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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.

In this section we review some well-known phenomena in order to motivate subsequent developments and provide a background of the phenomena and terminology.

Volatility smiles. In a nutshell, vanilla options with different maturities and strikes have different volatilities implied by the well-known formula of [

For many years, practitioners and academics have tried to analyse the volatility smile phenomenon and under- stand its implications for derivatives pricing and risk management. In [

From a hedging perspective, traders who use the Black-Scholes model must continuously change the volatility assumption in order to match market prices. Their hedge ratios change accordingly in an uncontrolled way: the models listed above bring some order into this chaos. In the course of time, the general consensus, as advocated by practitioners and academics, is to choose a model that produces hedging strategies for both vanilla and exotic options resulting in profit and loss distributions that are sharply peaked at zero. We argue that a model recovered from option prices by no means explains the phenomenon.

Convexity adjustments. One of many well-known adjustments is the convexity adjustment; the implied yield of a futures and the equivalent forward rate agreement contracts are different. This phenomenon implies that market participants need to be paid more (or less) premium.

The common approach, as used by most practitioners and academics, is to adjust futures quotes such that they can be used as forward rates. Naturally, this approach depends on an model that is used for this purpose. For the extended Vasicek known as [

Parallel derivative markets. In an economic system, a financial market consists of a risk-free money account, primary and parallel markets. Examples of primary markets are stocks and bonds, and examples of parallel markets are derivatives such as forward, futures, vanilla options whose values are derived from the same primary asset. Market makers can trade and make prices for derivatives in a parallel market without references to another.

The framework is as follows: a complete probability space

martingale representing the price process of a risky asset.

The absence of arbitrage opportunities implies the existence of a probability measure

For simplicity, we consider only one horizon of uncertainty

generally,

In financial terms, every contingent claim can be replicated by means of a trading strategy (or inter- changeably known as hedging strategy or a replication portfolio) which is a portfolio consisting of the primary asset

and an adapted process, respectively.

tively, held at time

for

for

where

The constant value

Thus far, we have presented the well-known mathematical construction of a hedging strategy in a complete market where every contingent claim is attainable. In a complete market, derivative prices are unique—no arbitrage opportunities exist. Derivatives cannot be valuated in a parallel market at any price other than

From financial and economic point of view, the phenomena imply that the market is incomplete, arbitrage opportunities exist and may not be at all eliminated. A derivative can be valued at different prices and hedged by mutually exclusively trading in risky assets (or derivatives) in parallel markets where market makers engage in market activities: investments, speculative trading, hedging, arbitrage and risk management. In addition, market makers expose themselves to market conditions such as liquidity, see for instance [

In general, market incompleteness is a principle under which every contingent claim bears intrinsic risks. Let us postulate an assumption as a basis for subsequent reasonings and discussions.

Assumption. The market is incomplete and there exist intrinsic risks inherent in every contingent claim.

While the assumption is theoretical, it is rather realistically a proposition with the phenomena as proof.

In a mathematical context, let

The superscript indicates the dependence of a particular contingent claim

We now introduce the Kunita-Watanabe decomposition

where

where

example [

From a mathematical point of view, market incompleteness implies that there exists in the set

In this section we propose a continuous time financial market consisting of a primary price process

Let

where

where

We expand the portfolio value process (2) as follows:

where

and

Here,

and

for

and

It is not hard to see that the price process

Note that the martingale measure

The expectation is taken under the measure

It is important to note that in the risk-neutral world the essential theoretical assumptions are: 1) the true price process (10) is correctly specified and 2) prices of derivatives

We now consider the representation (7) in a continuous time framework where the measure of intrinsic risk (6) can be defined, without loss of generality, in terms of changes in values in a future time interval

Definition. A measure of intrinsic risk in a time interval

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

where

and

Under

Consequently the fair value of a contingent claim is given by the formula

From a pragmatic standpoint, what is needed in determining prices of derivatives and managing their risks is to allow sources of uncertainty that are epistemic (or subjective) rather than aleatory in nature. In theory, the value of a derivative can be perfectly replicated by a combination of other derivatives provided that these derivatives are uniquely determined by Formula (16). In practice, prices of derivatives (such as futures, vanilla options) on the same primary asset are not determined by (16) from statistically or econometrically observed model (10), but made by individual market makers who, with little, if not at all, knowledge of the true price process, have used their personal perception of the future. We argue further on this point as follows. If we let

asset),

where

which is finite for finite time

with

where

Here, we see the concurrence of the SDEs (19) and (22), the source of randomness

An important note here is that the trading strategy (17) is equivalent to the risk-free money account, that is the growth of portfolio value (2) is at the risk-free rate

In this section, we shall first discuss some problems related to asset models in parallel markets so as to provide some background for subsequent applications.

In the light of intrinsic risk, the SDE (21) in practice may represent a risky asset price process in parallel markets such as: 1) futures price process, or 2) an implied price process recovered from option prices where

Market makers indeed have dispensed with the correct specification (10) and directly use an implied price process as a tool to prescribe the dynamics of the implied volatility surface. A practice of recovering an implied price process from observed derivative prices (such as vanilla option prices) and use it to price derivatives is known as instrumental approach, described in [

Note that this hedge error resembles the term (23). Clearly, the hedging error is an intrinsic price of risk presented as traded asset in the hedging strategy (17), but not in (12).

Before we illustrate a number of applications for pricing and hedging with specific form of the measure of intrinsic risk, let us state a general result for derivative valuation.

We have established the risk-subjective valuation Formula (20) where the risk-subjective price process is given by (19).

Theorem 1. The risk-subjective value

is a unique solution to

with

Proof. The result is obtained by directly applying the Feynman-Kac formula.

We have shown that the trading strategy (17) yields the risk-free rate of return on the value of a derivative, and also the intrinsic risk is perfectly hedged by delta-hedging represented in (9) and (17).

As unpredictable as a market, prices in a parallel market (such as futures and corresponding vanilla options) may not be driven by the same source of randomness that drives the primary asset (such as stock and bond). Motivated by results (23) and (24), in the present framework it makes sense to formulate

where

the parameter

Remark. While the diffusion term

In practice, forward contracts are necessarily associated with the primary asset (such as stock and bond) and therefore their prices are determined by (16) and hedged by (12). As was illustrated in the previous section, can be determined by (20) which includes a measure of intrinsic risk,

Hedgers holding the primary asset in their hedging portfolio would receive dividends which are assumed to be a continuous stream of payments, whereas hedgers holding other hedge instruments (such as futures, vanilla options) do not receive dividends. In this case,

Suppose that

As an exogenous variable to the risk-subjective price process (19),

With reference to the liquidity preference theory or the preferred habitat theory of [

It is well-known among both academics and practitioners that the standard complete market framework often fails, see for example [

a structure is what needed to be imposed on the mutual movements of the primary and derivative markets so that, at least, the pricing and hedging derivatives (such as swaps and caplets) can be undertaken on a consistent basis. Apart from such conceptual aspect, the measure

where

In view of the last financial crises, the market has evolved and there is an apparent need, both among practitioners and in academia, to comprehend the problems caused by an excessive dependence on a specific asset modeling approach, by ambiguous specification of risks and/or by confusions between risks and uncertainties. We approach the problems by presenting a continuous time framework. This framework brings unity, simplicity and consistency to two important aspects: pricing with a correctly specified model for the primary asset, and risks must be correctly understood and specified. In addition, the framework proposed in this article is rigorous in the sense that the true meanings of properties and relationships of intrinsic risk and volatility are self-consistent such that their values are not arbitrarily assigned nor should their properties be misused by ignorance.