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Abstract:
Selected dynamic investment strategies are analyzed within a unifying theoretical framework. We suggest a Kolmogorov-type partial differential equation for a profit and loss (P&L) distribution of strategies contingent on the current value of the basic asset as well as on a balance of a trading account ?"P&L-to-date". This gives a possibility to study much wider class of strategies than is usually done in the literature and practical applications. Using our equation we build dynamic efficient frontier and demonstrate that an attempt to minimize variance for a given expected profit leads to a contrarian trading strategy, also known as the "St.Petersburg paradox". Similar analysis is performed for the Black-Jones-Perold constant proportion portfolio insurance (CPPI). It is shown that both, dynamic efficient frontier and CPPI, belong to a special class of power-option-replicating strategies. Despite its small Sharpe ratio CPPI has an advantage of controlled downside, as its PL distribution is far from gaussian. Conversely, Sharpe-optimal dynamic efficient frontier has an uncontrollable downside. We show how to blend the advantages of these two strategies. A special part is devoted to discrete analysis and risk capital considerations for non-replicating strategies when P&L-to-date is not a single-valued function of the current asset value and path-dependency is essential.

Abstract:
Hedging a derivative security with non-risk-neutral number of shares leads to portfolio profit or loss. Unlike in the Black-Scholes world, the net present value of all future cash flows till maturity is no longer deterministic, and basis risk may be present at any time. The key object of our analysis is probability distribution of future P&L conditioned on the present value of the underlying. We consider time dynamics of this probability distribution for an arbitrary hedging strategy. We assume log-normal process for the value of the underlying asset and use convolution formula to relate conditional probability distribution of P&L at any two successive time moments. It leads to a simple PDE on the probability measure parameterized by a hedging strategy. For risk-neutral replication the P&L probability distribution collapses to a delta-function at the Black-Scholes price of the contingent claim. Therefore, our approach is consistent with the Black-Scholes one and can be viewed as its generalization. We further analyze the PDE and derive formulae for hedging strategies targeting various objectives, such as minimizing variance or optimizing distribution quantiles. The developed method of computing the profit and loss distribution for a given hedging scheme is applied to the classical example of hedging a European call option using the "stop-loss" strategy. This strategy refers to holding 1 or 0 shares of the underlying security depending on the market value of such security. It is shown that the "stop-loss" strategy can lead to a loss even for an infinite frequency of re-balancing. The analytical method allows one to compute profit and loss distributions without relying on simulations. To demonstrate the strength of the method we reproduce the Monte Carlo results on "stop-loss" strategy given in Hull's book, and improve the precision beyond the limits of regular Monte-Carlo simulations.

Abstract:
Hedging a derivative security with non-risk-neutral number of shares leads to portfolio profit or loss. Unlike in the Black-Scholes world, the net present value of all future cash flows till maturity is no longer deterministic, and basis risk may be present at any time. The key object of our analysis is probability distribution of future P&L conditioned on the present value of the underlying. We consider time dynamics of this probability distribution for an arbitrary hedging strategy. We assume log-normal process for the value of the underlying asset and use convolution formula to relate conditional probability distribution of P&L at any two successive time moments. It leads to a simple PDE on the probability measure parameterized by a hedging strategy. For risk-neutral replication the P&L probability distribution collapses to a delta-function at the Black-Scholes price of the contingent claim. Therefore, our approach is consistent with the Black-Scholes one and can be viewed as its generalization. We further analyze the PDE and derive formulae for hedging strategies targeting various objectives, such as minimizing variance or optimizing distribution quantiles. The developed method of computing the profit and loss distribution for a given hedging scheme is applied to the classical example of hedging a European call option using the "stop-loss" strategy. This strategy refers to holding 1 or 0 shares of the underlying security depending on the market value of such security. It is shown that the "stop-loss" strategy can lead to a loss even for an infinite frequency of re-balancing. The analytical method allows one to compute profit and loss distributions without relying on simulations. To demonstrate the strength of the method we reproduce the Monte Carlo results on "stop-loss" strategy given in Hull's book, and improve the precision beyond the limits of regular Monte-Carlo simulations.

Abstract:
Methods of functional analysis are applied to describe collectively fluctuating default-free pure discount bonds subject to trading-related noise which generates arbitrage opportunities. Two key elements of the model are: (i) the naturally incorporated fixed bond price at maturity which is achieved by making use of only those fluctuating parts of price motion which terminate at a specified final condition, and (ii) the most attractive arbitrage opportunities between bonds with close maturities, with modeled a local linear approximation. The Black-Scholes equation for contingent claims is derived, and a connection with the conventional methods of option valuation is indicated.

See also a related paper by A.N.Adamchuk, S.Adamchuk and S.E.Esipov
"Arbitrage Relaxation of Instruments with Temporal Constraints"

Abstract:
The classical "no-arbitrage" argument assumption is based on perfect replication of instruments. In fixed-income markets there are many instruments for which replication is costly and/or time-consuming. This work is based on a modeling approach where finite times are needed to perform arbitrage. Termed "arbitrage times" they provide measures of market liquidity. From a mathematical stance, the models in complete markets can be recovered in the limit of vanishing arbitrage times.

There are indications that pricing in fixed-income markets will soon be based on a VaR or actuarial approach to profit and loss distributions modeled with static or dynamic positions in fixed-income instruments. Such models require a "seed" process similar to risky log-normal motion for the underlying security which is widely used in conventional derivatives.

A mechanism of almost certain local arbitrage between bonds with close maturities is suggested which is effective in forming the term structure of interest rates. A "seed" interest rate model is proposed and calibrated using 1994 U.S. interest rates. Current term structure (i.e., market information about future) is incorporated as one of the boundary conditions in this model, while the "arbitrage times" come from historical data. Unlike in risk-neutral term structure models the fluctuating forward rates explicitly reflect market fundamentals.

Finally, we emphasize that there is no real difference between future and past in the model since the stochastic process is updated in additional time dimension, called "artificial time." We think that this is the minimal theoretical structure which accommodates initial, final and running boundary conditions.

See related paper by A.N. Adamchuk, and S.E. Esipov "Collectively Fluctuating Assets in the Presence of Arbitrage Opportunities and Option Pricing"

Abstract:
One of the fundamental properties of most markets is the existence of more than one asset price or distribution of prices for a given asset. In many markets this distribution is clearly bi-modal, and can be related to the so-called "bid" and "ask" positions. The underlying detailed "microscopic" kinetics of bids, offers, and their matches can be complex and includes information flow along with capital and inventory balances. The reduction of a micro-model to the observed bid-ask spread determines the parameters of interest. We study a dynamic modification of the Garman model without inventory shortages, and obtain a relationship between the bid-ask spread, Value-at-Risk of the market maker, required returns, and the rate of arrivals of orders (measure of liquidity). Our formula for the bid-ask spread can be parametrized by historical data and used as a key ingredient in the information systems and/or automatic trading systems. The formula is derived in the risk-premium pricing framework suggested earlier by one of us and Guo. The bid-ask spread and liquidity costs are computed theoretically and compared with estimates for the most traded equity stocks. The remainder of the paper explores the benefits of dynamic bid-ask trading strategies.

Abstract:
This article proposes a portfolio-based pricing method to evaluate risk and systematically consider risk premium. The risk premium is charged to satisfy risk management and return on risk capital requirements. The P&L distributions are priced based on Value-at-Risk and return on capital approach. The pricing generating process is explicitly presented using an example of Standard and Poor 500 index (SPX) put options.

While the complete list of risk factors is extensive, here we focus on three major concerns an option underwriter has to consider: (1) non-log-normal distribution of the underlying asset, such as stochastic volatility and/or jumps, (2) transaction costs in dynamic hedging, (3) risk premium for unhedgeable remaining basis risk. With these complications, the final profit and loss to the option underwriter is no longer certain. The underwriter has to develop a formal procedure which would allow him/her to relate the P&L distribution at maturity to a (dollar) value of the derivative security.

We can exactly fit the implied volatility "smile" or "smirk" curve of the one-year SPX options (liquid Exchange market) in term of slope and volatility values. It is then used to price five-year out-of-the-money put (non-liquid over-the-counter market). In total, the implied volatility is about 10% to 40% higher than the historical one. This numbers are consistent with the behavior of S&P500 market.