Anchoring And Adjustment Heuristic In Option Pricing
University of Queensland
December 30, 2015
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Based on experimental and anecdotal evidence, an anchoring-adjusted option pricing model is developed in which the volatility of the underlying stock return is used as a starting point that gets adjusted upwards to form expectations about call option volatility. I show that the anchoring price lies within the bounds implied by risk-averse expected utility maximization when there are proportional transaction costs. The anchoring model provides a unified explanation for key option pricing puzzles. Two predictions of the anchoring model are empirically tested and found to be strongly supported with nearly 26 years of options data.
Anchoring And Adjustment Heuristic In Option Pricing – Introduction
A call option is widely considered to be a stock surrogate by market professionals as their payoffs are closely related by construction, and move in sync perhaps more than any other pair of assets in the market. This built-in similarity between the payoffs of the two instruments has given rise to a popular trading strategy known as the “stock replacement strategy”, which includes replacing the underlying stock in a portfolio with a corresponding call option.
Where would you start if you need to form a risk judgment about a given call option? One expects the risk of a call option to be related to the risk of the underlying stock. In fact, a call option creates a leveraged position in the underlying stock. Hence, a natural starting point is the risk of the underlying stock, which needs to be scaled-up. Defining as the standard deviation of stock returns, and as the standard deviation of call returns, one expects the following to hold:
where (1+?) is the scaling-up factor.
The Black-Scholes model specifies a particular value for ?, which is equal to ?1 where is the call price elasticity with respect to the underlying stock price. Any other value of ? creates a risk-less arbitrage opportunity under Black-Scholes assumptions. For example, if ?<?1, the call option gets overpriced in comparison with the cost of replicating portfolio if the risk premium on the underlying stock is positive. One may buy the replicating portfolio and write the call option to make riskless arbitrage profits. Hence, the only plausible value is the correct one.
If one allows for a little ‘sand in the gears’ in the form of transaction costs while keeping the other assumptions of the Black-Scholes model the same, a whole range of values of ? different from the Black-Scholes implied value become plausible. That is, it becomes possible to support incorrect beliefs in equilibrium because such beliefs cannot be arbitraged away.
With transaction costs, the belief about ? can be on either side of the correct value in equilibrium; however, there are strong cognitive and psychological reasons to expect that it falls consistently on one side. Due to their strongly similar payoffs, underlying stock volatility is a natural starting point, which is adjusted upwards to form call volatility judgments. Beginning with the early experiments in Tversky and Kahneman (1974), over 40 years of research has demonstrated that starting from (often self-generated) initial values, adjustments tend to be insufficient. This is known as the anchoring bias (see Furnham and Boo (2011) for a literature review). From self-generated anchors, adjustments are insufficient because people tend to stop adjusting once a plausible value is reached (see Epley and Gilovich (2006) (2001) and references therein). Hence, assessments remain biased towards the starting value known as the anchor.
A few examples illustrate the anchoring and adjustment heuristic quite well. Studies (Epley and Gilovich (2006) among others) have shown that when asked, most respondents do not know the year George Washington became the first president of America. However, they know that it had to be after 1776 as the declaration of independence was signed in 1776. So, while guessing their answer, they tend to start with 1776 to which they add a few years and then stop once a plausible value has been reached. Such a cognitive process implies that their answers remain biased towards the self-generated anchor as people typically stop adjusting at the edge of plausible values on the side of the anchor.
Another example of this thinking process is provided by the freezing point of Vodka. Most respondents know that Vodka freezes at a temperature below the freezing point of water, so they start from 32 degree Fahrenheit (0 Celsius) and adjust downwards. However, such adjustments are typically insufficient (Epley and Gilovich (2006)).
What is the fair price of a 3-bedroom house in the Devon neighborhood of Chicago? If you know the sale price of a 4-bedroom house in the same neighborhood but in a slightly better location, then you would likely start with that price and adjust downwards for size, location, and other differences. The above examples illustrate that the defining feature of a self-generated anchor is its strong relevance to the problem at hand. Without such anchors, one expects people to choose around the middle value within the set of values deemed plausible by them. Self-generated anchors change this as the process of adjustment stops once a value deemed plausible has been reached. Hence, location of the anchor relative to the set of plausible values becomes an important factor.
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