Building on the general equilibrium setup solved in the last week, this lecture looks in depth at the relationships between productivity, patience, prices, allocations, and nominal and real interest rates. The solutions are given to three of Fisher’s famous examples: What happens to interest rates when people become more or less patient? What happens when they expect to receive windfall riches sometime in the future? And, what happens when wealth in an economy is redistributed from the poor to the rich?
PREREQUISITES
Building on the general equilibrium setup solved in the last week, this lecture looks in depth at the relationships between productivity, patience, prices, allocations, and nominal and real interest rates. The solutions are given to three of Fisher’s famous examples: What happens to interest rates when people become more or less patient? What happens when they expect to receive windfall riches sometime in the future? And, what happens when wealth in an economy is redistributed from the poor to the rich?
ECON251 Interest Rate Theory HELP（EXAM HELP， ONLINE TUTOR）
问题 1.
Assume the equilibrium equation shown below. What is the return on the zero-beta portfolio and the return on the market assuming the zero-beta model holds? $$ \bar{R}_i=0.04+0.10 \beta_i $$
问题 2.
Given the model shown below, what is the risk-free rate if the posttax equilibrium model describes returns? $$ \bar{R}_i=0.05+0.10 \beta_i+0.24 \delta_i $$
问题 3.
Given the following situation: $$ \begin{aligned} \bar{R}_M=15 & \sigma_M=22 \ R_Z=5 & \sigma_Z=8 \ R_F=3 & \end{aligned} $$ draw the minimum variance curve and efficient frontier in expected return standard deviation space. Be sure to give the coordinates of all key points. Draw the security market line.
问题 4.
You have just lectured two tax-free institutions on the necessity of including taxes in the general equilibrium relationship. One believed you and one did not. Demonstrate that if the model holds, the one that did could engage in risk-free arbitrage with the one that did not in a manner such that: A. Both parties believed they were making an arbitrage profit in the transaction. B. The one who believed in the posttax model actually made a profit; the other institution incurred a loss.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely available through the university library here) • Essentials of Stochastic Processes, Third Edition by Durrett (freely available through the university library here) To reiterate, the textbooks are freely available through the university library. Note that you must be connected to the university Wi-Fi or VPN to access the ebooks from the library links. Furthermore, the library links take some time to populate, so do not be alarmed if the webpage looks bare for a few seconds.
The financial institutions offering derivative products and services control and monitor their counterparty risk with the trading counterparty. Some products such as a deposit or a structured note imply one-way counterparty risk where the end investor takes the counterparty risk of the financial institution but the financial institution has no risk from the end investor. However, in some other products such as forward and swap, both parties are taking counterparty risk. The growth of OTC derivatives business prompted the standardization for the contract terms as well as the counterparty credit risk management which are widely used by the institutions and corporates actively involved in financial market.
ISDA (International Swaps and Derivatives Association) Master Agreement, initially developed in the 1980 s to cover the IRS and currency swaps, has been progressively updated to include the derivatives such as forward, swap and option linked to equities, commodities and funds. It sets out standard terms applied to all trades between the two parties.
In general, two parties set up their trading relationship for OTC derivatives by negotiating the applicability and eventual adjustment of the standard terms. The document containing the agreed terms is the ISDA Master Agreement signed by both parties.
CSA (Credit Support Annex) is the document for credit support (i.e. collateral) for derivative transactions. It defines the acceptable collaterals with the “haircuts”. In particular, it defines the “Threshold” which is the consolidated MtM level of all the trades to trigger the margin call. The “Independent Amount (IA)” is the initial margin (collateral) required by one party (usually the dealer) to the other party (usually the end user) for mitigating the counterparty risk linked to an OTC transaction. It is returned only after the termination of the transaction. Its level depends on the volatility of the mark-to-market value of the trade as well as the credit worthiness of the counterparty. During the life of the trade, “variation margin” will be exchanged according to its mark-to-market.
The institutions and corporates actively transacting derivatives usually establish an ISDA/CSA Master Agreement with their counterparties. Normally, the credit agreement between two financial institutions is a two-way CSA in which both parties may post margins for their OTC trades. Between a financial institution and a corporate (especially the small ones), the credit agreement may be a one-way CSA, meaning that only the corporate posts margins to the financial institution. Under a master agreement (ISDA/CSA or any bespoke master agreement), the specific terms and conditions of each OTC derivative trade will take a short form called term sheet or transaction supplement. The individual investors, small corporates and other nonactive entities involved in financial market normally trade with financial institutions with a hespoke agreement or a long form confirmation which contains all terms and conditions for each trade.
金融代写|金融衍生品代写Financial derivatives代考|Securities Borrowing & Lending and Repo
A repurchase agreement is a contract for the sale of a security (e.g. stock or bond) with a commitment by the seller to buy the same security back from the buyer at a specified price at a future date. During the tenor of the trade, the seller (also called the lender) of the security surrenders the legal ownership of the security. There are two activities based on the repurchase agreement: Securities Lending and Repo.
Securities Borrowing \& Lending $(S B L)$ transaction allows the lender to lend securities to the borrower on either “Open” (i.e. anytime callable) or “Term” (a fixed tenor) basis. Upon the trade termination, the securities will be returned to the lender. The borrower posts collateral with daily adjustment and pays fees to the lender. The fee rate depends on the borrow supply/demand for the underlying security. The eligible collateral can be cash or other securities negotiated by the parties. The cash collateral level is usually $\sim 105 \%$ of the latest closing price of the security. The International Securities Lending Association has developed a standard agreement called Global Master Securities Lending Agreement (GMSLA) which is followed by most of the institutions. The motivation for the borrower includes short position recovering, hedging of derivatives, corporate action arbitrage, etc.
In a sale and repurchase agreement (Repo), one counterparty (the repo seller) is borrowing money and providing collateral (mostly fixed-income assets) for the loan. See Fig. 1.1 for reference. The seller gains access to funds at lower funding costs than are typically available elsewhere as the loan is collateralized. The collateral eligibility and haircuts are negotiable between the repo counterparties. The standard agreement for Repo is Global Master Repurchase Agreement (GMRA), published by the International Capital Market Association (ICMA). A Reverse Repo is the opposite transaction seen by the other counterparty of the Repo trade. Some central banks use Repo/Reverse Repo operations to regulate the money supply in the financial system.
If the collateral is held at a third party, usually a custodian bank or an international central securities depository, the transaction is call a Tri-Party Repo or Tri-Party Securities Lending. The third party will provide services such as the valuation and adjustment of the collateral. The risk in a Tri-Party Repo transaction is the correlation of the default probability of the counterparty and the value of the collateral in custody.
Although most Repo activities take place on the OTC market, there exists Stock Exchange Repo (e.g. Shanghai Stock Exchange Repo) whereby the exchange determines the collateral pool and haircuts, standardizes the contract features such as size and tenor, and facilitates clearing and pledge of collateral.
A financial market designates the aggregate of participants, organizations and facilities in which people trade financial securities (e.g. stocks and bonds), currencies and commodities at prices that reflect supply and demand. The participants include financial institutions (such as banks, insurance companies, pension funds, mutual funds, hedge funds), individual investors and corporations. The financial market can be detailed by the type of services it offers: Capital markets: which provide financing through the issuance of shares and debts, and enable the subsequent trading thereof. Capital markets include debt and equity markets.
Equity market (also known as Stock market) allows the participants to buy and sell stocks of (publicly traded) companies. The value of a stock reflects the view about the expected dividend payments, future earnings, and resources that the company will control.
Debt market (or Fixed-income market) includes bond market which deals in government, corporate and other bonds for long term financing, and money market for short term (up to 1 year) debt securities such as bank deposits, treasury bills, certificates of deposit, commercial papers, etc. Foreign exchange market: where currencies are bought and sold. Commodity market: where commodities such as precious metals, industrial metals, energy products and agricultural products are traded. Futures contracts are the most convenient instruments for commodities trading activities. A futures contract may be sold out before the commodity is due to be delivered. Derivatives market: where futures, swaps, options and other derivatives are transacted. The financial market can also be classified with other criteria, such as Primary market: where new issues are first sold through IPOs (Initial Public Offerings). The primary market business for debts and stocks is covered respectively by DCM (Debt Capital Market) and ECM (Equity Capital Market) divisions of the Corporate Finance entity in the investment banks. Secondary market: for all subsequent trading after IPO between market participants. It constitutes the support for the financial products for investment and risk management.
As an OTC derivative involves potential payments between the parties in the future, the counterparty risk that one party does not pay as obligated in the contract can not be neglected. To mitigate the counterparty risk, the long and short parties of a derivative contract may transact with the “Central Clearing House” which covers the risk by a collateral deposit system known as the margining system.
For illustration purpose, we take the case of futures contracts transacted at a Futures Exchange serving as the central clearing house.
The buyer and seller should have their “margin account” in place before trading futures.
At the contract inception, both parties will deposit an “initial margin”, fixed by the exchange according to the type and price of futures, as collateral which are typically cash or government bonds.
At the end of each trading session, each party will have their margin account debited or credited for the daily P/L (Profit and Loss, or PnL).
A minimum margin level called “maintenance margin” is required for every margin account. If the account value is below this level, a “margin call” will be issued for bringing back the account to the level of initial margin.
金融代写|金融衍生品代写Financial derivatives代考|Investment Returns and Risks
The investment return of a non-dividend paying asset for the period between time $s$ and time $t(s<t)$, is defined as: $R=\frac{P_t-P_s}{P_s} \quad$ or $\quad R=\frac{P_t}{P_s}-1, \quad$ where $P_t$ is the price of the asset at time $t$.
The return rate is an annualized concept in general. There are different types of return rates. For instance, the IRR (Internal Rate of Return) is the value of $r$ which makes the following equality: $P_t=P_s(1+r)^\tau$, where $\tau$ represents the number of years for the period $[s, t]$. Usual measures for the rate of return of an asset or a portfolio are Price return: it is measured by the portfolio’s value at the beginning and the end of the period. The dividend payments during the period are ignored. Total return: it is obtained with all the dividends re-invested back into the assets of the portfolio with the same proportion. It represents the return of a fully funded portfolio. Excess return: It is defined as the portfolio’s total return minus the financing cost or a relevant interest rate reference. It represents the return of a self-financed portfolio.
Example 1.1 Consider two assets: Asset 1 and Asset 2. Asset 2 pays a $\$ 1$ dividend per unit at time $t$. The financing cost is $1 \%$ on the borrowed amount.
There are many indicators that are used for measuring the performance of an investment strategy or an investment fund (aggregated form of investment) other than its total return $R$. The most popular ones include
Sharpe ratio $=(R-r) /$ “standard deviation of $(R-r)$ “, where $r$ is the return of a risk-free investment. ${ }^3$ It is the most used measure for risk-adjusted excess return. Information ratio $=\left(R-R_{B M}\right) /$ “standard deviation of $\left(R-R_{B M}\right)$ “, where $R_{B M}$ is the return of the relevant benchmark (e.g. an equity index, if the investment portfolio is composed of stocks). Sortino ratio $=(R-r) /$ /downside deviation”, where “downside deviation” is the standard deviation obtained with negative returns only. ${ }^4$ This measure is preferred by people who consider that the Sharpe ratio punishes the “good risk” as the positive returns are also used in its standard deviation calculation. Maximum drawdown: the peak-to-trough decline in percentage during the considered period of an investment. For the considered period $\left[t_1, t_2\right]$, it is defined as $\frac{m\left(\hat{t}, t_2\right)}{M\left(t_1, t_2\right)}-1$, where $M\left(t_1, t_2\right)$ is the highest price which is achieved at time $\hat{t}$, and $m\left(\hat{t}, t_2\right)$ is the lowest price after $\hat{t}$.
金融代写|期权定价理论代写Option Pricing Theory代考|Complete models versus incomplete models
From (1.13), the market model (1.1) is complete if there exists a unique $\mathcal{F}{t^{-}}$ adapted vector $\lambda_t \in \mathbb{R}^d$ such that $$ \forall i \in \text { asset, } \quad b_i\left(t, X_t\right)-r_t X_t^i=\sum{j=1}^d \sigma_{i, j}\left(t, X_t\right) \lambda_t^j $$ The unique ELMM $\mathbb{Q}$ is then given by $$ \frac{d \mathbb{Q}}{\left.d \mathbb{P}^{h i s t}\right|{\left.\right|{\mathcal{F}T}}} \equiv Y_T(-\lambda ; 1)=\prod{j=1}^d e^{-\int_0^T \lambda_t^j d W_t^j-\frac{1}{2} \int_0^T\left(\lambda_t^j\right)^2 d t} $$ The unique arbitrage-free price is $$ \mathcal{B}t\left(F_T\right)=\mathcal{S}_t\left(F_T\right)=\mathbb{E}^{\mathbb{Q}}\left[D{t T} F_T \mid \mathcal{F}t\right] $$ An inspection of (1.14) reveals that if the market is complete, then the rank of $\sigma\left(t, X_t\right)$ is equal to $d$ a.s. (which implies that #assets $\geq d$ ). In the case where #assets ${i, j}\left(t, X_t\right)\right){i \in \text { asset }, 1 \leq j \leq d}$ is invertible, then the market is complete. So, provided the volatility matrix $\sigma{i, j}\left(t, X_t\right)$ is correctly estimated, there is a unique arbitrage-free price.
Examples of complete models that are commonly used by practitioners include Dupire’s local volatility model [95]; Libor market models with local volatilities, e.g., BGM with deterministic volatilities [67]; and Markov functional models [138] (see also [11], Chapter 2).
Common examples of incomplete models are stochastic volatility models (in short SVMs). Here #assets $<d$. An example of stochastic volatility model is the double lognormal SVM, which has attracted the attention of practitioners in equity markets $[113,137]$. The dynamics of the underlying, denoted by $X_t$, reads under a risk-neutral measure $\mathbb{Q}^0 \sim \mathbb{P}^{\text {hist }}$ as with $\chi^2=1-\rho_{\mathrm{XV}^0}^2-\frac{\left(\rho_{\mathrm{XV}}-\rho \rho_{\mathrm{XV}}\right)^2}{1-\rho^2}$ and $W_t^1, W_t^2, W_t^3$, three uncorrelated standard $\mathbb{Q}^0$-Brownian motions. $V_t$ is the instantaneous variance, and $V_t^0$ plays the role of a moving long-term average value for $V_t$. Neither $V_t$ nor $V_l^0$ are tradable instruments.
金融代写|期权定价理论代写Option Pricing Theory代考|Pricing in practice
In practice, the seller’s price at time $t$ is computed by picking out a particular ELMM $\mathbb{Q}$ : $$ u_t \equiv \mathbb{E}^{\mathbb{Q}}\left[D_{t T} F_T \mid \mathcal{F}_t\right] $$ Under this measure $\mathbb{Q}$, the drift for an asset $X^i$ is fixed to $b_i\left(t, X_t\right) \equiv r_t X_t^i$ (see Remark 1.1). In an incomplete market, $\mathbb{Q}$ does not necessary achieve the supremum in Theorem 1.3, and we lose the superhedging strategy paradigm. Selling options becomes a risky business. However, it seems that the idea of a “true price” (based on a “true model”) is still vivid in the community of structurers and sales people (see the quote at the beginning of this chapter). In practice, picking a particular ELMM simplifies a lot the pricing problem: it becomes a linear problem, i.e., the price of the (European) payoff $F_T^1+F_T^2$ equals the sum of the prices of the (European) payoffs $F_T^1$ and $F_T^2$.
We will always assume that there exists a (deterministic) function $r$ such that $r_t=r\left(t, X_t\right)$. Then $$ D_{t_1 t_2}=\exp \left(-\int_{t_1}^{t_2} r\left(s, X_s\right) d s\right) $$ If there exists $g$ such that $F_T=g\left(X_T\right)$, we speak of a vanilla option. In such a case, by the Markov property of $X$, $$ u_t=\mathbb{E}^{\mathbb{Q}}\left[\exp \left(-\int_t^T r\left(s, X_s\right) d s\right) g\left(X_T\right) \mid \mathcal{F}_t\right] \equiv u\left(t, X_t\right) $$ is a function $u$ of $\left(t, X_t\right)$. Below, we recall that $u(t, x)$ is a solution to a linear second order parabolic PDE, the so-called Black-Scholes pricing PDE.
金融代写|期权定价理论代写Option Pricing Theory代考|Arbitrage and arbitrage-free models
Let us now introduce the notion of arbitrage. An arbitrage is a self-financing strategy that is worth zero initially and yields a positive gain without any risk.
DFFINITION $1.2$ Arhitrage A self-financing admissihle portfolio is called an arbitrage if the corresponding value process $\pi_t$ satisfies $\pi_0=0$ and $$ \pi_T \geq 0 \quad \mathbb{P}^{\text {hist }}-\text { a.s } \quad \text { and } \quad \mathbb{P}^{\text {hist }}\left(\pi_T>0\right)>0 $$ Arbitrageurs are a special kind of trader. Their role is precisely to detect and take full advantage of arbitrage opportunities as soon as they appear in the market. This impacts market prices: arbitrage opportunities tend to disappear as soon as they arise. Absence of arbitrage opportunities is therefore a natural modeling assumption. The next lemma gives a sufficient condition under which we exclude arbitrage opportunities in our market model. LEMMA 1.1 Sufficient condition excluding arbitrage Suppose there exists a measure $\mathbb{Q}$ on $\left(\Omega, \mathcal{F}T\right)$ such that ${ }^3 \mathbb{Q} \sim \mathbb{P}^{\text {hist }}$ and such that, for all asset $X^i$, the discounted price process $\left{\tilde{X}_t^i\right}{t \in[0, T]}$ is a local mar-tingale with respect to $\mathbb{Q}^4{ }^4$ Then the market $\left{X_t\right}_{t \in[0, T]}$ has no arbitrage. Note that the assumption of Lemma $1.1$ bears only on assets $X^i$ only, not on non-tradable components of $X$, such as instantaneous interest rates, instantaneous stochastic volatility, etc.
Let us assume that, at time $t$, we buy and delta-hedge a European option ${ }^5$ written on $m$ assets, say $X_t^1, \ldots, X_t^m$, with maturity $T$ and payoff $F_T$, at the price $z$. In general, the payoff $F_T$ is a function of the paths $\left(X_t^i, 0 \leq t \leq T\right)$ followed by the prices of the $m$ assets between times 0 and $T$. The final value of the buyer’s portfolio, discounted at time 0 , is $$ \begin{aligned} \tilde{\pi}T^B & =-D{0 t} z+\sum_{i=1}^m \int_t^T \Delta_s^i d \tilde{X}s^i+D{0 T} F_T \ & =-D_{0 t} z+\int_t^T \Delta_s \cdot d \tilde{X}s+D{0 T} F_T \end{aligned} $$ Wè can then define the buyer’s super-réplication price at time $t$ as the greatest price $z$ such that the value of the buyer’s portfolio $\tilde{\pi}T^B$ is $\mathbb{P}^{\text {hist }}$-a.S. nonnegative. To be precise, we introduce the following: DEFINITION $1.4$ Buyer’s price $\mathcal{B}_t\left(F_T\right)=\sup \left{z \in \mathcal{F}_t \mid\right.$ there exists an admissible portfolio $\Delta$ such that $$ \left.\tilde{\pi}_T^B \equiv-D{0 t} z+\int_t^T \Delta_s \cdot d \tilde{X}s+D{0 T} F_T \geq 0 \mathbb{P}^{\text {hist }}-a . s .\right} $$ The price $z$ must be $\mathcal{F}t$-measurable, denoted by $z \in \mathcal{F}_t$, i.e., we cannot look into the future. Similarly, we can define the seller’s super-replication price as: DEFINITION $1.5$ Seller’s price $\mathcal{S}_t\left(F_T\right)=\inf \left{z \in \mathcal{F}_t \mid\right.$ there exists an admissible portfolio $\Delta$ such that $$ \left.\tilde{\pi}_T^S \equiv D{0 t} z+\int_t^T \Delta_s \cdot d \tilde{X}s-D{0 T} F_T \geq 0 \mathbb{P}^{\text {hist }}-a . s .\right} $$
金融代写|期权定价理论代写Option Pricing Theory代考|Models of financial markets
Let us consider a filtered probability space $\left(\Omega,\left(\mathcal{F}t\right){0 \leq t \leq T}, \mathbb{P}^{\text {hist }}\right)$. Here $\mathbb{P}^{\text {list }}$ is the historical or real probability measure under which we model our market. A market model is defined by an $n$-dimensional stochastic differential equation $(\mathrm{SDE})$ $$ d X_t^i=b_i\left(t, X_t\right) d t+\sum_{j=1}^d \sigma_{i, j}\left(t, X_t\right) d W_t^j, \quad i \in{1, \ldots, n} $$ and by another positive stochastic process $B_t$, called the money-market account, representing the value of cash, which satisfies $$ d B_t=r_t B_t d t, \quad B_0=1 $$ i.e., $$ B_t=\exp \left(\int_0^t r_s d s\right) $$
$r_t$ is the short term interest rate. It is adapted to $\mathcal{F}t$, which is the (natural) filtration generated by the $d$-dimensional uncorrelated standard Brownian motion $\left{W_t^j\right}{1 \leq j \leq d}$. In order to ensure that SDE (1.1) admits a unique strong solution (see e.g., [13]), we assume that $b$ and $\sigma$ satisfy:
Assum(SDE): The functions $b$ and $\sigma$ are Lipschitz-continuous in $x$ uniformly in $t$, and satisfy a linear growth condition: there exists a positive constant $C$ such that for all $t \geq 0, x, y \in \mathbb{R}^n$, $$ \begin{aligned} |b(t, x)-b(t, y)|+|\sigma(t, x)-\sigma(t, y)| & \leq C|x-y| \ |b(t, x)|+|\sigma(t, x)| & \leq C(1+|x|) \end{aligned} $$ We set $$ D_{t u} \equiv B_t B_u^{-1}=\exp \left(-\int_t^u r_s d s\right) $$ which is the discount factor from date $u$ to date $t$. Throughout the book, we will denote by $\tilde{Y}t \equiv D{0 t} Y_t$ the discounted value of any price process $Y_t$. Certain market components $X^i$ may not be sold or bought in the market, such as the short term interest rate, or a stochastic volatility. Throughout this book, a market component $X^i$ that can be sold and bought in the market is called an “asset.”
Let us assume that we have a portfolio consisting of $m$ assets, say $X_t^1, \ldots, X_t^m$, and the money-market account $B_t$. It is convenient to use the notation $X^0$ for $B$. The portfolio at a time $t$ is composed of $\Delta_t^i$ assets $X_t^i$ and $\Delta_t^0$ units of $X_t^0$ (cash). The $\Delta_t^i$ ‘s must be $\mathcal{F}t$-measurable, i.e., we cannot look into the future. The portfolio value $\pi_t$ is $$ \pi_t \equiv \sum{i=0}^m \Delta_t^i X_t^i $$ As time passees, wẽ can readjust the allocations $\Delta_t^i$, but no cash is éver injéctéd or removed from the portfolio: between $t$ and $t+d t$, the variation in the portfolio value is only due to the variation of the values of the assets, i.e., $$ d \pi_t=\sum_{i=0}^m \Delta_t^i d X_t^i $$ We then speak of a self-financing portfolio. In terms of discounted values, this rears $$ d \tilde{\pi}t=\sum{i=0}^m \Delta_t^i d \tilde{X}t^i=\sum{i=1}^m \Delta_t^i d \tilde{X}_t^i $$
because for any price process $Y_t, d \tilde{Y}t=D{0 t}\left(d Y_t-r_t Y_t d t\right)$, concluding that ${ }^2$ $$ \tilde{\pi}t=\pi_0+\sum{i=1}^m \int_0^t \Delta_s^i d \tilde{X}_s^i $$ We may also write this as $$ \tilde{\pi}_t=\pi_0+\int_0^t \Delta_s \cdot d \tilde{X}_s $$ where $\cdot$ denotes the usual scalar product in $\mathbb{R}^m$. As a technical condition, we need to introduce the notion of admissible portfolio:
DEFINITION 1.1 Admissible portfolio $\left(\Delta_t, 0 \leq t \leq T\right)$ defines an admissible portfolio if $\tilde{\pi}_t$ is bounded from below for all $t \mathbb{P}^{\text {hist }}$-a.s., i.e., there exists $M \in \mathbb{R}$ such that $$ \mathbb{P}^{\text {hist }}\left(\forall t \in[0, T], \tilde{\pi}_t \geq M\right)=1 $$
金融代写|交易策略作业代写Trading strategy代考|Arbitrage Between Own Spot Spread and Future Spread
Power generation costs consist of fixed costs (e.g., equipment depreciation costs, labor costs, and maintenance costs) and variable costs (e.g., fuel costs and exhaust processing costs). However, the variable cost is almost the fuel cost. Therefore, the unit cost of gas-fired generation expressed as Cost and the corresponding natural gas procurement cost expressed as Gas satisfy the following equation: $$ \text { Cost }=\alpha_0 \times \text { Gas }+\alpha_1, $$ where $\alpha_0$ and $\alpha_1$ are the coefficients. Although the Henry Hub futures price HenryHub $b_{f, t}$ and the PJM futures price $P J M_{f, t}$, both of which are unit root processes, are cointegrated, the long-term equilibrium equation, which is a stochastic process, can have large outliers. Then, when the own spot spread, that is, the difference between the power generation unit cost and the corresponding gas procurement unit price, is smaller than the future spread, that is, the future price difference between the PJM and Henry Hub, we swap the spot spread, Spread $_s$ and the future spread, Spread $_f$, which we express as $$ \begin{gathered} \text { Spread }s=\alpha_0 \times \text { HenryHub }{f, t}+\alpha_1-\text { HenryHub }{f, t} \ \text { Spread }_f=\text { PJM }{f, t}-\text { HenryHub } b_{f, t} . \end{gathered} $$ Therefore, the difference between these spreads is $$ \text { Spread }s-\text { Spread }_f=\alpha_0 \times \text { HenryHub } b{f, t}+\alpha_1-P J M_{f, t} . $$ In the following equation: $$ \alpha_0 \times \text { HenryHub } b_{f, t}+\alpha_1-P J M_{f, t}<0 . $$ If we take the Henry Hub long position and the PJM short position corresponding to the electric energy planned for generation, we can lock in profit.
By estimating the long-term equilibrium equation of $H e n r y H u b_{f, t}$ and $P J M_{f, t}$ in a cointegration relationship, we can determine whether the futures spread on a candidate trading date is wider or narrower than the expected spread. This determination enables statistical arbitrage trading between Henry Hub and PJM.
Since the prices in period $t$ are not available for trading in period $t$, we estimate the following long-term equilibrium equation using the price series up to period $t-1$ : $$ P J M_{f, t}=\beta_{f, 0} \times \text { HenryHub} b_{f, t}+\beta_{f, 1^{\circ}} . $$ If the futures spread is higher than the expected value, then we express it as $$ P J M_{f, t}>\beta_{f, 0} \times \text { HenryHub } b_{f, t}+\beta_{f, 1} . $$ We can consider that the PJM price is higher and the Henry Hub price is lower; therefore, we take the PJM short position and Henry Hub long position. Then, the condition for closing these arbitrage positions is $$ \begin{aligned} \text { PJM }{f, t} &-\text { avgShort } P J M_f+\operatorname{avg} \text { LongHenryHub } \ &-\text { HenryHub } b{f, t}>0 \end{aligned} $$ where avg Short $P J M_f$ is the average price of the PJM futures short positions taken, and avgLong Henry Hub $b_f$ is the average price of the Henry Hub futures long positions taken. The clearance of all these futures positions under this condition leads to profit.
Conversely, if the futures spread is below the expected value, then we express it as $$ P J M_{f, t}<\beta_{f, 0} \times \text { HenryHub} b_{f, t}+\beta_{f, 1} . $$ We determine that the PJM price is lower and the Henry Hub price is higher; therefore, we take the PJM long position and Henry Hub short position.
We can estimate the cointegrating vectors by using dynamic OLS (DOLS). OLS estimates the following equation with lag terms for the explanatory variables to climinate autocorrelation: $$ x_{v, t}=\varphi_0+\sum_{i=1}^{v-1}\left(\beta_i \varphi_{i, t}+\sum_{j=-K}^K \phi_{i, j} \Delta x_{i, t-j}\right) . $$ Since Sect. 2.2.4 utilizes a two-variable model, the model for estimating the longterm equilibrium is $$ P J M_t=\varphi_0+\varphi_1 \text { HenryHub}t+\sum{j=-K}^K \phi_j \Delta \text { Henry Hub } b_{t-j} . $$ The lag order $K$ was determined using SBIC. The long-term equilibrium equation for future prices is The long-term equilibrium equation for the spot prices is $$ P J M_{\text {spot }}=11.142 \times \text { HenryHub }_{\text {spot }}+5.732 . $$
The only way to profit by trading goods is to “buy at a lower price and sell at a higher price.” If we trade only one item, then price forecasting is the most important matter. Is this realistically possible? A market is efficient if the information that affects the market price is comprehensive, constant, and has a timely effect on the price. Markets for securities and commodities listed on exchanges are almost efficient and depend on liquidity. In other words, we cannot forecast the price because the price already reflects all the currently available information, and any information that affects the price will occur independently of the price. Unfortunately, it is impossible for market participants to earn returns above the market average. Certainly, a “fully efficient market” is theoretical or virtual. Therefore, some investors and speculators try to collect information before it is reflected in the price. However, these actions make the market more efficient. Because the stationary hypothesis for most energy prices is rejected by the unit root test using daily data, energy companies should consider energy markets as efficient, and energy prices as unpredictable.
In general, power companies procure various types of fuels from various markets, produce electricity using various power generation methods, and sell the power through various sales channels. Section $2.3$ assumes a simple model of purchasing natural gas at the Henry Hub price and selling electricity at the PJM price, as Fig. $2.8$ illustrates. We propose two trading strategies. Section $2.4$ will simulate these methods using actual historical data. Both focus not on these prices but on the price difference between Henry Hub prices and PJM prices. We cannot expect profit owing to market efficiency, even if we analyze each price in detail. On the other hand, we demonstrate the potential to make a profit by investigating price differences, which is a stationary process. When buying the gas required to produce one unit of electricity and selling it, the gross margin is often called the spark spread.
The trading strategy introduced in Sect. 2.3.1 is the arbitrage between the futures market spreads and a company’s spreads expected from its power generation efficiency. This takes advantage of the spread of futures as a stochastic process. All we have to do is take the Henry Hub long position and the PJM short position to secure profits when a favorable futures spread occurs stochastically. The strategy proposed in Sect. 2.3.2 is statistical arbitrage utilizing the cointegration relationship between Henry Hub prices and PJM prices in the futures market. Making use of the longterm equilibrium equation in the futures market that expresses the futures spread, the lower PJM long positions and the higher Henry Hub short positions are expected to yield profit in the narrower spreads than the market when the spread approaches the long-term equilibrium.
Before conducting various analyses and simulations, it is extremely important to interpret the representative statistics of the data. Table $2.1$ provides the summary statistics of the Henry Hub and the PJM.
Considering that each future has a maturity of one month, we set each spot price to January 29, 2021 and each future to December 30, 2020 to simulate the spot-future arbitrage described later in Sect. 2.3.1. Because we extract only the days when both the Henry Hub and PJM data are available, we have 1511 and 1477 observations for the futures and spot prices, respectively.
The mean and median are numerical values located in the center of the economic variables. The mean $\bar{x}$ of the series $\left(x_i \mid i=1,2, \ldots, N\right)$ is calculated as $$ \bar{x}=\frac{1}{N} \sum_{i=1}^N x_i . $$ On the other hand, the median is a value located in the center of each series arranged in descending order. The medians of these futures and spot series are at the 756th and 739th values, respectively. If the number of observations is even, then the median is the average of the two data points in the center. Thus, the median is a more stable index expressing the middle than the mean because outlier values have less effect. Figure $2.1$ shows three distribution examples with the same mean, but different medians. Table $2.1$ indicates that both the mean and median of each future are higher than those of each spot. In other words, both Henry Hub and PJM tend to be contango. We can infer that the supply and demand are not very tight during this period. We can express the relationship between the future price $p_f$ and its spot price $p_s$ as $$ p_f=p_s e^{c_c \Delta T}, $$ where $C_c$ is the cost of carry expressed in terms of yield and $\Delta T$ is the period from the present to maturity. The cost of carry is the sum of the risk-free interest rate and holding cost, expressed as yield minus the convenience yield. Therefore, if their supply and demand remained tight during the period, then the utility of holding their spots would be increasing. Thus, their costs of carry should become negative, and their futures should become lower than their spots. In addition, the medians of both the Henry Hub future and spot prices are higher than their respective means. Therefore, we can expect to find many outliers in the left tail of each distribution. On the contrary, the medians of both the PJM future and spot prices are lower than their respective means. Therefore, we can expect to find many outliers in the right tail of each distribution.
金融代写|交易策略作业代写Trading strategy代考|Cointegration Test
Figures $2.1$ and $2.2$ bring to mind the long-term equilibrium relationship between Henry Hub and the PJM in both futures and spot markets. However, as all four variables accept the unit root hypothesis, we must suspect a spurious regression. Engle and Granger [7] introduced the concept of “cointegration,” which connects multiple unpredictable stochastic variables with a unit root. If a linear combination of multiple unit root processes is stationary, then these variables have a cointegrated relationship. In other words: suppose that the following vector consists of $v$ variables in a unit root process: $$ \mathbf{X}t={ }^T\left(x{1 t}, x_{2 t}, \ldots, x_{v t}\right) $$ The following linear combination is derived from the inner product of the $v$ dimensional coefficient vector and $\mathbf{X}t$ : $$ \boldsymbol{\beta} \mathbf{X}_t=\left(\beta_1, \beta_2, \ldots, \beta_v\right)^T\left(x{1 t}, x_{2 t}, \ldots, x_{v t}\right) $$ If $\boldsymbol{\beta} \boldsymbol{X}t$ is a stationary process, then $x{1 t}, x_{2 t}, \ldots, x_{v t}$ have a cointegrated relationship. Additionally, $$ \boldsymbol{\beta}=\left(\beta_1, \beta_2, \ldots, \beta_v\right) $$ is the cointegrating vector. If there is cointegration between some variables, then the deviation of the observed values from their long-term equilibrium is a stable stochastic process. Because many economic variables have unit roots, this concept is very often applied in a wide range of fields to examine the relationships between economic variables.
Therefore, we test whether the Henry Hub and PJM prices are cointegrated and expect to use this cointegrated relationship in the trading strategies.
Engle and Granger’s [7] proposed test for cointegration has limitations. First, it does not expect a system with three or more variables to have two or more cointegration relationships. Second, the test results may change when the variables are interchanged.
