In a continuous-time model with multiple assets described by cadlag processes, this paper characterizes superhedging prices, absence of arbitrage, and utility maximizing strategies, under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. With such frictions, dual elements correspond to a pair of a shadow execution price combined with an equivalent martingale measure. For utility functions defined on the real line, optimal strategies exist even if arbitrage is present, because it is not scalable at will.
Nonlinear Price Impact and Portfolio Choiceguasoni
In a market with price-impact proportional to a power of the order flow, we derive optimal trading policies and their implied welfare for long-term investors with constant relative risk aversion, who trade one safe asset and one risky asset that follows geometric Brownian motion. These quantities admit asymptotic explicit formulas up to a structural constant that depends only on the price-impact exponent. Trading rates are finite as with linear impact, but they are lower near the target portfolio, and higher away from the target. The model nests the square-root impact law and, as extreme cases, linear impact and proportional transaction costs.
Never selling stocks is optimal for investors with a long horizon and a realistic range of preference and market parameters, if relative risk aversion, investment opportunities, proportional transaction costs, and dividend yields are constant. Such investors should buy stocks when their portfolio weight is too low, and otherwise hold them, letting dividends rebalance to cash over time rather than selling. With capital gain taxes, this policy outperforms both static buy-and-hold and dynamic rebalancing strategies that account for transaction costs. Selling stocks becomes optimal if either their target weight is low, or intermediate consumption is substantial.
Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogeneous...guasoni
We solve a general equilibrium model of an incomplete market with heterogeneous preferences, identifying first-order and second-order effects. Several long-lived agents with different absolute risk-aversion and discount rates make consumption and investment decisions, borrowing from and lending to each other, and trading a stock that pays a dividend whose growth rate has random fluctuations over time. For small fluctuations, the first-order equilibrium implies no trading in stocks, the existence of a representative agent, predictability of returns, multi-factor asset pricing, and that agents use a few public signals for consumption, borrowing, and lending. At the second-order, agents dynamically trade stocks and no representative agent exist. Instead, both the interest rate and asset prices depend on the dispersion of agents' preferences and their shares of wealth. Dynamic trading arises from agents' intertemporal hedging motive, even in the absence of personal labor income.
Leveraged and inverse ETFs seek a daily return equal to a multiple of an index' return, an objective that requires continuous portfolio rebalancing. The resulting trading costs create a tradeoff between tracking error, which controls the short-term correlation with the index, and excess return (or tracking difference) -- the long-term deviation from the levered index' performance. With proportional trading costs, the optimal replication policy is robust to the index' dynamics. A summary of a fund's performance is the \emph{implied spread}, equal to the product of tracking error and excess return, rescaled for leverage and average volatility. The implies spread is insensitive to the benchmark's risk premium, and offers a tool to compare the performance of funds on the same benchmark, but with different multiples and tracking errors.
Nonlinear Price Impact and Portfolio Choiceguasoni
In a market with price-impact proportional to a power of the order flow, we derive optimal trading policies and their implied welfare for long-term investors with constant relative risk aversion, who trade one safe asset and one risky asset that follows geometric Brownian motion. These quantities admit asymptotic explicit formulas up to a structural constant that depends only on the price-impact exponent. Trading rates are finite as with linear impact, but they are lower near the target portfolio, and higher away from the target. The model nests the square-root impact law and, as extreme cases, linear impact and proportional transaction costs.
Never selling stocks is optimal for investors with a long horizon and a realistic range of preference and market parameters, if relative risk aversion, investment opportunities, proportional transaction costs, and dividend yields are constant. Such investors should buy stocks when their portfolio weight is too low, and otherwise hold them, letting dividends rebalance to cash over time rather than selling. With capital gain taxes, this policy outperforms both static buy-and-hold and dynamic rebalancing strategies that account for transaction costs. Selling stocks becomes optimal if either their target weight is low, or intermediate consumption is substantial.
Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogeneous...guasoni
We solve a general equilibrium model of an incomplete market with heterogeneous preferences, identifying first-order and second-order effects. Several long-lived agents with different absolute risk-aversion and discount rates make consumption and investment decisions, borrowing from and lending to each other, and trading a stock that pays a dividend whose growth rate has random fluctuations over time. For small fluctuations, the first-order equilibrium implies no trading in stocks, the existence of a representative agent, predictability of returns, multi-factor asset pricing, and that agents use a few public signals for consumption, borrowing, and lending. At the second-order, agents dynamically trade stocks and no representative agent exist. Instead, both the interest rate and asset prices depend on the dispersion of agents' preferences and their shares of wealth. Dynamic trading arises from agents' intertemporal hedging motive, even in the absence of personal labor income.
Leveraged and inverse ETFs seek a daily return equal to a multiple of an index' return, an objective that requires continuous portfolio rebalancing. The resulting trading costs create a tradeoff between tracking error, which controls the short-term correlation with the index, and excess return (or tracking difference) -- the long-term deviation from the levered index' performance. With proportional trading costs, the optimal replication policy is robust to the index' dynamics. A summary of a fund's performance is the \emph{implied spread}, equal to the product of tracking error and excess return, rescaled for leverage and average volatility. The implies spread is insensitive to the benchmark's risk premium, and offers a tool to compare the performance of funds on the same benchmark, but with different multiples and tracking errors.
Shortfall aversion reflects the higher utility loss of a spending cut from a reference point than the utility gain from a similar spending increase, in the spirit of Prospect Theory's loss aversion. This paper posits a model of utility of spending scaled by a function of past peak spending, called target spending. The discontinuity of the marginal utility at the target spending corresponds to shortfall aversion. According to the closed-form solution of the associated spending-investment problem, (i) the spending rate is constant and equals the historical peak for relatively large values of wealth/target; and (ii) the spending rate increases (and the target with it) when that ratio reaches its model-determined upper bound. These features contrast with traditional Merton-style models which call for spending rates proportional to wealth. A simulation using the 1926-2012 realized returns suggests that spending of the very shortfall averse is typically increasing and very smooth.
This presentation provides and overview of the paper "Jump-Diffusion Risk-Sensitive Asset Management." The paper proposes a solution to a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure,
with drifts that are functions of an auxiliary diffusion ‘factor’ process.
Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model...Volatility
1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs
2) Examine the connection between the realized variance and the realized P&L
3) Find approximate solutions for the P&L volatility and the expected total transaction costs
4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance
5) Consider hedging strategies to minimize the jump risk
Shortfall aversion reflects the higher utility loss of a spending cut from a reference point than the utility gain from a similar spending increase, in the spirit of Prospect Theory's loss aversion. This paper posits a model of utility of spending scaled by a function of past peak spending, called target spending. The discontinuity of the marginal utility at the target spending corresponds to shortfall aversion. According to the closed-form solution of the associated spending-investment problem, (i) the spending rate is constant and equals the historical peak for relatively large values of wealth/target; and (ii) the spending rate increases (and the target with it) when that ratio reaches its model-determined upper bound. These features contrast with traditional Merton-style models which call for spending rates proportional to wealth. A simulation using the 1926-2012 realized returns suggests that spending of the very shortfall averse is typically increasing and very smooth.
This presentation provides and overview of the paper "Jump-Diffusion Risk-Sensitive Asset Management." The paper proposes a solution to a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure,
with drifts that are functions of an auxiliary diffusion ‘factor’ process.
Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model...Volatility
1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs
2) Examine the connection between the realized variance and the realized P&L
3) Find approximate solutions for the P&L volatility and the expected total transaction costs
4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance
5) Consider hedging strategies to minimize the jump risk
Stochastic Local Volatility Models: Theory and ImplementationVolatility
1) Hedging and volatility
2) Review of volatility models
3) Local volatility models with jumps and stochastic volatility
4) Calibration using Kolmogorov equations
5) PDE based methods in one dimension
5) PDE based methods in two dimensions
7) Illustrations
Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshop
Talk of Michael Samet, entitled "Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models" at the International Conference on Computational Finance (ICCF)", Wuppertal June 6-10, 2022
I am Bing Jr. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab Deakin University, Australia. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signal Processing.
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Investing on behalf of a firm, a trader can feign personal skill by committing fraud that with high probability remains undetected and generates small gains, but that with low probability bankrupts the firm, offsetting ostensible gains. Honesty requires enough skin in the game: if two traders with isoelastic preferences operate in continuous-time and one of them is honest, the other is honest as long as the respective fraction of capital is above an endogenous fraud threshold that depends on the trader’s preferences and skill. If both traders can cheat, they reach a Nash equilibrium in which the fraud threshold of each of them is lower than if the other one were honest. More skill, higher risk aversion, longer horizons, and greater volatility all lead to honesty on a wider range of capital allocations between the traders.
American student loans are fixed-rate debt contracts that may be repaid in full by a certain maturity. Alternatively, income-based schemes give borrowers the option to make payments proportional to their income above subsistence for a number of years, after which the remaining balance is forgiven but taxed as ordinary income. The repayment strategy that minimizes the present value of future payments takes two possible forms: For a small loan balance, it is optimal to make maximum payments until the loan is fully repaid, forgoing both income-based schemes and loan forgiveness. For a large balance, enrolling in income-based schemes is optimal either immediately or after a period of maximum payments. Overall, the benefits of income-based schemes are substantial for large loan balances but negligible for small loans.
Compared with existing payment systems, Bitcoin’s throughput is low. Designed to address Bitcoin’s scalability challenge, the Lightning Network (LN) is a protocol allowing two parties to secure bitcoin payments and escrow holdings between them. In a lightning channel, each party commits collateral towards future payments to the counterparty and payments are cryptographically secured updates of collaterals. The network of channels increases transaction speed and reduces blockchain congestion. This paper (i) identifies conditions for two parties to optimally establish a channel, (ii) finds explicit formulas for channel costs, (iii) obtains the optimal collaterals and savings entailed, and (iv) derives the implied reduction in congestion of the blockchain. Unidirectional channels costs grow with the square-root of payment rates, while symmetric bidirectional channels with their cubic root. Asymmetric bidirectional channels are akin to unidirectional when payment rates are significantly different, otherwise to symmetric bidirectional.
Reference Dependence: Endogenous Anchors and Life-Cycle Investingguasoni
In a complete market, we find optimal portfolios for an investor whose satisfaction stems from both a payoff's intrinsic utility and its comparison with a reference, as specified by Koszegi and Rabin. In the regular regime, arising when reference-dependence is low, the marginal utility of the optimal payoff is proportional to a twist of the pricing kernel. High reference-dependence leads to the anchors regime, whereby investors reduce disappointment by concentrating significant probability in one or few fixed outcomes, and multiple personal equilibria arise. If stocks follow geometric Brownian motion, the model implies that younger investors have larger stocks positions than older investors, highlighting the suggestion that reference-dependence helps explain this typical recommendation of financial planners.
A monopolist platform (the principal) shares profits with a population of affiliates (the agents), heterogeneous in skill, by offering them a common nonlinear contract contingent on individual revenue. The principal cannot discriminate across individual skill, but knows its distribution and aims at maximizing profits. This paper identifies the optimal contract, its implied profits, and agents' effort as the unique solution to an equation depending on skill distribution and agents' costs of effort. If skill is Pareto-distributed and agents' costs include linear and power components, closed-form solutions highlight two regimes: If linear costs are low, the principal's share of revenues is insensitive to skill distribution, and decreases as agents' costs increase. If linear costs are high, the principal's share is insensitive to the agents' costs and increases as inequality in skill increases.
Should Commodity Investors Follow Commodities' Prices?guasoni
Most institutional investors gain access to commodities through diversified index funds, even though mean-reverting prices and low correlation among commodities returns indicate that two-fund separation does not hold for commodities. In contrast to demand for stocks and bonds, we find that, on average, demand for commodities is largely insensitive to risk aversion, with intertemporal hedging demand playing a major role for more risk averse investors. Comparing the optimal strategies of investors who observe only the index to those of investors who observe all commodities, we find that information on commodity prices leads to significant welfare gains, even if trading is confined to the index only.
Asset Prices in Segmented and Integrated Marketsguasoni
This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.
We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs are the unique solution to a system of integral equations, which reduce to a linear matrix equation under suitable representations of the underlying probabilities. Even when implied volatilities are all higher than historical volatilities, it can be optimal to sell options on some assets while buying options on others, as hedging demand outweighs demand for asset-specific returns.
Health-care slows the natural growth of mortality, indirectly increasing utility from consumption through longer lifetimes. This paper solves the problem of optimal dynamic consumption and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect the Gompertz' law. Optimal consumption and healthcare imply an endogenous mortality law that is asymptotically exponential in the old-age limit, with lower growth rate than natural mortality. Health spending steadily increases with age, both in absolute terms and relative to total spending. Differential access to healthcare with isoelastic effects can account for observed longevity gains across cohorts.
When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to the asset's volatility and liquidity. In a continuous-time model with one safe and one risky asset with constant investment opportunities and proportional transaction costs, we find the efficient portfolios that maximize long term expected returns for given average volatility. As leverage and volatility increase, rising rebalancing costs imply a declining Sharpe ratio. Beyond a critical level, even the expected return declines. For funds that seek to replicate multiples of index returns, such as leveraged ETFs, our efficient portfolios optimally trade off alpha against tracking error.
USDA Loans in California: A Comprehensive Overview.pptxmarketing367770
USDA Loans in California: A Comprehensive Overview
If you're dreaming of owning a home in California's rural or suburban areas, a USDA loan might be the perfect solution. The U.S. Department of Agriculture (USDA) offers these loans to help low-to-moderate-income individuals and families achieve homeownership.
Key Features of USDA Loans:
Zero Down Payment: USDA loans require no down payment, making homeownership more accessible.
Competitive Interest Rates: These loans often come with lower interest rates compared to conventional loans.
Flexible Credit Requirements: USDA loans have more lenient credit score requirements, helping those with less-than-perfect credit.
Guaranteed Loan Program: The USDA guarantees a portion of the loan, reducing risk for lenders and expanding borrowing options.
Eligibility Criteria:
Location: The property must be located in a USDA-designated rural or suburban area. Many areas in California qualify.
Income Limits: Applicants must meet income guidelines, which vary by region and household size.
Primary Residence: The home must be used as the borrower's primary residence.
Application Process:
Find a USDA-Approved Lender: Not all lenders offer USDA loans, so it's essential to choose one approved by the USDA.
Pre-Qualification: Determine your eligibility and the amount you can borrow.
Property Search: Look for properties in eligible rural or suburban areas.
Loan Application: Submit your application, including financial and personal information.
Processing and Approval: The lender and USDA will review your application. If approved, you can proceed to closing.
USDA loans are an excellent option for those looking to buy a home in California's rural and suburban areas. With no down payment and flexible requirements, these loans make homeownership more attainable for many families. Explore your eligibility today and take the first step toward owning your dream home.
how to swap pi coins to foreign currency withdrawable.DOT TECH
As of my last update, Pi is still in the testing phase and is not tradable on any exchanges.
However, Pi Network has announced plans to launch its Testnet and Mainnet in the future, which may include listing Pi on exchanges.
The current method for selling pi coins involves exchanging them with a pi vendor who purchases pi coins for investment reasons.
If you want to sell your pi coins, reach out to a pi vendor and sell them to anyone looking to sell pi coins from any country around the globe.
Below is the contact information for my personal pi vendor.
Telegram: @Pi_vendor_247
how to sell pi coins at high rate quickly.DOT TECH
Where can I sell my pi coins at a high rate.
Pi is not launched yet on any exchange. But one can easily sell his or her pi coins to investors who want to hold pi till mainnet launch.
This means crypto whales want to hold pi. And you can get a good rate for selling pi to them. I will leave the telegram contact of my personal pi vendor below.
A vendor is someone who buys from a miner and resell it to a holder or crypto whale.
Here is the telegram contact of my vendor:
@Pi_vendor_247
Currently pi network is not tradable on binance or any other exchange because we are still in the enclosed mainnet.
Right now the only way to sell pi coins is by trading with a verified merchant.
What is a pi merchant?
A pi merchant is someone verified by pi network team and allowed to barter pi coins for goods and services.
Since pi network is not doing any pre-sale The only way exchanges like binance/huobi or crypto whales can get pi is by buying from miners. And a merchant stands in between the exchanges and the miners.
I will leave the telegram contact of my personal pi merchant. I and my friends has traded more than 6000pi coins successfully
Tele-gram
@Pi_vendor_247
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
BYD SWOT Analysis and In-Depth Insights 2024.pptxmikemetalprod
Indepth analysis of the BYD 2024
BYD (Build Your Dreams) is a Chinese automaker and battery manufacturer that has snowballed over the past two decades to become a significant player in electric vehicles and global clean energy technology.
This SWOT analysis examines BYD's strengths, weaknesses, opportunities, and threats as it competes in the fast-changing automotive and energy storage industries.
Founded in 1995 and headquartered in Shenzhen, BYD started as a battery company before expanding into automobiles in the early 2000s.
Initially manufacturing gasoline-powered vehicles, BYD focused on plug-in hybrid and fully electric vehicles, leveraging its expertise in battery technology.
Today, BYD is the world’s largest electric vehicle manufacturer, delivering over 1.2 million electric cars globally. The company also produces electric buses, trucks, forklifts, and rail transit.
On the energy side, BYD is a major supplier of rechargeable batteries for cell phones, laptops, electric vehicles, and energy storage systems.
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
Poonawalla Fincorp and IndusInd Bank Introduce New Co-Branded Credit Cardnickysharmasucks
The unveiling of the IndusInd Bank Poonawalla Fincorp eLITE RuPay Platinum Credit Card marks a notable milestone in the Indian financial landscape, showcasing a successful partnership between two leading institutions, Poonawalla Fincorp and IndusInd Bank. This co-branded credit card not only offers users a plethora of benefits but also reflects a commitment to innovation and adaptation. With a focus on providing value-driven and customer-centric solutions, this launch represents more than just a new product—it signifies a step towards redefining the banking experience for millions. Promising convenience, rewards, and a touch of luxury in everyday financial transactions, this collaboration aims to cater to the evolving needs of customers and set new standards in the industry.
Exploring Abhay Bhutada’s Views After Poonawalla Fincorp’s Collaboration With...beulahfernandes8
The financial landscape in India has witnessed a significant development with the recent collaboration between Poonawalla Fincorp and IndusInd Bank.
The launch of the co-branded credit card, the IndusInd Bank Poonawalla Fincorp eLITE RuPay Platinum Credit Card, marks a major milestone for both entities.
This strategic move aims to redefine and elevate the banking experience for customers.
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
Empowering the Unbanked: The Vital Role of NBFCs in Promoting Financial Inclu...Vighnesh Shashtri
In India, financial inclusion remains a critical challenge, with a significant portion of the population still unbanked. Non-Banking Financial Companies (NBFCs) have emerged as key players in bridging this gap by providing financial services to those often overlooked by traditional banking institutions. This article delves into how NBFCs are fostering financial inclusion and empowering the unbanked.
Introduction to Indian Financial System ()Avanish Goel
The financial system of a country is an important tool for economic development of the country, as it helps in creation of wealth by linking savings with investments.
It facilitates the flow of funds form the households (savers) to business firms (investors) to aid in wealth creation and development of both the parties
Hedging, Arbitrage, and Optimality with Superlinear Frictions
1. Superlinear Frictions
Hedging, Arbitrage, and Optimality
with Superlinear Frictions
Paolo Guasoni1;2 Miklós Rásonyi3;4
Boston University1
Dublin City University2
MTA Alfréd Rényi Institute of Mathematics, Budapest3
University of Edinburgh4
Analyse stochastique pour la modélisation des risques
CIRM, September 10th, 2014
2. Superlinear Frictions
Market Depth
Depth:
the size of an order flow innovation required to change prices a given
amount (Kyle, 1985)
Documented empirically:
Amihud (2002), Admati and Pfleiderer (1988), Cho (2007)
In Illiquid Portfolio Choice:
Rogers and Singh (2010), Garleanu and Pedersen (2013)
In Optimal Liquidation:
Almgren and Chriss, (2001), Bertsimas and Lo (1998), Schied and
Schöneborn (2009)
Unlike frictionless markets, trading affects prices.
Unlike transaction costs, prices depend on direction and speed.
3. Superlinear Frictions
Theory
Usual questions: arbitrage, hedging, optimality.
Discrete time:
Astic and Touzi (2007), Pennanen and Penner (2010),
Dolinsky and Soner (2013)
Continuous Time:
Cetin, Soner, and Touzi (2010), Cetin and Rogers (2007).
Which trading strategies?
Attainable payoffs?
4. Superlinear Frictions
Results in a Nutshell
Risky asset: cadlag process St .
Number of shares t =
R t
0 sds. Effect of friction G on wealth Xt :
dXt = tdSt G(t )dt
G superlinear. Trading twice as fast for half the time more expensive.
Typical example G(x) = x2 from Kyle’s equilibrium model.
Absolutely continuous strategies are closed.
(Limits of their payoffs are also their payoffs.)
Market bound: all payoffs dominated by one random variable!
Martingale (not local) property of (shadow) wealth processes.
No doubling strategies.
Superhedging: price is sup of expected shadow values, minus a penalty.
Reduces to frictionless and transaction costs for G = 0 and G(x) = jxj.
5. Superlinear Frictions
Feasible Strategies
Zero safe rate. S0 1. Risky assets Si
t cadlag adapted processes.
Definition
A feasible strategy is a process in the class
A :=
(
: is a Rd -valued, optional process;
Z T
0
)
:
jujdu 1 a.s.
Unlike usual admissible strategies...
...the definition of feasible strategy does not depend on the asset
...and wealth can be unbounded from below
...but the number of shares must change at a finite rate.
6. Superlinear Frictions
Friction
Assumption (Friction)
Let G :
[0; T] Rd ! R+ be a O
B(Rd )-measurable function, such that
G(!; t; ) is convex with G(!; t; x) G(!; t; 0) for all !; t; x.
Gt (x) short for G(!; t; x)
Positions in risky assets and cash
Vi
t (z; ) :=zi +
Z t
0
i
udu 1 i d;
V0
t (z; ) :=z0
Z t
0
uSudu
Z t
0
Gu(u)du:
Doing something costs more than doing nothing. Participation cost.
Convexity: speed expensive. Patience pays off.
For one risky asset and Gt (0) = 0, equivalent to execution price equal to:
~St = St + Gt (t )=t
7. Superlinear Frictions
Superlinear Friction
Assumption
There is 1 and an optional process H such that
inf
t2[0;T]
Ht 0 a.s.;
Gt (x) Ht jxj; for all !; t; x;
Z T
0
sup
jxjN
!
dt 1 a.s. for all N 0;
Gt (x)
sup
t2[0;T]
Gt (0) K a.s. for some constant K:
Superlinearity: trading twice as fast (uniformly) more than twice expensive.
Frictions never disappear...
...but remain finite in finite time.
8. Superlinear Frictions
Market Bound
Lemma
Under the superlinearity assumption, any feasible 2 A satisfies
V0
T (z; ) z0 +
Z T
0
G
t (St )dt 1 a.s.
G
t (y) := supx2Rd (xy Gt (x)) is the dual friction (Dolinsky and Soner, 2013).
Proof.
z0
Z t
0
uSudu
Z t
0
Gu(u)du z0
t +
Z t
0
G
u(Su)du
t (y) supr2R (ry Ht jr j) = 1
and G
1
1H
1
1
t jyj
1 .
Without frictions or with transaction costs, no-arbitrage conditions make
the payoff space bounded in L0. Superlinear frictions do even better.
All payoffs below the market bound. Almost surely.
9. Superlinear Frictions
Shadow Probabilities
Definition
P denotes the set of probabilities Q P such that
EQ
Z T
0
H
13. )dt 1:
~ P denotes the set of probability measures Q 2 P such that
EQ
Z T
0
jSt jdt 1 and EQ
Z T
0
sup
jxjN
Gt (x)dt 1 for all N 1:
For a (possibly multivariate) random variable W, define
P(W) := fQ 2 P : EQjWj 1g; ~ P(W) := fQ 2 ~ P : EQjWj 1g:
Think of these sets as martingale probabilities for some execution price ~St
– and integrable enough.
14. Superlinear Frictions
Trading Volume Bound
Lemma
Let Q 2 P and 2 A be such that EQ 1, where
:=
Z T
0
Stt dt
Z T
0
Gt (t )dt:
Then
EQ
Z T
0
jt j
16. dt 1:
Excessive trading may be hazardous for your wealth.
Bounded Losses imply bounded trading volume...
Follows from careful use of Hölder’s inequality.
17. Superlinear Frictions
Closed Payoff Space
Space of payoffs superhedged by feasible strategies
C := [fVT (0; ) : 2 Ag L0(Rd+1
+ )] L0(Rd+1)
Proposition
The set C L1(Q) is closed in L1(Q) for all Q 2 P such that
R T
0 jSt jdt is
Q-integrable.
Corollary
T the set C is closed in probability.
Absolutely continuous strategies are the only strategies.
Similar to proof that fh :
R 1
0 (h_ t )2dt 1g is relatively compact in L1.
18. Superlinear Frictions
Superreplication
For x 2 Rd+1, define x 2 Rd by xi = (xi=x0)1fx06=0g.
Theorem
Let W 2 L0(Rd+1), z 2 Rd+1. There exists 2 A such that VT (z; ) W a.s. if
and only if
Z0z EQ(ZTW) EQ
Z T
0
Z0
t G
t (Zt St )dt; (1)
for all Q 2 P and for all Rd+1
+ -valued bounded Q-martingales Z with Z0
0 = 1
satsifying Zi
t = 0, i = 1; : : : ; d on fZ0
t = 0g.
Multivariate claims: assets not convertible instantaneously.
Take Q = P for simplicity. Then any positive martingale Z gives a shadow
price, penalized for how far Z is from S.
If Gt (0) = 0 and Z = S (i.e. SZ0 is a martingale), then penalty is zero.
19. Superlinear Frictions
Examples
Superreplication is expensive.
Example
Let 2 R, ; S0 0, St := S0e(2=2)t+Wt , Gt (x) = 2
Stx2, where Wt is a
Brownian motion. Then a cash payoff equal to ST cannot be superreplicated
from any initial capital.
Cannot hedge what may exceed the market bound!
What is superreplicable, then?
Example
Let St 0 a.s. for all t and Gt (x) := 2
Stx2. Then, for all k 0, the contract
that at time T pays 1
p
1 + 2k=St 1)dt units of the risky asset is
R T
0 (
superreplicable from initial cash position kT .
Buy asset at rate k and get that payoff.
20. Superlinear Frictions
Arbitrage
Superreplication theorem does not require absence of arbitrage...
...so it helps characterize it.
Definition
An arbitrage of the second kind is a strategy 2 A, such that VT (c; ) 0 for
some c 0.
in other words, a negative superreplication price for a positive payoff.
Theorem
Absence of arbitrage of the second kind holds if and only if, for all 0, there
exists Q 2 P and an Rd+1
-valued Q-martingale Z with ZT + 2 L
(Q) such that
EQ
R T
0 Z0
t (Zt St )dt , where 0
22. + 1=
= 1.
t G
Enough to find shadow prices that are very close to martingale measures.
Processes with conditional full support satisfy this criterion for H constant.
23. Superlinear Frictions
Utility Maximization
Theorem
Let U : R ! R be concave and nondecreasing, W a random variable, and let
EjU(c + B +W)j 1 hold for the market bound B =
R T
0 G
t (St )dt. There is
2 A0(U; c) such that
EU(V0
T (c; ) +W) = sup
2A0(u;c)
EU(V0
T (c; ) +W);
where
A0(U; c) = f 2 A : ViT
(c; ) = 0; i = 1; : : : ; d; EU(V0
T (c; ) +W) 1g:
Optimal strategies exist under general conditions.
Friction generates compactness. Cannot buy too much of anything.
24. Superlinear Frictions
First-order Condition
Theorem
a) Let U be concave with U0 strictly decreasing, and for some C 0 and 1
U(x) Cjxj; x 0;
b) assume that ~U0 is strictly increasing, where ~U is the conjugate of U,
c) let W be a bounded random variable;
d) let Q 2 P be such that dQ=dP 2 L;where (1=) + (1=) = 1;
e) let G0
t () exist continuous P Leb-a.s. and be strictly increasing;
f) let Z be a càdlàg process with ZT 2 L
0 for some
0
and let be a
feasible strategy such that, for some y 0, the following conditions hold:
i) Z is a Q-martingale;
ii) U0(V0
T (x; ) +W) = y(dQ=dP) a.s.;
iii) Zt = St + G0
t (t
) a.s. in P Leb;
iv) EQ
V0
T (x; )
R T
0 G
t (Zt St )dt
= x.
Then is optimal for the problem max2A0(U;c) E
U(V0
T (x; ) +W)
:
25. Superlinear Frictions
Cash is a Martingale
Milestone for optimality: pass from a.s. upper bound
V0
T (x; ) x
Z T
0
Ztt dt +
Z T
0
G
t (Zt St )dt
to risk-neutral upper bound
0 EQ
x V0
T (x; ) +
Z T
0
G
t (Zt St )dt
!#
:
Lemma
Under the assumptions of the previous Theorem, any 2 A0(U; c) satisfies
EQ
Z T
0
tZt dt = 0:
Shadow wealth is a martingale.
26. Superlinear Frictions
Conclusion
Superlinear frictions:
execution prices increase arbitrarily with trading speed.
Market bound: one payoff dominates them all.
Finite losses imply finite volume.
Absolutely continuous strategies generate closed payoff space.
Utility maximization: first order conditions for payoff and for shadow price.
