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barefootpilgrim89
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Posts: 16
Joined: April 3rd, 2020, 9:33 pm

Is it not necessary to use any risk premium when valuing a stock option?

April 7th, 2020, 10:55 am

Dear Quant Finance Community,

"The attempts at valuation before 1973 basically determined the expected value of a stock option at expiration and then discounted its value back to the time of evaluation. Such an approach requires taking a stance on which risk premium to use in the discounting. This is because the value of an option depends on the risky path of the stock price, from the valuation date to maturity. But assigning a risk premium is not straightforward. The risk premium should reflect not only the risk for changes in the stock price, but also the investors attitude towards risk. And while the latter can be strictly defined in theory, it is hard or impossible to observe in reality. This year's laureates resolved these problems by recognizing that it is not necessary to use any risk premium when valuing an option. This does not mean that the risk premium disappears, but that it is already incorporated in the stock price." - https://www.nobelprize.org/prizes/economic-sciences/1997/advanced-information/

Does anyone have insight regarding the assumption that the risk premium is already incorporated in the stock price? This seems to be related to the controversial efficient markets hypothesis?

Assuming that the assumption "recognized" by Black, Scholes and Merton are correct; it would seem that the risk premium of an option (which doesnt disappear) is being calculated based on a constant volatility assumption of the underlying, the numerical value of which gets derived from historical data. If this is correct, then option premiums should only be calculated based on historical stock price movement data and the current price of the stock?

If the risk premium doesn't disappear but is already incorporated in the stock price, where did implied volatility really come from? Are practitioners (buy-side and sell-side) making the assumption that the stocks risk premium is already incorporated in the stock price? Or are practitioners reintroducing their own subjective or mathematical estimations of what they believe the future may hold, thereby ignoring the premise of BSM that it is not necessary to use any risk premium when valuing an option?

It would seem that in practice, implied volatility represents the "plug" that practitioners can use to satisfy their own subjective views or mathematical estimations of what they believe the future may hold in terms of the underlying stocks unknown pricing trajectory. This would suggest that the option risk premium does reflect not only the risk for changes in the stock price, but also the investors attitude towards risk? 
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