Production Planning

1 Hedging
Consider the following policy for hedging. Let R0 be its return (net present value). If the price of
the stock increases more than 3 percent in the first six months, buy a European call option for the
next six months. Otherwise, buy a European put option for the next six months. Set the exercise
price of the put option at $97 and assume the price of the put option is $2.35. Assume the cost of
the call option is $3.40, sample size N = 1000. Use the following parameters for the geometric
random walk model for the stock price movement.
S0= 100 ;% Current Price S0
m= 0.08;% Drift m
s= 0.1; % Volatility s
T= 0.5 ;% Expiration Time T
Dt= 0.5; % Time interval for observing stock price
r= 0.059;% Risk-free Rate r
Kp= 97;% Exercise Price K for put option
ic= 0.03;% Percentage Increse for call option
Kc= 103;% Strike price for call option
cc= 3.4;% Price of a call option
cp= 2.35;% Price of a put option
N= 10ˆ3; % Sample size
Questions
1. What is the distribution (histogram) of R0? Compute the relative frequency of R0 based on the
simulated sample, and graph the histogram.
2. What is the expect return of the hedging strategy? And its 90% confidence interval?
3. What is the probability that R0 < 0? What is the probability that 1 < R0 < 2? 4. What is the probability of purchasing a put option in a year? 1 2 Production Planning Consider the daily operation of a supply chain consisting of one supplier and 10 buyers, i = 1, . . . , 10. In each day, each buyer i has probability pi = 0.7 to place an order (to the supplier) with order quantity Di following a discrete uniform distribution U[1, 10]. buyers’ ordering decisions are independent of each other. The supplier’s unit production cost is $1K. In each day, the supplier first produces x = 50 units of the product, then receives all the orders from the buyers and satisfies them as much as possible. The unit selling price is $10K, and each unit of leftover costs the supplier $1K to dispose. Let V (x) be the supplier’s expected daily profit under production quantity x. Assume sample size N = 365. 1 (a) What is the distribution (histogram) of the daily total order quantities received by the supplier? Graph it. (b) What is the 90% confidence interval of the expected daily profit for the supplier? (c) What is the supplier’s probability of shortage? And what is the probability that the supplier has leftover? (d) What is the profit-maximizing production quantity x ∗ for the supplier? Graph V (x) against x to support your conclusion. (e) Buyer Collusion: To gain bargaining power, 10 buyers consolidate their procurement and reduce the selling price to $ 7K. They coordinate their ordering process as follows: in each day, each supplier i first observes their individual demand Di ∼ U[0, 10]. Then if the consolidated demand ∑10 i=1 Di ≥ 35, all orders will be placed; otherwise, no order will be placed. The supplier’s operation and other parameters remain the same. What is the profit-maximizing production x ∗ for the supplier under buyer collusion? 1Besides your data sheets, you need to have a designated sheet only for answers in your Excel file. 2