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**Unformatted text preview: **MAFS525 – Computational Methods for Pricing Structured Prod- ucts Topic 1 – Lattice tree methods 1.1 Binomial option pricing models • Risk neutral valuation principle • Continuous limit of the binomial model • Multiperiod extension • Dynamic programming procedure • Estimating delta and other Greek letters • Discrete dividend models • Pricing of lookback options- Hull-White scheme- Cheuk-Vorst algorithms 1 1.2 Trinomial schemes • Discounted expectation approach • Multistate extension – Ritchken-Kamrad’s approach 1.3 Forward shooting grid algorithms (strongly path dependent options) • Cumulative Parisian feature of knockout provision • Call options with strike reset feature • Alpha-quantile options • Floating strike arithmetic averaging call • Accumulators 2 1.1 Binomial option pricing models Risk neutral valuation principle Discrete model of the dynamics of the underlying price process Under the binomial random walk model, the asset prices after one period Δ t will be either uS or dS with probability q and 1- q , respectively. We assume u > 1 > d so that uS and dS represent the up-move and down- move of the asset price, respectively. The proportional jump parameters u and d will be related to the asset price dynamics. Let R denote the growth factor of riskless investment over one period so that $1 invested in a riskless money market account will grow to $ R after one period. In order to avoid riskless arbitrage opportunities, we must have u > R > d . 3 Formation of replicating portfolio By buying the asset and borrowing cash (in the form of riskless invest- ment) in appropriate proportions, one can replicate the position of a call. Suppose we form a portfolio which consists of α units of asset and cash amount M in the form of riskless investment (money market account). After one period of time triangle t , the value of the portfolio becomes braceleftBigg αuS + RM with probability q αdS + RM with probability 1- q. 4 The portfolio is used to replicate the long position of a call option on a non-dividend paying asset. As there are two possible states of the world: asset price goes up or down, the call price is dependent on asset price, thus it is a contingent claim. Suppose the current time is only one period triangle t prior to expiration. Let c denote the current call price, and c u and c d denote the call price after one period (which is the expiration time in the present context) corresponding to the up-move and down-move of the asset price, respectively. 5 Let X denote the strike price of the call. The payoff of the call at expiry is given by braceleftBigg c u = max( uS- X, 0) with probability q c d = max( dS- X, 0) with probability 1- q....

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