Delta hedging, replicating portfolio, and risk neutral valuation pricing in a multiple time-step setting

The delta hedging and replicating portfolio perspectives for pricing options are important in that they enable us to think carefully as well as deeply concerning the economics of option pricing. For example, the delta hedging approach illustrates that an appropriately hedged portfolio consisting of either long-short call-share positions or long-long put-share positions is riskless and consequently must produce a riskless rate of return. Similarly, the replicating portfolio approach reminds us that a call option represents a “synthetic” margined stock investment, whereas a put option represents a “synthetic” short sale of the share combined with lending money. As we saw in our study earlier this semester of forward contracts, if there is any difference between the value of a derivative and its replicating portfolio, then one can earn profits with zero net investment and no exposure to risk. Thus, the “arbitrage-free” price for the derivative (option or forward) corresponds to the value of its replicating portfolio.

An important result shown in the Binomial Option Pricing Model (single-period) reading which we will cover tomorrow is that the delta hedging and replicating portfolio approaches to pricing options both imply that a “risk-neutral” valuation relationship exists between the derivative and its underlying asset.  This insight provides a deceptively simple pricing equation which enables us to price options by discounting the risk-neutral expected value of the option payoff (i.e., its certainty-equivalent) at the riskless rate of interest.  During tomorrow’s class meeting, I also hope to extend the pricing model from a single timestep to a multiple timestep setting. To accomplish this task, we rely upon backward induction. Backward induction involves beginning at the very end of the binomial tree and working our way back to the beginning. We’ll continue with the numerical example introduced during class today and illustrate the backward induction method for the delta hedging, replicating portfolio and risk neutral valuation approaches. We’ll discover that multiple timesteps imply that the delta hedging and replicating portfolio methods imply “dynamic” trading strategies which require portfolio rebalancing as the price of the underlying asset changes over time (cf. the Dynamic Delta Hedging Numerical Example (calls and puts) and Dynamic Replicating Portfolio Numerical Example (calls and puts) readings).

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