with probability 1−q. Hence, assuming k up-states and n−k down-states, the value of the stock at maturity date T = n∆t,n ∈ lN, is given by Sn k:= uk dn−k S . Theorem 2.2 (Cox-Ross-Rubinstein Model) The price C0 of a European call-option in the n-period model i
Write down the equilibrium price of a put for an n period binomial model. BINOMIAL OPTION PRICING MODEL Assumptions and Notation The price of the stock at period zero is denoted by S0 There are only two periods: 0 and 1. The are only two possibilities. The price of the stock in period one can either go up to uS0, u > 1, or down to dS0, d < 1. The price of a put option at period zero is denoted. N-period binomial model as the number of steps N goes to inﬁnity. We also derive the Black-Scholes partial diﬀerential equation, and we verify that the Black-Scholes formulas are solutions of the Black-Scholes partial diﬀerential equation. We discuss the Greeks, the partial derivatives of the function given by the Black-Scholes formulas The binomial model illustrates option pricing ideas in a discrete-time, discrete-space setting. Suppose time runs from 0 to T in N discrete, equal steps of length δt. The market consists of 3 assets: stock S, cash B and derivative V. The dynamics for the evolution of these three assets through time are speciﬁed as follows: The stock-price binomial tree is constructed in the same way as above if we decide to split the time horizon Tinto nbinomial periods, each one of length h= T=n. We set up our binomial asset-pricing model by positing uand d
2. The Binomial Model We begin by de ning the binomial option pricing model. Suppose we have an option on an underlying with a current price S. Denote the option's strike by K, its expiry by T, and let rbe one plus the continuously compounded risk-free rate. We model the option's price using a branching binomial tree over ndiscrete time peri-ods. Let urepresent one plus a positive return on the underlying's value over a singl The binomial option pricing model values options using an iterative approach utilizing multiple periods to value American options. With the model, there are two possible outcomes with each..
In an n-period binomial model, the time to maturity [0, T] is partitioned into n equal time steps Ai = T/n. If the stock price is S in this period, then it is assumed to jump either upward to uS with proba bility p or downward to dS with probability 1 ? p in the next period, where 0 < d < er?t < u and 0 < p < 1. The binomial model is completely determined by the jump sizes u and d and the risk. n-period recombining binomial tree with n+1 leaves, n= 3 However, if we want to price some path dependent option, then, if we want to do this with Theorem 2.1 by using the recursion relations (2.14), we have to consider the following non recombining binary tree structure with 2n leaves at time step t n: n-period binary tree with 2n leaves, n= 3 Since for example 2250 ≈ 1075.26 is not that. Consider an N-period binomial model for a non-dividend paying stock where the true probability of an up-move in each period is given by p= :5. The initial value of the stock is S 0 = $100. Let C 0 denote the time t= 0 price of a European call option on the stock with strike Kthat expires after Nperiods. Now suppose that some extremely favorable news about the stock has just been announced so. We have seen that the sampling space N= fH;Tg underlies the N-period binomial model for the stock-price process. Elementary event != ! 1:::! N 2 is interpreted as complete information of the development of the stock over the time N. Likewise, ! 1:::! nis seen as partial information about the stock available by time n. By time ninvestor has observed n'up' or 'down' moves of the stock. Delta Hedging in the Binomial Model . In the 2-period binomial model, suppose you hold one put option. Construct a trading strategy that lets you hedge the risk of this put using the stock. At each node, explain how the portfolio values are calculated. To conduct this exercise, run the Binomial Tree Module from the Virtual Classroom page. For the default data make sure Put Option is.
Binomial Model. In an N-period binomial model, the stock price process is speci ed by three parameters: p;u;d: From time nto time n+1, the price jumps from S nto uS nwith probability pand to dS n with probability 1 np. Equivalently, ln S +1 Sn takes value lnuwith probability pand takes value lndwith probability 1 p. Put X n+1 = ln S n+1 Sn lnu+lnd 2 lnu lnd 2: Then X n+1 = 8 <: 1; with. M: Pricing in the N Period Binomial Model / Common Short Rate Models W: Common Short Rate Models and Pricing F: Mid-Semester Break. No class. Week #9: Oct 22 - 26 Assignments: M: Introduction to Credit Derivatives W: Common Short Rate Models and Pricing F: Installation of Farnam Jahanian as CMU President. No Class. Week #10: Oct 29 - Nov 2. 3.1.1 One period binomial model with collateral..21 3.1.2 n period binomial model with collateral.. 25 3.1.3 Partial collateralisation..27 3.2 Black-Scholes PDE with collateral and dividends..28 3.3 Showing that the PDE approach and the binomial lattice approach are equivalent.. 30 3.4 Numerical implementation..34 3.4.1 PDE approach..35 3.4.2 Binomial lattice. The Binomial Model for N-Period European Call Option Price Musa, Marwan; Massalesse, J.; Diaraya; Abstract. Options are rights to buy or sell underlying assets for an exercise price (strike price), which is fixed by the terms of the option contract. In this research, option price fluctuations determined by stocks as its underlying asset. In binomial model, stock price fluctuations modeled by. 106 (f) - Dynamic Replication in N-Period Binomial Model 35m Describes how to create a replicating portfolio in N-Period Binomial Model. Share with friends Facebook Twitter Email Share on Facebook Share on Twitter Share via Email Load More. Start your free trial. or. Sign in. Help Terms.
106 (f) - Dynamic Replication in N-Period Binomial Model. Describes how to create a replicating portfolio in N-Period Binomial Model. 26:43 107 - Risk Neutral Measure and No Arbitrage 107 - Risk Neutral Measure and No Arbitrage. Describes how to guarantee no arbitrage in a binomial model. 27:42 108 - Cash Flow Valuation 108 - Cash Flow Valuation. Describes how to value multiple cash flows in a. Binomial model, lognormal distribution approach, and the Black-Scholes model can be used to price an option. Similar results can be obtained by any of them if we assume some additional assumptions. Binomial model The binomial option pricing model derived by Rendleman and Bartter and Cox et al. is one of the most used models to price options. In binomial model settings, stock price S either. We start from the basic binomial lattice for the spot price. If this discrete-time model is to approach the underlying continuous-time process, the size of the jumps at each time step along with their probabilities must fulfill certain properties. From the one-period lattice we then move to the n-period setting. We consider initially a non. Binomial models (and there are several) are arguably the simplest techniques used for option pricing. The mathematics behind the models is relatively easy to understand and (at least in their basic form) they are not difficult to implement. This tutorial discusses the general mathematical concepts behind the binomial model with particular attention paid to the original binomial model.
In an n-period binomial model, the time to maturity (from today t to maturity date T) is divided into n subintervals, to = t < t1 << tn = T. The length of all of the subintervals is AT=ti+1-ti=(T-t)ln. The call price at each node of the standard binomial tree, Sij=SeiU{s:;+j(r-u'/2)!lT, is computed recursively from the discounted expected payoff relationship, equation (2.5). The binomial. The model that we will discuss is called the Black-Scholes Model and it is constructed from the \( n \) period binomial model by taking the limit as \( n \) goes to infinity. Before giving the definition of the model, and before listing all the assumptions that we will be making, let us first analyze what kind of restrictions we need to make if we are about to recycle the smart ideas we came.
Binomial model's Radon-Nikodym derivative. Ask Question Asked 3 years, 5 months ago. Active 3 years, 2 months ago. Viewed 295 times 3 $\begingroup$ Related: Dumb question: is risk-neutral pricing taking conditional expectation? In the one-step binomial model... For $\frac{d \mathbb Q}{d \mathbb P}$, I think it's $\frac{d \mathbb Q}{d \mathbb P} = \frac{q_u}{p_u}1_u + \frac{q_d}{p_d}1_d$, so it. Title: Microsoft PowerPoint - 10b_Multiperiod_Options.pptx Author: markus Created Date: 12/17/2009 3:13:32 P
Proof of optimal exercise time theorem for American derivative security in N-period binomial asset-pricing model. At least two textbooks (Shreve's Stochastic Calculus for Finance - I, theorem 4.4.5 or Campolieti & Makarov's Financial Mathematics, proposition 7.8) prove the optimal exercise theorem that says asset-pricing binomial stopping-time. asked Aug 14 '16 at 16:09. zer0hedge. 1,592 1. Chapter 3 Binomial Model: Two-Period Analysis. Chapter 4 Dynamic Hedging Strategies . Chapter 5 N-Period Binomial Option Pricing Model. Chapter 6 The Black-Scholes Option Pricing Model. Chapter 7 Hedge Parameters and Comparative Statics. Chapter 8 Option Trading Strategies. Chapter 9 Dividend Yield Model. Chapter 10 Currency Options. Chapter 11 Options on Futures. Chapter 12 A Unified Approach. www.ntu.edu.s 20 Jan 2017 - 36 min - Uploaded by FinMath SimplifiedDescribes how to create a replicating portfolio in N-Period Binomial Model. What you say is false. Exercising American call options. The binomial option pricing model assumes that the price of the underlying asset follows a. Monte Carlo simulation approach can be very helpful in solving large Binomial steps. Derivatives pricing in the.
The binomial option pricing model is an options valuation method developed in 1979. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or. A Foreign Exchange Model 7.1 The assets and the risk-neutral measures Steve E. Shreve October 10, 2005 In this section we consider an N-period binomial model, but with two currencies. In particular, there is a domestic interest rate r ≥ 0 and a foreign interest rate rf ≥ 0. The domestic interest rate leads to a domestic money market account whose price at each time n is M n = (1+r)n, n = 0. 3The Black-Scholes model, or the Cox-Ross-Rubinstein binomial model are examples of this approach. 1. stock price in some 'nice' way frequently lead to inconsistency (see, for example, [1], [2] and the references therein.) This becomes most evident when we try to model a stock with a discrete set of dividend payments at dates t1 < t2 < :::. This is a very real problem (all stocks have.
A Multi-Step Binomial Model To produce a more practical model, we need to value the option under the assumption that there are many revaluations of the stock between the time t = 0, when the option is written, and the time t = τ, when it expires. This will given rise to a wide range of possible eventual prices. To begin the generalisation to a multistep model, we may consider a model with two. Binomial trees are frequently used to approximate the movements in the price of a stock or other asset under the Black-Scholes-Merton model. There are several approaches to building the underlying binomial tree, such as Cox-Ross-Rubinstein, Jarrow-Rudd, and Tian. In all of the approaches above, the lattice is designed so as to minimize the discrepancy between the approximate (discrete) and.
Consider the N-period binomial model and processes Y0,Y1,...YN and C0,C1,...CN. Suppose that Y is Markov and C is deterministic (i.e., Cn does not depend on the coin flips ù). a) Show: If Cn doesn't equal 0 for all n, then C0Y0,C1Y1,....,CNYN is Markov. b) Give an example of processes C and Y such that at least one Cn is zero and C0Y0,C1Y1,....,CNYN is not Markov In the binomial model, we assume the current stock price S either rises to Su or falls to Sd at the end of the next period, and if n is the number of periods, we use the notation C(n) for the n-period binomial model call option price and set t = . The ﬁrst proofs that the binomial option price converges to the Black-Scholes price as the number of time periods tends to inﬁnity were given. Check Pages 1 - 5 of Lecture 5: American Derivatives - Columbia University in the flip PDF version. Lecture 5: American Derivatives - Columbia University was published by on 2015-09-16. Find more similar flip PDFs like Lecture 5: American Derivatives - Columbia University. Download Lecture 5: American Derivatives - Columbia University PDF for free THE LEAST COST SUPER REPLICATING PORTFOLIO IN THE BOYLE-VORST MODEL WITH TRANSACTION COSTS. By Ken Palmer. Arbitrage Theory in Continuous Time. By Anita Rossi. Introduction to the Economics and Mathematics of Financial Markets. By Pablo Jimenez Olea. LEVERAGE AND DEFAULT IN BINOMIAL ECONOMIES: A COMPLETE CHARACTERIZATION. By John D Geanakoplos. A numerical algorithm for indifference pricing. This article develops a flexible binomial model with a tilt parameter that alters the shape and span of the binomial tree. A positive tilt parameter shifts the tree upward while a negative tilt parameter does exactly the opposite. This simple extension of the standard binomial model is shown to converge with any value of the tilt parameter. More importantly, the binomial tree can be.
THE LEAST COST SUPER REPLICATING PORTFOLIO IN THE BOYLE-VORST MODEL WITH TRANSACTION COSTS. International Journal of Theoretical and Applied Finance, 2008. Ken Palmer. PDF. Download Free PDF. Free PDF. Download PDF. PDF. PDF. Download PDF Package. PDF. Premium PDF Package. Download Full PDF Package. This paper . A short summary of this paper. 36 Full PDFs related to this paper. READ PAPER. 11 Multi-period binomial tree model. The one-period and two-period binomial trees presented in Chapters 9 and 10 had the advantage of introducing important concepts and procedures, such as replication, portfolio dynamics and risk-neutral formulas, in fairly simple setups. Now, we seek to generalize these ideas to a model with more than two time steps. The main objective of this chapter is to.
2. I saw a lot of examples on tex.exchange.com about how to construct a 2-period or 3-period binomialtree in latex. I tried to modify these examples to construct a generel n-period binomialtree, but it didn't work. What I'm looking for is something like this: Any help is appreciated. Thanks in advance. I tried the edit the following code Use tikz to draw a hierarchical model (mult-level model from statistics) 2. n-period binomialtree/lattice in latex. 4 {genealogytree} How to model my Cousins. 3. How do I get these specific pathlines to nodes? 1. Binomial trees using forest package. 2. Modifying a decision tree terminal nodes using `tikz` 1. Connect adjacent nodes in hierarchy in LaTeX? 1. How to draw this tree of extensive. Black / Scholes Model. I. Limiting Case of N-Period Binomial Model. A. If we let N , split the time until exp. (e.g. 1 year) into shorter intervals, until S moves continuously. Then Binomial Model converges to Black / Scholes Model. B. More formally, IF we make the following assumptions: 1. Stock price (S . T) follows the lognormal distribution; S follows a random walk; If S. T has lognormal.
2.3. n-Period Binomial Model with Collateral. Next, the one-period model is extended to an n-period model. A two-period binomial asset pricing model with discrete dividends is illustrated in Figure 3. Figure 3. Two-period binomial asset pricing model with discrete dividends. Figure 3. Two-period binomial asset pricing model with discrete dividends. The price of a collateralized derivative. the theoretical price of the option under an n-period Binomial model with volatility ˙. Using a CRR parameterization and a large enough n, determine the so-called implied volatility smile, which is the graph of ^˙(K) versus K. Note: Need to create a program that automatically is able to recover the implied volatility ^˙(K) given nand (K). Then, use this program to obtain the implied. N-Period Model: C = S. 0 B[ a,N,P' ] - K (1+r)-N B[ a,N,P ]. where a = number of increases out of N trials. required for the call to finish in-the-money (S>K); P' = (u / r) P; B[ a,N,P ] = cumulative probability of getting at least a increases. out of N trials ( probability that call will finish ITM ). 4. Black / Scholes Option Pricing Model; Limiting case of N-Period Binomial Model. C = S.
In an N-period binomial model setup, the process {E, 0 < n < N}, where Yn — S2 +1 adapted to the filtration where is the a-field containing the sets determined by the first k tosses. TRUE FALSE No, the answer is incorrect. Score: 0 Accepted Answers: FALSE 3) Consider an Al-period binomial model setup with the filtration where is the a-fiel The n-Period Binomial Option Pricing Model; EXTENSION 18.1: Linking the Binomial Model to the Black-Scholes-Merton Model; Extending the Binomial Model . Known Dividends; Valuing American Options; Spreadsheet Applications . A Two-Period Binomial Example; A Sixteen-Period Example; Summary; Cases; Questions and Problems ; No Access. 19: The Black-Scholes-Merton Model. Pages: 422-461; https. Q2. In a (Cox-Ross-Rubinstein) binomial-tree model with discount rate 1+ˆ per period, 'up' and 'down' factors 1 + u, 1 + d and 'up' and 'down' proba-bilities q, 1 q, ﬁnd the condition for q to be the risk-neutral probability. Describe how to price an American put with strike K in an N-period binomial-tree model
Binomial. In a given period of n years, the probability of a given number r of events of a return period is given by the binomial distribution as follows. (=) = (). This is valid only if the probability of more than one occurrence per year is zero. Often that is a close approximation, in which case the probabilities yielded by this formula hold. Q. 2 (Binomial model gymnastics, 25 pts). Consider an N -period binomial model with up factor u, down factor d, risk-free return r, and probability p ∈ (0, 1) of up. Assume that the principle of no-arbitrage holds in this model. For each n ∈ {0, . . . , N }, let Sn be the stock price at time n. a. (10) Calculate the expected value of Sn. b.
Financial Derivative, Problem Sheet 3 Deadline: Sunday noon,12pm, week 4, MT16, 30th Oct. Question 1 Consider a binomial tree model for the stock price process {S t: 0 ≤ t ≤ 3} and ∆t = 1.Let S0 = 100 and let the price rise or fall by 10 % at each time step. We assume that the interest rate r = 5%.The contract we wish to price is an American pu The model is then compared to what the data suggests and conclusions are made. Comparison of N-period Binomial and Trinomial Asset Pricing Models, Thomas Devine, Myles Ellis, Branndon Marsical. Advisor: David Handron Abstract: The trinomial model is used to model stock prices and compute the present value of options. The factors that measure up, middle and down movements are u, m, and d. Its straightforward to see that we get the same results for an n period process as well. What the result means is that the discounted stock price [or forward price] for the (n+1)th period depends only on the stock price of nth period. Markov property. Within the binomial asset pricing model. Let X 0, X 1,....X N be a random process. If for every n between 0 and N-1 and f(x), there exists.
Option valuation via the generalised n-period binomial model 4. Understand the concept of no arbitrage and the risk-neutral approach to valuing derivatives securities. a. Explain the concept of no arbitrage when comparing actual and synthetic calls, or when comparing actual and synthetic puts. b. Explain the concepts underlying the risk-neutral approach to valuing derivatives securities in the. Q3. In a (Cox-Ross-Rubinstein) binomial-tree model with discount rate 1+ˆ per period, 'up' and 'down' factors 1 + u, 1 + dand 'up' and 'down' proba-bilities q, 1 q, nd the condition for qto be the risk-neutral probability. Describe how to price an American put with strike K in an N-period binomial-tree model Bounds for the Upper Prices of Options and Overhedging Strategies in n-period Models. Angewandte Mathematik und Informatik 19/00-S Wrede, M.: The Binomial Model: A Guide to Upper Prices and Overhedging Strategies in Time-discrete Models. Angewandte Mathematik und Informatik 26/00-S Außerdem wurde in der Reihe ``Skripten zur Mathematischen Statistik'' als Band 35 herausgegeben Dümbgen, L. Consider the binomial experiment with probabilityof success p,0 < p < 1. We let Nk denote the number of successes after k independent trials. Deﬁne the discrete process Yk by Nk −kp, the excess number of successes above the mean kp. Show that Y k is a martingale. 5. Consider the two-period securities model in the lecture note ofTopic3, p.15. Suppose the riskless interest rate r violates. As binomial models, trinomial model of n period is built based on trinomial model of one period. The trinomial model of one period is stock market (trading) model with one period (one time step), in other words, in this model there are only two trading time, which is t = 0 and t = 1. As discussed earlier, at the end of period or when t = 1, the stock price movement has three possible . First.
Consider the N-period binomial model and processes Y0,Y1,...YN and C0,C1,...CN. Suppose that Y is Markov and C is deterministic (i.e., Cn does not depend on the coin flips ù). a) Show: If Cn doesn't equal 0 for all n, then C0Y0,C1Y1,....,CNYN is Markov. b) Give an example of processes C and Y such that at least one Cn is zero and C0Y0,C1Y1,....,CNYN is not Markov. Jul 09 2020 09:16 PM. Expert. Q. 2 (Binomial model gymnastics, 25 pts). Consider an N -period binomial model with up factor u, dow; 5.9. You form a portfolio by combining shares of stocks for two different companies. The companies ha; 6.12. Which of the following firms has the smallest beta? a. Abercrombie & Fitch (Apparel) b. Val; 7.13. Which one of the following.
Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash ﬂow on an n-period stock price path. Path-dependent optionsare particularlydifﬁcultto value since the future cash ﬂow depends on the entire stock price path rather than. N-period binomial tree is PN t=0(t + 1) = (N + 2)(N + 1)/2 = O(N2), a quadratic growth in maturity N. this paper, the tremendously inﬂuential generalized autoregressive conditional het-eroskedastic (GARCH) model, is also bivariate. Bollerslev (1986) and Taylor (1986) independently proposed the GARCH proces
Pricing a Call Option with the Tree Model: 49: 3.3: Pricing an American Option: 52: 3.4: Pricing an Exotic Option--Knockout Options: 55: 3.5: Pricing an Exotic Option--Lookback Options: 59: 3.6: Adjusting the Binomial Tree Model to Real-World Data: 61: 3.7: Hedging and Pricing the N-Period Binomial Model: 66: 4: Using Spreadsheets to Compute. Moreover, we apply the discrete time formulization and analyse the quantum version of the Cox-Ross-Rubinstein Binomial Model. We find the limit of the N-period bond market, which convergences to planar Brownian motion and then we made an application to option pricing in planar Brownian motion compared with Levy models by Fourier techniques and Monte Carlo method. Furthermore, we analyse the. Using MCMC to Fit the Shifted-Beta-Geometric Customer Lifetime Value Model. Jan 19, 2015. Two professors of marketing, Peter Fader and Bruce Hardie, have developed probability models for estimating customer lifetime value (LTV). In their papers and example spreadsheets, they estimate the models using maximum-likelihood estimation (MLE) 14339W: February 5th 2021. The Pre-Limit i n-period Binomial TreesSn--y • •-e: ROOT I Slo). = sis.--the stock price: resulting from k upsteps • So p O T n periods e.g., in a forward tree Un. . . up factor un = e Cr-s)CTW tortadu. . . down factor du = ecr-8)(Tn)-Orth=D she = Slo)Unh-duh-k= Slo) (Udy)!din k corresponds to a realization of the binomial distribution w/ n trials and pin as.
The holder then has the same obligations as if a futures contract had originally been bought or sold) Strike Price Relationship to Current Futures Price Condition Put Option Call Option SP < futures Out-of-the money In-the money SP = futures At-the money At-the money SP > futures In-the money Out-of-the money History of Binomial Options Pricing The binomial options pricing model was developed. 3.2 Pricing a Call Option with the Tree Model 49 66; 3.3 Pricing an American Option 52 69; 3.4 Pricing an Exotic Option—Knockout Options 55 72; 3.5 Pricing an Exotic Option—Lookback Options 59 76; 3.6 Adjusting the Binomial Tree Model to Real-World Data 61 78; 3.7 Hedging and Pricing the N-Period Binomial Model 66 8 Finance Stoch (2007) 11:91-105 DOI 10.1007/s00780-006-0020-6 Smooth convergence in the binomial model Lo-Bin Chang · Ken Palmer Received: 17 August 2005 / Accepted: 18 August 2006
[Springer Finance] Stochastic Calculus for Finance I || The Binomial No-Arbitrage Pricing Model Shreve, Steven E. Volume: 10.1007/97. Year: 2004. Language: english. DOI: 10.1007/978--387-22527-2_1. File: PDF, 2.05 MB. Preview. Send-to-Kindle or Email . Please to your account first; Need help? Please read our short guide how to send a book to Kindle. Save for later. Post a Review . You. We study Bayesian decision making based on observations `Xn,t : t ∈ {0, Tn , 2 Tn , . . . , n Tn}´(T > 0, n ∈ N) of the discrete-time price dynamics of a financial asset, when the hypothesis a special n-period binomial model and the alternative is a different n-period binomial model. As the observation gaps tend to zero (i. e. n→∞), we obtain the limits of the corresponding Bayes risk.
Model Risk is the risk that a financial model used for pricing or risk management is an inappropriate, inaccurate model for the reality of the financial markets it aims to describe. Any financial position based on that particular financial model is consequently mispriced i.e. a mark to market value derived from the financial model i A binomial example--a call option on a 6 month T-bill p. 79 State prices p. 81 Risk-neutral probabilities p. 83 Linking state prices and probabilities p. 84 Multi-period valuation p. 87 A three-period binomial model p. 87 The n-period binomial model of Ho and Lee p. 91 The money-market account and the backward valuation formula p. 93 Questions p. 94 State prices, forward induction, and tree.