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Understanding the Binomial Option Pricing Model
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With applications in Equity and FX
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Previous Article Optimal treated mosquito bed nets and insecticides for eradication of malaria in Missira. The factors and tradables typically have nonempty intersection, so that if, for example, factor Ai t is also a tradable, denoted by S j t , i. Example 2. Nonredun- dancy outlaws perfectly correlated tradables, meaning that if in some theoretical or practical situations there is market redundancy, i. Also, if we start solving the same set of problems with too few factors, it will become transparent quickly that in such a setup there cannot be a solution.
That depends on two things. One is the market dynamics postulated in 2. Indeed, while the market dynamics affect the pricing PDE, the payoff structure will affect the terminal condition and, possibly, the right-hand side of the pricing PDE. The notation in 2. Observe that vectors, such as S t , are always understood as one-dimensional arrays, so that they cannot, and need not, be transposed. Otherwise, the market S t is said to be incomplete.
They are equivalent, so it is a matter of taste. Our choice was motivated by the structure of the pricing PDEs to be developed in the next chapter. We shall study both complete and incomplete markets, with emphasis on incomplete markets, while complete markets will be viewed simply as special cases of the incomplete ones.
Neutral and Indifference Portfolio Pricing, Hedging and Investing
Indeed, 2. This model is referred to as the log-normal or Black—Scholes model. We also note that the state space, i. One of them is by accounting for the randomness of the interest rates see [5, 7, 12, 16, 23, 32, 43, 46]. Here we embed the very well known Vasicek model  in the above simple- economy framework.
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We consider two cases. Consider the same model by declaring two factors, and derive the same conclusion as above. Exercise 2. Consider the same model by declaring two appropriately chosen factors, and derive the same conclusion as above. Indeed, inside Mathematica the data will look like the following Warning: data-providing web sites sometimes change the format.
For example, if decimal time counted in years and closing prices are extracted from the above data set, then the above, after little Mathematica programming MathematicaTime[x? We shall adopt the latter approach here. Consider two of the above models for stochastic volatility, 2. We observe that once the above minimizations 2. Sign In.
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Home Explore Srdjan Stojanovic auth. Like this book? You can publish your book online for free in a few minutes! Download PDF.
Discover the best professional documents and content resources in AnyFlip Document Base. Published by feqah. What is the effect of the dividend policy see Preface ix Section 6. Stojanovic Contents 1 Background Material. We observe 1. Nevertheless, while forward contracts can, and typically do, have nonzero value thereafter, futures contracts, since they are settled at the end of each trading day, and in the idealized world of stochastic calculus settled instantaneously, 1.
The plot presents 16 1 Background Material Auger, William F. Sheridan auth. Bass, James A. Birchler eds. The portfolio remains risk-free regardless of the underlying price moves. Supposing instead that the individual probabilities matter, arbitrage opportunities may have presented themselves. In the real world, such arbitrage opportunities exist with minor price differentials and vanish in the short term.
But where is the much-hyped volatility in all these calculations, an important and sensitive factor that affects options pricing? The volatility is already included by the nature of the problem's definition. But is this approach correct and coherent with the commonly used Black-Scholes pricing? Options calculator results courtesy of OIC closely match with the computed value:. Is it possible to include all these multiple levels in a binomial pricing model that is restricted to only two levels?
Yes, it is very much possible, but to understand it takes some simple mathematics.
Factor "u" will be greater than one as it indicates an up move and "d" will lie between zero and one. The call option payoffs are "P up " and "P dn " for up and down moves at the time of expiry. If you build a portfolio of "s" shares purchased today and short one call option, then after time "t":. Solving for "c" finally gives it as:. Note: If the call premium is shorted, it should be an addition to the portfolio, not a subtraction.
Overall, the equation represents the present day option price , the discounted value of its payoff at expiry. Substituting the value of "q" and rearranging, the stock price at time "t" comes to:. In this assumed world of two-states, the stock price simply rises by the risk-free rate of return, exactly like a risk-free asset, and hence it remains independent of any risk. Investors are indifferent to risk under this model, so this constitutes the risk-neutral model.
In real life, such clarity about step-based price levels is not possible; rather the price moves randomly and may settle at multiple levels.
Neutral and Indifference Portfolio Pricing, Hedging and Investing
To expand the example further, assume that two-step price levels are possible. We know the second step final payoffs and we need to value the option today at the initial step :. To get option pricing at number two, payoffs at four and five are used. To get pricing for number three, payoffs at five and six are used. Finally, calculated payoffs at two and three are used to get pricing at number one. Please note that this example assumes the same factor for up and down moves at both steps — u and d are applied in a compounded fashion.
Similarly, binomial models allow you to break the entire option duration to further refined multiple steps and levels. Using computer programs or spreadsheets, you can work backward one step at a time to get the present value of the desired option.