Munich Personal RePEc Archive

Options valuation.

ilya, gikhman (2005): Options valuation.

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Abstract

This paper deals with the option-pricing problem. In the first part of the paper we study in details the discrete setting of the option-pricing problem usually referred to as the binomial scheme. We highlight basic differences between the old and the new approaches. The main qualitative distinction of the new pricing approach from either binomial or Black Scholes’s is that it represents the option price as a stochastic process. This stochastic interpretation can not give straightforward advantage for an investor due to stochastic setting of the pricing problem. The new approach explicitly states that the options price is more risky than represented by binomial scheme or Black Scholes theory. To highlight the difference between stochastic and deterministic option price definitions note that if a deterministic value is interpreted as a perfect or fair price we can comment that the stochastic interpretation provides this number or any other with the probability that real world option value at maturity will be bellow chosen number. This probability is a pricing risk of the option. Thus with an investor’s motivation of the option pricing the stochastic approach gives information about the risk taking. The investor analyzing option price and corresponding risk makes a decision to purchase the option or not. Continuous setting will be considered in the second part of the paper following [1]. A significant conclusion can be drawn from the new approach. It is shown that either binomial or Black-Scholes solutions of the option pricing problem have serious drawbacks. In particular, the binomial scheme establishes the unique price for a stock that takes two values and strike price K, Sd < K < Su. According the binomial scheme this ‘fair’ price does not depends on real probabilities. Thus two options with that promise fixed income at maturity with probability close to 1 or 0 do have the same price. This of course does not have any sense. From this follows that there is no sense in using either neutral probabilities or ‘neutral world’ in options applications for valuation interest rates or credit derivatives either theoretically or numerically. Recall that Black Scholes’ approach was introduced in [2] and then later the binomial scheme was published [3]. Here we first represent discrete scheme. In several examples we discuss two-period plain vanilla option valuation. Note that the scheme can be applied for arbitrary states of a security over one step market. Then we extend the discrete scheme over an application to exotic option-pricing referred to as a compound option. The compound option in Black Scholes setting was first studied in [4] and then in [5,6].

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