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Donghan Kim
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An open market is a subset of an entire equity market composed of a certain fixed number of top capitalization stocks. Though the number of stocks in the open market is fixed, the constituents of the market change over time as each companys rank by its market capitalization fluctuates. When one is allowed to invest also in the money market, the open market resembles the entire closed equity market in the sense that the equivalence of market viability (lack of arbitrage) and the existence of numeraire portfolio (portfolio which cannot be outperformed) holds. When access to the money market is prohibited, some topics such as Capital Asset Pricing Model (CAPM), construction of functionally generated portfolios, and the concept of the universal portfolio are presented in the open market setting.
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We consider the problem of optimal inside portfolio $pi(t)$ in a financial market with a corresponding wealth process $X(t)=X^{pi}(t)$ modelled by begin{align}label{eq0.1} begin{cases} dX(t)&=pi(t)X(t)[alpha(t)dt+beta(t)dB(t)]; quad tin[0, T] X(0)&=x_0>0, end{cases} end{align} where $B(cdot)$ is a Brownian motion. We assume that the insider at time $t$ has access to market information $varepsilon_t>0$ units ahead of time, in addition to the history of the market up to time $t$. The problem is to find an insider portfolio $pi^{*}$ which maximizes the expected logarithmic utility $J(pi)$ of the terminal wealth, i.e. such that $$sup_{pi}J(pi)= J(pi^{*}), text {where } J(pi)= mathbb{E}[log(X^{pi}(T))].$$ The insider market is called emph{viable} if this value is finite. We study under what inside information flow $mathbb{H}$ the insider market is viable or not. For example, assume that for all $t<T$ the insider knows the value of $B(t+epsilon_t)$, where $t + epsilon_t geq T$ converges monotonically to $T$ from above as $t$ goes to $T$ from below. Then (assuming that the insider has a perfect memory) at time $t$ she has the inside information $mathcal{H}_t$, consisting of the history $mathcal{F}_t$ of $B(s); 0 leq s leq t$ plus all the values of Brownian motion in the interval $[t+epsilon_t, epsilon_0]$, i.e. we have the enlarged filtration begin{equation}label{eq0.2} mathbb{H}={mathcal{H}_t}_{tin[0.T]},quad mathcal{H}_t=mathcal{F}_tveesigma(B(t+epsilon_t+r),0leq r leq epsilon_0-t-epsilon_t), forall tin [0,T]. end{equation} Using forward integrals, Hida-Malliavin calculus and Donsker delta functionals we show that if $$int_0^Tfrac{1}{varepsilon_t}dt=infty,$$ then the insider market is not viable.
The objective of this paper is to introduce the theory of option pricing for markets with informed traders within the framework of dynamic asset pricing theory. We introduce new models for option pricing for informed traders in complete markets where we consider traders with information on the stock price direction and stock return mean. The Black-Scholes-Merton option pricing theory is extended for markets with informed traders, where price processes are following continuous-diffusions. By doing so, the discontinuity puzzle in option pricing is resolved. Using market option data, we estimate the implied surface of the probability for a stock upturn, the implied mean stock return surface, and implied trader information intensity surface.
Using agent-based modelling, empirical evidence and physical ideas, such as the energy function and the fact that the phase space must have twice the dimension of the configuration space, we argue that the stochastic differential equations which describe the motion of financial prices with respect to real world probability measures should be of second order (and non-Markovian), instead of first order models `a la Bachelier--Samuelson. Our theoretical result in stochastic dynamical systems shows that one cannot correctly reduce second order models to first order models by simply forgetting about momenta. We propose some simple second order models, including a stochastic constrained n-oscillator, which can explain many market phenomena, such as boom-bust cycles, stochastic quasi-periodic behavior, and hot money going from one market sector to another.
We consider thin incomplete financial markets, where traders with heterogeneous preferences and risk exposures have motive to behave strategically regarding the demand schedules they submit, thereby impacting prices and allocations. We argue that traders relatively more exposed to market risk tend to submit more elastic demand functions. Noncompetitive equilibrium prices and allocations result as an outcome of a game among traders. General sufficient conditions for existence and uniqueness of such equilibrium are provided, with an extensive analysis of two-trader transactions. Even though strategic behaviour causes inefficient social allocations, traders with sufficiently high risk tolerance and/or large initial exposure to market risk obtain more utility gain in the noncompetitive equilibrium, when compared to the competitive one.
Using the Donsker-Prokhorov invariance principle we extend the Kim-Stoyanov-Rachev-Fabozzi option pricing model to allow for variably-spaced trading instances, an important consideration for short-sellers of options. Applying the Cherny-Shiryaev-Yor invariance principles, we formulate a new binomial path-dependent pricing model for discrete- and continuous-time complete markets where the stock price dynamics depends on the log-return dynamics of a market influencing factor. In the discrete case, we extend the results of this new approach to a financial market with informed traders employing a statistical arbitrage strategy involving trading of forward contracts. Our findings are illustrated with numerical examples employing US financial market data. Our work provides further support for the conclusion that any option pricing model must preserve valuable information on the instantaneous mean log-return, the probability of the stocks upturn movement (per trading interval), and other market microstructure features.
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