代写Using Futures for hedging代写C/C++程序

PART A: Background

We  are  the  procurement  manager  for  a  large  construction  company  based  in  China.  Our company is a major consumer of steel rebar, which is a key input material for the concrete structures we build for residential and commercial projects.

Over the past year, we have noticed significant volatility in the spot price of steel rebar, which has impacted our construction costs and project profitability. Concerning about the company's exposure to fluctuations in the steel rebar market, we would like to find a way to hedge this price risk to provide more stability and predictability in your material costs.

As a kind of steel used in construction, the spot price of rebar usually shows obvious seasonal changes. September through November is the peak construction season for the building industry. Increased construction starts drive up demand for rebar, at which point rebar spot prices usually trend higher. In the early September, the price of steel rebar has shown an upward trend. To lock our construction costs and hedge against the upward risk, we are trying to use futures and options.

Objectives:

l Identify  an  appropriate  hedging  instrument  that  can  help  mitigate  the  risk  of  adverse movements in steel rebar prices.

l Develop  a  hedging  strategy  that  can  protect  our  company's  profitability  against potential downside risks in the rebar market.

l Analyze the effectiveness of the hedging strategy under different market scenarios, including both favorable and unfavorable price movements.

l Evaluate the costs associated with implementing the hedging program and compare the net benefits to the unhedged scenario.

PART B: Using Futures for hedging

1.   Introduction

Due  to  business  needs,  we  should  purchase   1,000  metric  tons  of  steel  rebar  to  start  a construction project at the end of October. We want to reduce the impact of price changes in the future. And we prefer to have a cash settlement rather than a physical delivery.

The strategy is using steel rebar futures to hedge the risk of price changes in the coming future. Our business is in China, therefore, we want to choose future contracts pricing in CNY. Through this strategy, we can hedge against the price change in steel rebar and, in the meantime, avoid physical delivery, which helps us control the costs, such as delivery fees and warehousing costs.

2.   Futures Selection

The  steel  rebar  futures  contracts  provided  by  the  Shanghai  Futures  Exchange  meet  our requirements well. The steel rebar futures contract has a size of 10 metric tons per lot and a minimum trade margin of 5% of the total value of the futures contract, moreover, those contracts are pricing in CNY.

The  chart  above  shows  the  term  structure  of  steel  rebar  futures  contracts  observed  on September 26th. From the chart, the futures market is showing contango, which indicated that the market’s expectation of the steel rebar is the price may increase in the future.

The relatively best choice is RB2411 contracts. The RB2411 futures contracts, with a maturity date of November  15th, 2024, and the delivery date is from November  18th  to November  19th, 2024.

PART C: Futures hedging Results

Our project needs 1,000 metric tons of steel rebar, the corresponding quantity is  100 lots of RB2411 contracts. The hedging period is from  September 26th  to October 25th. The hedging strategy is long 100 RB2411 futures contracts on September 26th  with the price of 3, 151 CNY per ton. And close position on October 25th. Buy the steel rebar from the spot market. The spot market and futures market prices data are in the chart below.

To start our position, the initial margin balance should be 157,550 CNY. According to the trade rule of mark-to-market daily, the maintenance margin should be calculated daily.

Before we start our strategy with realized prices. Let’s suppose two extreme scenarios. The historical data  shows that the lowest price is around 2,920 CNY per ton, and the highest is around 4,076 CNY for the past 12 months. Using the historical data as the extreme cases.

The first scenario is that the price goes down to the lowest price, which is 2,920 CNY. 231,000 CNY is required into margin account, which prevents the close of the position early. In that case, indicates that the  steel rebar’s  supply  in the  spot  market  is  increasing, we  should  close the position before, and at the same time, we should purchase steel rebar in the spot market, which may cause the warehousing cost.

The second scenario is when the price goes up to the highest price, which is 4076 CNY, indicating that the spot market may be short of the steel rebar supply. In that case, although we may realize a profit of 925,000 CNY, the supply in the spot market is decreasing, which may cause a higher cost to buy steel rebar in the spot market.

Take both scenario results into consideration, we should set limit to adjust the positions. The limit is that when price go down 5% compared with initial price, if decreased beyond 5%, the position should be closed early and should buy the steel rebar in the spot market directly. If the price goes up, we should maintain position until we close position on October 25th.

Using the realized price to daily mark-to-market settlement, the chart below shows the result.

On October 25th, the RB2411 futures contracts settled price is 3,246 CNY per ton. Compared buying steel rebar directly from the spot market, using futures contracts to hedge helps us save 95,000 CNY. The strategy require the initial cost is 157,550 CNY in the form of margin balance to start the position. And luckily, since the futures prices is increasing, we do not meet the limit and even do not receive any margin call. Due to close the position before the maturity day, the cost of warehousing cost and delivery fee are avoided.

PART D: Using Options for hedging

1. Suitability

l Flexibility:

The current spot price of rebar is (3,271. 14 CNY) and our selected strike price (3,400 CNY) is close to and above the current price. And the expiration time of the contract (October 25th, 2024) basically covers the period we expect the spot price of rebar to rise. By choosing the strike price expiration date, the option hedging strategy can be customized based on market conditions and risk  appetite.  In  addition,  the  dynamic  adjustment  of positions  (delta  hedging)  ensures  that hedging is optimized in real time in the face of price volatility.

l Defined Risk:

The maximum loss from purchasing a call option is the premium paid, and the risk is fixed at the time of purchase.

l Leverage:

Options control large amounts of underlying assets through leverage for a small premium payment.

l Profit:

If the price of rebar rises, the value of the call option will increase to cover potential losses from fluctuations in the price of the underlying asset. If the price goes down, there will be a loss of option premiums, but an increase in the price of the stock will result in a gain.

2.   Features of market

l Option Used：RB2411C3400 (Call Option)

l Contract：

Code	RB2411C3400
Strike Price	3400
Expire Date	25-Oct-2024
Type	American
Size	100 (10 tons/lot)

l Number of Contract:

Based on the Delta value, determine the number of call options to be purchased by using the formula:

l Delivery Date:

The contract expiration date is the same as the last trading day, which is the penultimate fifth trading  day  of the  first  month  preceding  the  delivery  month  (November)  of the  underlying futures contract. We have therefore selected October 25, 2024 as the contract expiration date.

3. Strategy

We use a dynamic hedging strategy with options to hedge a stock position.

l Identify the Position quantity and select the options contract:

Assume that our project needs 1,000 metric tons of steel rebars, and we choose RB2411C3400, which has the hedging period from September 26th to October 25th.

l Calculate the options value:

Use the Black-Scholes model to calculate the theoretical value of call option.

Input parameters include the spot price, risk-free rate, volatility, time to expiration, and strike price.

l Calculate the Delta:

Compute the partial derivative of the option value with respect to the spot price, which gives the delta. The Delta reflects the sensitivity of the option price to changes in the spot price. The formulation of delta is as followed:

l Repeat the calculation:

Recalculate the Delta and adjust the hedging position accordingly.

l Calculate the total return:

Calculate total return of option positions. If prices rise as  expected, the gains from the call option position can offset the losses, effectively hedging the downside risk. Finally, compare to unhedged position to evaluate effectiveness.

Hedging process

PART E: Option Hedging Results

Before  we  start  analyzing  the  hedging  results.  Let’s  suppose  two  extreme  scenarios. Calculating the historical price data of the underlying asset (rebar) for the past year, the highest single-day return is: μ1 = 3.2% , and a minimum single day return of: μ2  = −2. 1%. The highest volatility σ = 32.65%（annualized）.

l Using the historical data as the extreme cases: Increase scenario  (S1): µ =  3.2%, σ = 32.65%

Decrease scenario (S2): µ = −2. 1%, σ = 32.65%

l Model the price paths using the Geometric Brownian Motion (GBM) formulation. Daily price fluctuations are generated through a normal distribution, and the daily price paths of rebar are calculated  step-by-step,  culminating  in the prices  for  the up  scenario  (S1)  and  the  down scenario (S2). The formulation of GBM is as followed:

P(t) = P(0) × e[(μ−0.5σ2)∗t+σ∗t∗Z]

* P(t): price at time t; P(0): initial price (current spot price is RMB 3,271. 14/ton); µ: expected rate of return ( μ1 =  3.2%  for up scenario and  μ2 =  − 2. 1%  for  down  scenario);  σ: volatility (32.65%); t: time (252 trading days in a year, i.e. t = 1/252); Z: standard normally distributed random variable.

l Calculate the following data in the table based on the actual returns of the dynamic hedging strategy and assess the effectiveness of the hedging strategy by comparing the hedged and unhedged results under different scenarios (actual data and extreme price scenarios):

Scenario	Hedged returns	Unhedged returns	Hedge effect
Real data (S)	6.00%	4.46%	2.46%
Increase scenario (S1)	6.43%	5.50%	0.93%
Decrease scenario (S2)	-3.46%	-2.84%	-0.62%

Conclusion: In the rising price scenario, hedging slightly reduces returns, mainly due to the cost of royalties. In the falling price scenario, the hedging strategy effectively controls losses, but at a higher overall cost. Based on the above analysis, we believe that the option strategy is effective in controlling risk, but there is a need to weigh the costs and benefits.

PART F: Trading Cost

1.   Calculate Black-Scholes prices

In our hedging strategy, we use the one-month Treasury yield as the risk-free rate, risk_free =0.0141. The formula for calculating the call option price using the Black-Scholes-Merton model is as follows:

The real call option price we used in our hedging is the column ‘c’ of the graph. Given the realized path of underlying prices from real world, the Black-Scholes prices ofthe call option we used in our hedging are given in the column ‘call option Theoretical Price’ of the graph below.

During the hedging process, the delta is tracked daily to adjust the position of the call option used for hedging, under the realized path of underlying prices in real world given, we give delta in column ‘delta’ in the graph below.

Given the realized path of underlying prices S1 and S2, the Black-Scholes prices of the call option we used in our hedging are given in the columns ‘call option price-BSM-1 ’ and ‘call option price-BSM-2 ’ of the graph below.

2.   Comparison of hedging costs

Also, for the two extreme cases of scenario analysis, given the realized path of underlying prices S1, S2, we give delta in columns ‘delta1 ’ and ‘delta2 ’ in the graph below.

Given the realized path of underlying prices from the date of purchase until the end of the hedge, our realized trading cost is 55.34 CNY. If the call option prices follow BSM model, our trading cost will be 189,887.8 CNY.