Abstract
We review the nature of some well-known phenomena such as volatility smiles, convexity adjustments and parallel derivative markets. We propose that the market is incomplete and postulate the existence of intrinsic risks in every contingent claim as a basis for understanding these phenomena. In a continuous time framework, we bring together the notion of intrinsic risk and the theory of change of measures to derive a probability measure, namely risk-subjective measure, for evaluating contingent claims. This paper is a modest attempt to prove that measure of intrinsic risk is a crucial ingredient for explaining these phenomena, and in consequence proposes a new approach to pricing and hedging financial derivatives. By adapting theoretical knowledge to practical applications, we show that our approach is consistent and robust, compared with the standard risk-neutral approach.
Highlights
We review some well-known phenomena in order to motivate subsequent developments and provide a background of the phenomena and terminology.Volatility smiles
We have shown that the trading strategy (17) yields the risk-free rate of return on the value of a derivative, and the intrinsic risk is perfectly hedged by delta-hedging represented in (9) and (17)
It is well-known among both academics and practitioners that the standard complete market framework often fails, see for example [37]
Summary
We review some well-known phenomena in order to motivate subsequent developments and provide a background of the phenomena and terminology. In relation to the hedging strategy (7), the measure of intrinsic risk shall be considered as the value of all possible future capital which, required to control the risk incurred by the market maker (such as hedger) and invested in the primary asset, makes the contingent claim acceptable, and its valuation fair. As was represented earlier in (7), the evolution of a trading strategy shall be adaptable to adjust for the measure of intrinsic risk which can be considered an additional/less capital required in a time interval dt , that is dV= (t ) α (t )(dX (t ) + dG (t )) +ν (t ) β (t ) D (t )dt. In terms of pricing and hedging, the presence of intrinsic risk imposes an internal consistency and implies that possible arbitrage exists in the market (the primary market and its associated derivative markets)
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