Intrinsic Prices Of Risk

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.


Introduction
This section has two purposes. Firstly, we review the well-known phenomena in order to motivate the subsequent development. After that, we provide a background of the phenomena with some notation, terminology and notions.
1.1. Phenomena. Volatility smiles. In a nutshell, vanilla options with different maturities and strikes have different volatilities implied by the well-known formula of [Black & Scholes(1973)]. Implied volatility is quoted as the market expectation about the average future volatility of the underlying asset over the remaining life of the option. Thus compared to historical volatility it is the forward looking approach.
For many years, practitioners and academics have tried to analyse the volatility smile phenomenon and understand its implications for derivatives pricing and risk management. In [Cox & Ross(1976)], their link between the real-world and risk-neutral processes of the underlying would be complete by non-traded sources of risk. [Scott(1987)] found that the dynamics of the risk premium, when volatility is stochastic, is not a traded security. A number of models and extensions of, or alternatives to, the Black-Scholes model, have been proposed in the literature: the local volatility models of [Dupire(1994)], [Derman & Kani(1994)] and Rubinstein (1994); a jump-diffusion model of [Merton(1976)]; stochastic volatility models of [Hull & White(1988)], [Heston(1993)] and others; mixed stochastic jump-diffusion models of [Bates(1996)] and others; universal volatility models of [Dupire(1996)], [Morgan(1999)], [Britten-Jones & Neuberger(2000)], [Blacher(2001)] and others; regime switching models, etc.
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. Such a model, if recovered (or implied) from option prices, by no means nearly explains this phenomenon, but is a means only to describe the implied volatility surface.
Convexity adjustments. One of many well-known adjustments is the convexity adjustmentthe 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 the actual model used for this purpose. For the extended Vasicek known as [Hull & White(1990)] and [Cox et al.(1985)] model, explicit formulae can be derived. The situation is different for models whose continuous description gives the short rate a log-normal distribution such as the [Black et al.(1990)] and [Black & Karasinski(1991)] models: for these, in their analytical form of continuous evolution, futures prices can be shown to be positively infinite [Heath et al. (1992)] and [Sandmann & Sondermann(1994)]. In subsequent developments, we shall offer a different approach to this phenomenon.
Parallel derivative markets. In an economic system, a financial market consists of a riskfree money account, primary and parallel markets. Examples of primary markets are stocks and bonds, and of parallel markets for derivatives such as futures, vanilla options, credits which are derived from the same primary asset. Market makers can trade and make prices for derivatives in a parallel market without references to another.

1.2.
Background. The framework is as follows: a complete probability space (Ω, F, P) with a filtration F = F(t) satisfying the usual conditions of right-continuity and completeness. T ∈ R denotes a fixed and finite time horizon; furthermore, we assume that F(0) is trivial and that F(T ) = F. Let X = X(t) be a continuous semimartingale representing the price process of a risky asset.
The absence of arbitrage opportunities implies the existence of an probability measure Q equivalent to the probability measure P (the real world probability), such that X is a Qmartingale. Denote by Q the set of coexistent equivalent measures Q. A financial market is considered such that Q = ∅. Uniqueness of the equivalent probability measure Q implies the market is complete. The fundamental theorem of asset pricing establishes the relationship between the absence of arbitrage opportunities and the existence of an equivalent martingale measure and in a basic framework was proved by [Harrison & Kreps(1979)], [Harrison & Pliska(1981), Harrison & Pliska(1983)]. The modern version of this theorem, established by [Delbaen & Schachermayer(2004)], states that the absence of arbitrage opportunities is "essentially" equivalent to the existence of an equivalent martingale measure under which the discounted price process is a martingale.
For simplicity, we consider only one horizon of uncertainty [0, T ]. A contingent claim, or a derivative, H = H(ω) is a payoff at time T , contingent on the scenario ω ∈ Ω. The derivative has the special form H = h(X(T )) for some function h. Here, X is referred to as the primary (or the 'underlying'). More generally, H depends on the whole evolution of X up to time T and is a random variable (1.1) H ∈ L 2 (Ω, F, P).
In financial terms, every contingent claim can be replicated by means of a trading strategy (or interchangeably known as hedging strategy or a replication portfolio) which is a portfolio consisting of the primary asset X and a risk-free money account D = D(t). Let α = α(t) and β = β(t) be a predictable process and an adapted process, respectively. α(t) and β(t) are the amounts of asset and money account, respectively, held at time t. In this section, for ease of exposition, we assume that D(t) = 1 for all 0 ≤ t ≤ T . The value of the portfolio at time t is given by for 0 ≤ t ≤ T . It can be shown that the trading strategy (α, β) is admissible such that the value process V = V (t) is square-integrable and have right-continuous paths and is defined by For Q-almost surely, every contingent claim H is attainable and admits the following representation . Moreover, the strategy is self-financing, that is the cost of the portfolio (also known as derivative price) is a constant V 0 The constant value V 0 represents a perfect replication or a perfect hedge.
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 V 0 .
From a financial point of view, the phenomena described above imply that the market is incomplete. In parallel markets derivatives are themselves risky assets and their prices are, without particular references to the primary asset, determined by market makers who engage in activities: investments, speculative trading, hedging, arbitrage and risk management. In addition, there are market-inherent problems such as liquidity preferences, default risk of assets, risk preferences, regulatory compliances, attitudes toward risk, asymmetric information (see for example [Back(1993)]). In essence, we argue that level of these activities, severity of market-inherent problems and generally uncertain future events constitute a basis of arbitrage on which derivatives are evaluated. We shall call this basis of arbitrage the intrinsic risk. In relation to the hedging strategy (1.5), the measure of intrinsic risk can be used as the minimum (extra or less) capital which, required to control the risk anticipated by market makers and invested in the primary asset, makes the corresponding contingent claim acceptable.
In an economic sense, a derivative can be valued at different prices and hedged by mutually exclusively trading in risky assets in parallel markets. Arbitrage opportunities exists internally in an incomplete market. In general, market incompleteness is a principle and every contingent claim bears an intrinsic risk. Let us postulate a proposition as a basis for subsequent reasonings and discussions.
Proposition 1. The market is incomplete and there exists an intrinsic risk embedded in every contingent claim.
From a mathematical point of view, market incompleteness implies that there exists an equivalent measure, not necessarily a martingale measure, in the set Q assigned to a parallel market. As a basic object of our study, intrinsic risk shall therefore be a random variable on the set of states of nature at any future time t ∈ [0, T ] and its measure shall be interpreted as a possible future capital required to maintain the replicating portfolio. Let G be an intrinsic risk. Generally, G depends on the evolution of the asset X up to time T and may also depend on the contingent claim H.
The superscript indicates the dependence of a particular contingent claim. This leads to a new representation of H We now introduce the Kunita-Watanabe decomposition where V * 0 = V 0 + G 0 and α * = α + α H . This representation of H have been extensively dealt with, see for example [Follmer & Schweizer(1991)]. By incompleteness, the derivative value V * 0 manifests an intrinsic value G 0 . Intrinsic risk may depend on the derivative, as such it takes many forms some of which we shall propose in section 3. We shall not discuss further on the representations (1.7) and (1.9), but use them as a way of introducing the notion of intrinsic risk and make this notion concrete in a continuous time framework.

Market, Portfolio, Arbitrage and Intrinsic Price of Risk
In this section we propose a financial market with a specific form for the primary price process X and the risk-free money account D. We shall propose an explicit form for the intrinsic risk and show that explicit self-financing strategies can be constructed. We also show that the existence of intrinsic risk provides an internal consistency in pricing and hedging a contingent claim.
Let B = B(t) be a Brownian motion on the complete probability space (Ω, F, P). The underlying price process of X satisfies the SDE (the generalised Black-Scholes model) where µ = µ(t) and σ = σ(t) are Lipschitz continuous functions so that a solution exists. µ and σ can be functions of X. The price process of D is given by where ν = ν(t) is a Lipschitz continuous function.
We expand the portfolio value process (1.2) as follows: where W = W (t) is a Q-Brownian motion and is defined by and Here, Q is some martingale measure. Indeed, the theory of the Girsanov change of measure, see for example [Karatzas & Shreve (1998)], shows that there exists such a martingale measure Q equivalent to P and which excludes arbitrage opportunities. More precisely, there exists a probability measure Q ≪ P such that and X is a Q-martingale. Such a martingale measure Q is determined by the right-continuous square-integrable martingale for 0 ≤ t ≤ T . And explicitly and λ satisfies Novikov's condition It is not hard to see that the price process X under Q is given by Note that the martingale measure Q and λ are, if unique, theoretically and practically wellknown as the risk-neutral measure and the market price of risk, respectively. The risk-neutral valuation formula is given by The expectation is taken under the measure Q.
It is important to note that in the risk-neutral world the essential theoretical assumptions are: (1) the true price process (2.1) is correctly specified and (2) prices of derivatives H are drawn from this correct price process, that is derivative prices are uniquely determined by formula (2.7). These assumptions, if not violated, lead to a complete market and the trading strategy (2.3) and the measure Q are unique. However, in reality, these assumptions are strongly violated; as a result market completeness and uniqueness of derivative prices are no longer valid. That is Q is no longer risk-neutral, but only an equivalent measure in the set Q.
As proposed in (1), there exists an intrinsic risk embedded in every contingent claim. How does this intrinsic risk impact the hedging strategy (2.3) with X being the primary asset? In the present framework, it makes sense to formulate the intrinsic risk expressed per unit time as a fraction of the risky asset X. In mathematical terms, let the intrinsic risk be a gain/loss stream accumulated in a time interval dt being ζXdt, where ζ is a fraction of the primary asset.
Let ζ = ζ(t, T ) be a continuous adapted process representing a continuous time rate of intrinsic risk. The intrinsic risk, G = G(t, T ), in a time interval dt is given by dG(t, T ) = ζ(t, T )X(t)dt.
As represented in (1.7), the evolution (2.3) of a self-financing portfolio must be modified to incorporate an intrinsic risk as follows: where Z = Z(t) is a S-Brownian motion and is given by and S is a measure equivalent to P. Thus, S ∈ Q. Analogously, ζ/σ is defined as an intrinsic price of risk.
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 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 the formula (2.7). In reality, prices of derivatives (such as futures, vanilla options) on the same primary asset are not determined by (2.7) from statistically or econometrically observed model (2.1), 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. If we let Y = Y (t) be the price process of a derivative in a derivative market (such as futures), Y must have an abstract dynamics and is assumed to satisfy a SDE where T denotes a fixed time horizon larger than or equal to the maturity of any contingent claim,σ is a Lipschitz continuous function so that a solution exists. We now show that Z is a S-Brownian motion -the source of randomness that drives the price process Y . We introduce a change of time, see for example [Klebaner(2012)]. Let U (t) be a positive function such that which is finite for finite time t ≤ T and increases almost surely. Define τ (t) = U −1 (t), let Y be a replacement of X, i.e. X(t) = Y (τ (t), τ (T )) whose solution is given by with X(0) = Y (0). Rearranging the drift term gives where (2.14) Here, we see the concurrence of the SDEs (2.10) and (2.13), the source of randomness Z is the very S-Brownian motion (2.9). The measure S is subjective in the sense that if a derivative (such as futures) exists in a parallel market as a primary asset, its value is determined (2.11) and hedged by (2.3). If not, it can be determined by (2.7) and hedged by (2.3). We shall call the measure S the risk-subjective measure. The relationship between the risk-subjective measure and the risk-neutral measure described by (2.9) is far more precise than that found in [?].
An important note here is that the trading strategy (2.8) is equivalent to the risk-free money account, that is the growth of portfolio value (1.2) is at the risk-free rate ν. The presence of intrinsic risk imposes an internal consistency in the market with respect to pricing and hedging and no arbitrage exists between the primary market and its associated derivative markets.

Applications
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 (2.12) in reality 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 whereσ is the implied volatility. Attempts of recovering the implied price process were pioneered, for examples, by [Schonbucher(1999)], [Cont et al.(2002)], [Le(2005)], [Carr & Wu(2010)] and references therein.
Market makers indeed have dispensed with the correct specification (2.1) 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 [Rebonato(2004)]. A practical point that is more pertinent to the instrumental approach is that the prices of exotic derivatives are given by the price dynamics that can take into account or recover the volatility smile. With reference to intrinsic risk, an implied price process is a mis-specification for the primary asset, this was discussed in [El Karoui et al.(1998)] and was shown that successful hedging depends entirely on the relationship between the mis-specified volatilityσ and the true local volatility σ, and the total hedging error is given by, assuming zero risk-free rate, Note that this hedge error resembles the term (2.14). Clearly, the hedging error is an intrinsic price of risk presented as traded asset in the hedging strategy (2.8), but not in (2.3).
Remark. While the existence of intrinsic risk appears to undermine the true probability distribution of the underlying, it emphasises its important role in determining the values of derivatives. It ensures maximal consistency in pricing and hedging contingent claims that are path-dependent/independent and particularly derivatives on volatility (such as variance swap, volatility swap). It insists on a realistic dynamics for the underlying asset as far as delta hedge is concerned.
Before we illustrate a number of applications for pricing and hedging with specific form of intrinsic prices of risk, let us state a general result for derivative valuation.
3.1. Risk-subjective valuation. We have established the risk-subjective valuation formula (2.11) where the risk-subjective price process is given by (2.10).
Theorem 1. The risk-subjective value V of a contingent claim H = h(X(T )) given by is a unique solution to Proof. The result is obtained by directly applying the Feynman-Kac formula.
3.2. Parallel derivative models. 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 (2.14) and (3.1), in the present framework it makes sense to formulate ζ by an abstract form where σ is the volatility of the underlying asset,σ the volatility of a risky asset in a parallel market. We propose that ζ takes a general form of an exponential family the parameter θ = {σ,σ} and X(t) = x. As a result, (3.4) is a special case.
3.3. Valuation of forward and futures contracts. In reality, forward contracts are necessarily associated with the primary asset (such as stock and bond) and therefore their prices are determined by (2.7) and hedged by (2.3). Although the futures contracts is a contingent claim on the primary asset, it is itself a primary asset in the futures market, and therefore can be determined by (2.11) which includes a component of intrinsic risk, ζ, as a convexity adjustment.
3.4. Contingent claims on dividend paying assets with default 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, ζ can be considered as dividend yield and ζXdt is the amount of dividend received in a time interval dt. ζ may be a non-negative function representing the hazard rate of default in a time interval dt, this well-known approach was proposed in [Linetsky (2006)] and references therein.
3.5. Interest rate models. As an exogenous variable to the risk-subjective price process (2.10), ζ of a particular form would become a mean of reversion. This is a desirable feature in a number of well-known interest rate models such as extended model of [Hull & White(1990)], [Black & Karasinski(1991)] model.
With reference to the liquidity preference theory or the preferred habitat theory of [Keynes(1964)], a term premium for a bond can be represented as an intrinsic risk.

Concluding remarks
It is well-known among both academic and practitioners that the standard complete market framework often failed, see for example [Mehra & Prescott(1985)]. Incomplete market framework becomes crucial to explain well-known market anomalies. In this article we have introduced the notion of intrinsic risk and derived the risk-subjective measure S equivalent to the real-world measure P, where S ∈ Q. At a conceptual level, the theory of Girsanov change of measure allows us to recognise that the crucial role of S, rather than the expectation E S [H], is assigned to the price of a derivative (such as futures, vanilla option). In addition, the intrinsic risk as 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 S does not undermine the role of the measure P in that a lot of knowledge about the primary market is known at any given time t. More precisely, the market's expectation (predictions) in terms of a measure S at time t is given by the conditional probability distribution where F(t) is the information available given by the primary market at time t, and F S (t) is the information generated by derivatives (such as vanilla options) with maturities T > t.
A final remark: In view of the recent financial crisis, 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 (volatilities). As a result, we presented a continuous time framework that, we believe, brings unity, simplicity and consistency to two important aspects: pricing with correctly specified model for a primary risky asset, and hedging risks that can 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.