金融工程综合实验与实践

吴克坤

课程介绍

关于金融与金融工程

“金融”的意义

• 黄洪：
  • 金融一词并非古已有之。‘金’与‘融’这两个字，是极古老的中国字，可是这两个字连在一起组成一个既不能单独用‘金’字解释也不能单独用‘融’字解释的词—金融—则不见于任何古籍。《康熙字典》（1716）及其以前的所有辞书均无金与融连用的记栽。作为一个词条，最早见于1915年初版的《辞源》和1937年初版的《辞海》。这说明，至迟在19世纪下半叶，金与融这两个字组成的词已经定型并在经济领域中相当广泛地使用；两部辞书的释义均指通过信用中介机构的货币资金融通。”
  • 连起来的‘金融’始于何时，无确切考证，最大可能是来自明治维新的日本。那一阶段，有许多西方经济学的概念就是从日本引进的—直接把日语翻译西文的汉字搬到中国来。

金融活动

金融课程

金融研究

• 学术期刊

• 学会/协会

什么是金融？

• 金融学研究资源如何有效地在不确定的条件下跨期分配

• 金融学研究的几个重要问题：

  • 资产定价
  • 风险管理
  • 资产配置

什么是金融工程？

兹维博迪的定义

Financial engineering is the application of science-based mathematical models to decisions about saving, investing, borrowing, lending, and managing risk. -- by Zvi Bodie

• 金融工程是金融理论与工程方法的结合
• 金融工程不是一个工程学科
• 金融工程的发展基于现代金融理论
• 金融工程是一个应用学科

金融工程的研究领域

金融工程的核心问题

• 核心问题

  • 资产定价(基础资产、衍生工具、复杂的结构化产品)
  • 组合管理/优化
  • 风险管理

• 原理与方法

  • 合成与复制
  • 无套利动态过程
  • 风险中性

金融工程：合成与复制

• BofA Finds Way to Gain From Alibaba IPO Even After Advisory Snub

  • 思路：做多阿里巴巴的主要股东（Softbank、Yahoo），同时做空非阿里巴巴收入的成分
  • BAML的做法：买Soft bank，卖Sprint，卖Yahoo Japan，卖KDDI

• 期权定价：用含有期权的组合复制无风险证券

  • 做空一份期权合约
  • 做多若干股股票

金融工程：无套利动态过程

• 套利机会存在的条件

  • 存在两个不同的资产组合，它们的未来现金流相同，但它们的成本却不同
  • 存在两个成本相同的组合，第一个组合在所有可能状态下的现金流都不低于第二个组合，而且至少存在一种状态，在此状态下第一个组合的现金流大于第二个组合
  • 一个组合的成本为零，但在所有可能状态下，这个组合的损益都不小于零，而且至少存在一种状态，在此状态下这个组合的损益要大于零

• 套利的实现

  • 买入被低估的资产（组合），卖出被高估的资产（组合）
  • 通过融资/融券使初始投资为零
  • 在期末获得正的利润

• 无套利动态过程

Source: Beder T S, Marshall C M. Financial engineering: the evolution of a profession[M]. John Wiley & Sons, 2011.

金融工程与大数据

• 金融数据

• 大数据(4v)
  • Volume: The amount of data collected in files, records, and tables is very large, representing many millions, or even billions, of data points （GB->TB->PB->EB->ZB）.
  • Velocity: The speed with which the data are communicated is extremely great. Real- time or near- real- time data have become the norm in many areas.
  • Variety: The data are collected from many different sources and in a variety of formats, including structured data (e.g., SQL tables or CSV files), semi-structured data (e.g., HTML code), and unstructured data (e.g., video messages).
  • Value: Low value density.

Model Building for Financial Forecasting Using Big Data

Source: 2020 CFA Program curriculum Reading 8

关于金融工程教育

• 金融工程的学历教育项目
  • 几乎没有金融工程的本科项目
  • 有许多（授课式/专业型）硕士研究生项目 (Financial Engineering, Quantitative Finance, Mathematical Finance, Financial Mathematics)
  • 有一些博士项目专注该领域
  • 学位由工程学院、数学系、统计系、运筹与管理科学系等院系授予
  • Ranking & Placement: Quantnet Ranking, Program Websites, LinkedIn

• 课程与技能

  • 经济学与金融学
  • 数学、理论物理
  • 运筹学与统计学
  • 信息理论、编程

• 金融工程师的专业背景

  • 火箭科学家、应用数学家、运筹学、计算机科学家
  • 金融工程项目的毕业生

金融工程职业

Source: Beder T S, Marshall C M. Financial engineering: the evolution of a profession[M]. John Wiley & Sons, 2011.

我校金融工程专业的历史沿革及基本概况

• 历史沿革
  • 90年代中期周骏教授积极倡导在金融学大框架内增设金融工程专业方向
  • 2003年成立金融工程教研室开始招收硕士研究生
  • 2005年开始在硕士研究生教育中设立金融工程专业，全面面向金融专业本科生讲授金融工程课程
  • 2007年开始招收本科生，2009年成立金融工程系
  • 2019年被评为首批国家级一流本科专业建设点
  • 2018年开始大类招生，分流后每年学生规模在50人左右
• 专业排名
  • 《中国大学及学科专业评价报告》：20/258（2020），10/259（2021），8/264（2022），4/259（2023）
  • 校友会2022中国大学金融工程专业排名（研究型）：全国第2（2022），全国第2（2023）

金融工程的应用：量化投资

理解量化投资

Developing a Quant Strategy

• Specify the trading rules in a definitive, objective, and computer-testable form

• Tranform a rouph idea into a set of trading rules

  • Entry & Exit
  • Risk Management
  • Position Sizing

• Tranform the trading rules into computer programs

• Example: a simple MA strategy

The Human Language Pseudocode

1. Calculate a fast moving average
2. Calculate a slow moving average
3. Go long when yesterday the fast moving average was below the slow moving average and today the fast moving average is above the slow moving average
4. Once long, stay long until a sell entry occurs
5. Go short when yesterday the fast moving average was above the slow moving average and today the fast moving average is below the slow moving average
6. Once short, stay short until a buy entry occurs

Example: Coding with Backtrader

# 导入必要的库
import backtrader as bt
import matplotlib as mpl  
import matplotlib.pyplot as plt
import tushare as ts
import pandas as pd
import os
from datetime import datetime

定义策略

class SmaCross(bt.Strategy):
    # list of parameters which are configurable for the strategy
    params = dict(
        pfast=10,  # period for the fast moving average
        pslow=30   # period for the slow moving average
    )
    def __init__(self):
        super().__init__()
        sma1 = bt.ind.SMA(period=self.p.pfast)  # fast moving average
        sma2 = bt.ind.SMA(period=self.p.pslow)  # slow moving average
        self.crossover = bt.ind.CrossOver(sma1, sma2)  # crossover signal
    def next(self):
        if not self.position:  # not in the market
            if self.crossover > 0:  # if fast crosses slow to the upside
                self.order_target_size(target=1)  # enter long
            elif self.crossover < 0:  # in the market & cross to the downside
            self.order_target_size(target=0)  # close long position

策略执行

data_path = './data/'
if not os.path.exists(data_path):
    os.makedirs(data_path)
    mytoken='f3244*****8addef6e******ad68fa95b18c*****c3a5238533*****'
class Strategy_runner:
    def __init__(self, strategy, ts_code, start_date, end_date, data_path=data_path, 
	pro=False, token=mytoken):
        self.ts_code = ts_code
        self.start_date = start_date
        self.end_date = end_date
        # convert to datetime
        self.start_datetime = datetime.strptime(start_date,'%Y%m%d')
        self.end_datetime = datetime.strptime(end_date,'%Y%m%d')
        if pro:
            csv_name = f'pro_day_{str(ts_code)}-{str(start_date)}-{str(end_date)}.csv'
        else:
            csv_name = f'day_{str(ts_code)}-{str(start_date)}-{str(end_date)}.csv'
            csv_path = os.path.join(data_path,csv_name)
        if os.path.exists(csv_path):
            if pro:
                self.df = pd.read_csv(csv_path)
            else:
                self.df = pd.read_csv(csv_path,index_col=0)

        else:
            if pro:
                ts.set_token(mytoken)
                self.pro = ts.pro_api()
                self.df = self.pro.daily(ts_code=self.ts_code, 
				start_date=self.start_date, 
				end_date=self.end_date)
                if not self.df.empty:
                    self.df.to_csv(csv_path, index=False)
            else:
                self.df = ts.get_hist_data(self.ts_code, 
				str(self.start_datetime), 
				str(self.end_datetime))
                if not self.df.empty:
                    self.df.to_csv(csv_path, index=True)
            
        self.df_bt = self.preprocess(self.df, pro)
        print(self.df_bt)
        self.strategy = strategy
        self.cerebro = bt.Cerebro()

    def preprocess(self, df, pro=False):
        if pro:
            features=['open','high','low','close','vol','trade_date']
            convert_datetime = lambda x: pd.to_datetime(str(x))
            df['trade_date'] = df['trade_date'].apply(convert_datetime)
            print(df)
            bt_col_dict = {'vol':'volume','trade_date':'datetime'}
            df = df.rename(columns=bt_col_dict)
            df = df.set_index('datetime')
            # df.index = pd.DatetimeIndex(df.index)
        else:
            features=['open','high','low','close','volume']
            df = df[features]
            df['openinterest'] = 0
            df.index = pd.DatetimeIndex(df.index)

        df = df[::-1]
        return df

    def run(self):
        data = bt.feeds.PandasData(dataname=self.df_bt, fromdate=self.start_datetime, 
		todate=self.end_datetime)
        self.cerebro.adddata(data)  # Add the data feed
        self.cerebro.addstrategy(self.strategy)  # Add the trading strategy
        self.cerebro.broker.setcash(100000.0)
        self.cerebro.addanalyzer(bt.analyzers.SharpeRatio,_name = 'SharpeRatio')
        self.cerebro.addanalyzer(bt.analyzers.DrawDown, _name='DW')
        self.results = self.cerebro.run()
        strat = self.results[0]
        print('Final Portfolio Value: %.2f' % self.cerebro.broker.getvalue())
        print('SR:', strat.analyzers.SharpeRatio.get_analysis())
        print('DW:', strat.analyzers.DW.get_analysis())
        return self.results
    
    def plot(self, iplot=False):
        self.cerebro.plot(iplot=iplot)

ts_code='002456.SZ'
start_date='20240101'
end_date='20241231'
strategy_runner = Strategy_runner(strategy=SmaCross, ts_code=ts_code, 
    start_date=start_date, end_date=end_date, pro=True)
results = strategy_runner.run()
strategy_runner.plot()

欧菲光（002456.SZ）：2020-01-01 ~ 2020-12-31

欧菲光（002456.SZ）：2021-01-01 ~ 2021-12-31

欧菲光（002456.SZ）：2022-01-01 ~ 2022-12-31

欧菲光（002456.SZ）：2023-01-01 ~ 2023-12-31

海南机场（600515.SH）：2020-01-01 ~ 2020-12-31

海南机场（600515.SH）：2021-01-01 ~ 2021-12-31

海南机场（600515.SH）：2022-01-01 ~ 2022-12-31

海南机场（600515.SH）：2023-01-01 ~ 2023-12-31

Early Performance Measures Techniques

• Two Desirable Attributes

  • The ability to derive above-average returns for a given risk class. The superior risk-adjusted returns can be derived from either

    • Superior timing
    • Superior security selection

  • The ability to diversify the portfolio completely to eliminate

    • A completely diversified portfolio is perfectly correlated with the fully diversified benchmark portfolio

• Portfolio Evaluation before 1960

  • Evaluate portfolio performance almost entirely on the basis of the rate of return
  • Rate of return within risk classes, but don’t know how to measure it and could not consider it explicitly

• Peer Group Comparisons

  • Collects the returns produced by a representative universe of investors over a specific period of time
  • Potential problems
    • No explicit adjustment for risk
    • Difficult to form comparable peer group

"Modern" Performance Measures Techniques

Components of Investment Performance

Evaluateing Diversification

• The gross selectivity component can be broken into two parts
  • Net selectivity
  • Diversification

• : the return on the combination of the riskless asset and the market portfolio that has return volatility equivalent to that of the portfolio being evaluated

Performance Measurement with Downside Risk

• The Sortino measure is a risk-adjusted measure that differs from the Sharpe ratio in two ways
  • It measures the portfolio's average return in excess of a user-selected minimum acceptable return threshold
  • It captures just the downside risk (DR) in the portfolio rather than the total risk as in Sharpe measure
• The Form
  • : the minimum acceptable return threshold
  • : the downside risk coefficient for Portfolio

Factors That Affect Use of Performance Measures

• Market Portfolio Is Difficult to Approximate

  • Benchmark Portfolios

    • Performance evaluation standard
    • Usually a passive index or portfolio
    • May need benchmark for entire portfolio and separate benchmarks for segments to evaluate individual managers

  • Benchmark Error

    • Can effect slope of SML
    • Can effect calculation of beta
    • Greater concern with global investing
    • Problem is one of measurement

• Implications of the Benchmark Problems
  • Benchmark problems do not negate the value of the CAPM as a normative model of equilibrium pricing
  • There is a need to find a better proxy for the market portfolio or to adjust measured performance for benchmark errors
  • Multiple markets index (MMI) is major step toward a truly comprehensive world market portfolio

• Required Characteristics of Benchmarks

  • Unambiguous
  • Investable
  • Measurable
  • Appropriate
  • Reflective of current investment opinions
  • Specified in advance

• Selecting a Benchmark

  • A global level that contains the broadest mix of risky asset available from around the world
  • A fairly specific level consistent with the management style of an individual money manager

Software

Why I recommend Excel+VBA and Python?

• Excel+VBA

  • simple and powerful
  • good performance for dataset with moderate size
  • every company has office suit installed on their desktops

• Python: Life is short, you need Python!

  • best programming language for bigdata analytics
  • glue coding language
  • large and comprehensive standard library
  • smooth learning curve
  • highly readable and easy to code

Tushare

Tushare是一个免费、开源的python财经数据接口包。主要实现对股票等金融数据从数据采集、清洗加工 到 数据存储的过程，能够为金融分析人员提供快速、整洁、和多样的便于分析的数据，为他们在数据获取方面极大地减轻工作量，使他们更加专注于策略和模型的研究与实现上。考虑到Python pandas包在金融量化分析中体现出的优势，Tushare返回的绝大部分的数据格式都是pandas DataFrame类型，非常便于用pandas/NumPy/Matplotlib进行数据分析和可视化。

VNPY

• 丰富的Python交易和数据API接口，基本覆盖了国内外常规交易品种（证券、期货、期权、外汇、CFD）：
  • CTP（vn.ctp）：期货、期货期权
  • 飞创（vn.xspeed）：期货、期货期权
  • 飞马（vn.femas）：中金所的期货和期货期权
  • 金仕达期权（vn.ksotp）：期货、期货期权、证券期权
  • 金仕达黄金（vn.ksgold）：金交所贵金属
  • 飞鼠（vn.sgit）：期货、金交所贵金属
  • QDP极速柜台（vn.qdp）：期货、期货期权、金交所贵金属
  • OANDA（vn.oanda）：外汇、CFD
  • Interactive Brokers（vn.ib）：外盘股票、期货、期权、外汇等
  • 直达期货（vn.shzd）：外盘期货
  • OKCoin（vn.okcoin）：比特币、莱特币等
  • 通联数据（vn.datayes）：历史行情数据、基本面数据
    

• 事件驱动引擎（vn.event），用于实现Python在全局锁（GIL）限制下的高性能事件驱动编程
• 开发示例（vn.demo），通过简洁明了的代码展示如何使用API和事件驱动引擎开发交易程序
• 交易平台（vn.trader），整合了vn.py项目中所有的交易接口以及Interactive Brokers的三方接口（IbPy），围绕事件驱动引擎设计了针对策略算法和交易应用开发的上层API，使得交易员可以专注于解决交易业务需求而无需关注底层细节，平台中提供了一套完整的CTA策略模块（回测和实盘）作为开发参考
• RPC模块（vn.rpc），提供跨进程服务调用的RPC模块，同时支持服务端向客户端的主动数据推送，用于实现vn.py框架下模块的多进程解耦

金融工程的应用：风险管理

Market Risk

Value-at-Risk (VaR)

• VaR is the maximum loss over a target horizon such that there is a low, prespecified probability that actual loss will be larger.

• Formula

or,

Components of a Multivariate Risk Modeling Systems

• Risk system

  • portofolio position system
  • risk factor modeling system
  • aggregation system

• Describe joint movements in the risk factors

  • specify an analytical distribution
  • take the joint distribution from empirical observations

• Aggregation: VaR methods

  • delta-normal method
  • historical simulation method
  • Monte Carlo simulation method

Risk Mapping

Introduction

• Have assumed so far that each position has its own risk factor, which we model directly

  • Distinguish between positions and risk factors

• However, it is not always possible or desirable to model each position as having its own risk factor

• Might wish to map our positions onto some smaller set of risk factors

  • Might wish to map positions to () risk factors

Reasons for mapping

• Might not have enough data on our positions

  • E.g., might have small runs of Emerging Market data
  • Map to risk factors for which we do have data

• Might wish to cut down on the dimensionality of our covariance matrices

  • This is important!
  • With positions, covariance matrix has terms
  • As rises, covariance matrix becomes more unwieldy

• Need to keep dimensionality down to avoid computational problems too – rank problems, etc.

Stages of mapping

• Construct a set of benchmark instruments or factors

  • Might include key bonds, equities, etc.

• Collect data on their volatilities and correlations

• Derive synthetic substitutes for our positions, in terms of these benchmarks

  • This substitution is the actual mapping

• Construct VaR/CVaR of mapped porfolio

• Take this as a measure of the VaR/CVaR of actual portfolio

Selecting Core Instruments

• Usual approach to select key core instruments

  • Key equity indices, key zero bonds, key currencies, etc.

• Want to have a rich enough set of these proxies, but don’t want so many that we run into covariance matrix problems

• RiskMetrics core instruments

  • Equity positions represented by equivalent amounts in key equity indices
  • Fixed income positions by represented by combinations of cashflows of a limited number of maturities
  • FX positions represented by relevant amounts in `core' currenicies
  • Commodity positions represented by amounts of selected standardised futures positions

Mapping with Principal Components

• Can use PCA to identify key factors

• Small number of PCs will explain most movement in our data set

• PCA can cut down dramatically on dimensionality of our problem, and cut down on number of covariance terms

  • E.g., with 50 original variables, have separate covariance terms
  • With 3 PCs, have only 3 separate covariance terms

Mapping Positions to Risk Factors

• Most positions can be decomposed into primitive building blocks

• Instead of trying to map each type of position, we can map in terms of portfolios of building blocks

• Building blocks are

  • Basic FX
  • Basic equity
  • Basic fixed-income
  • Basic commodity

A General Example of Risk Mapping

• Replace each if the positions with a exposure on the risk factors. Define as the exposure of instruments to risk factor
• Aggregate the exposures across the positions in the portfolio,
• Derive the distribution of the portfolio return from the exposures and movements in risk factors, , using one of the three VaR methods

Example: Mapping with Factor Models

• Decompose stock return

  • a constant term (not important fot risk management purpose)
  • a component due to the market
  • a residual term

• The portfolio return

  • Mean:
  • Variance:
  • For equally weighted portfolio:
  • The mapping: on stock on index

• This approach is useful especially when there is no return history

Example: Mapping with Fixed-Income Portfolios

• Risk-free bond portfolio

  • maturity mapping: replace the current value of each bond by a position on a risk factor with the same maturity

  • duration mapping: maps the bond on a zero-coupon risk factor with a maturity equal to the duration of the bond

  • cash flow mapping: maps the current value of each bond payment on a zero-coupon risk factor with maturity equal to the time to wait for each cash flow

Corporate bond portfolio

• Decomposition:

• the movement in the value of bond price :

• the portfolio:
• aggregation:

• Variance:
• on bond on on

Choice of Risk Factors

It should be driven by the nature of the portfolio:

• portfolio of stocks that have many small positions well dispersed across sectors

• portfolios with a small number of stocks concentrated in one sector

• an equity market-neutral portfolio

Mapping Complex Positions

• Complex positions are handled by apply financial engineering theory
• Reverse-engineer complex positions into portfolios of simple positions
• Map complex positions in terms of collections of synthetic simple positions

• Some examples, using FE/FI theory:
  • Coupon-paying bonds: can regard as portfolios of zeros
  • FRAs: equivalent to spreads in zeros of different maturities
  • FRNs: equivalent to a zero with maturity equal to period to next coupon payment (because it reprices at par)
  • Vanilla IR swaps: equivalent to portfolio long a fixed-coupon bond and short a FRN
  • Structured notes: equivalent to combinations of IR swaps and conventional FRNs
  • FX forwards: equivalent to spread between foreign currency bond and domestic currency bond
  • Commodity, equity and FX swaps: combinations of spread between forward/futures and bond position

Dealing with Optionality

• All these positions can be mapped with linear based mapping systems because of their being (close to) linear

• These approaches not so good with optionality

  • Non-linearity of options positions can lead to major errors in mapping

• With non-linearity, need to resort to more sophisticated methods, e.g., delta-gamma and duration-convexity

Credit Risk

Spreads and Default Risk: Multiple Periods

We compound interest rates and default rates over each period.Let be the average annual default rate.

If we use the cumulative default probability

A very rough approximation:

Risk Premium

In the previous analysis we assume risk neutrality. As a result, is a risk neutral measure, which is not necessarily equal to the objective, physical probability of default.

Assuming and be the physical probability of default and the discount rate. We have the following

The risk premium () must be tied to some meaure of bond riskiness as well as investor risk aversion. In addition, this premium may incorporate a **liquidity premium and tax effects.

The Merton Model

• The Merton (1974) model views equity as akin to a call option on the assets of the firm, with an exercise price given by the face value of debt

• Consider a firm with total value that has one bond due in one period with face value

  • equity can be viewed as a call option on the firm value with strike price equal to the face value of debt
  • the current stock price embodies a forecast of default probability in the same way that an option embodies a forecast of being exercised
  • corporate debt can be viewed as risk-free debt minus a put option on the firm value

Pricing Equity and Debt

Firm value follows the geometric Brownian motion

The value of firm can be decompose in to the value of equity () and the value of debt (). The corporate bond price is obtained as

The equity value is

Stock Valuation

where

Firm Volatility

Bond Valuation

Risk-Neutral Dynamics of Default

Pricing Credit Risk

Credit Option Valuation

Applying the Merton Model

• the KMV approach: the company sells expected default frequencies (EDFs) for global firms

• Advantages

  • it relies on the price of equities, which are more actively traded than bonds
  • correlations between equity prices can generate correlations between bonds
  • it generates movements in EDFs that seems to lead changes in credit ratings

• Disadvantages

  • it can not be used to price sovereign credit risk
  • it relies on a static model of the firm's capital and risk structure
  • management could undertake new projects that increases not only the value if equity but also its volatility
  • the model fails to explain the magnitude of credit spreads we observe on credit-sensitive bonds

金融工程的应用：实物期权

An Alternative to the NPV Rule for Capital Investments

• Define stochastic processes for the key underlying variables and use risk-neutral valuation

• This approach (known as the real options approach) is likely to do a better job at valuing growth options, abandonment options, etc than NPV

The Problem with using NPV to Value Options

• Consider the example from Chapter 13: risk-free rate =4%; strike price = $21

• Suppose that the expected return required by investors in the real world on the stock is 16%. What discount rate should we use to value an option with strike price $21?

Correct Discount Rates are Counter-Intuitive

• Correct discount rate for a call option is 42.6%

  • , , , if required expected return is

• Correct discount rate for a put option is –52.5%

General Approach to Valuation

• Assuming

  • The market price of risk for a variable is
  • Suppose that a real asset depends on several variables . Let and be the expected growth rate and volatility of so that

• We can value any asset dependent on a variable by

  • Reducing the expected growth rate of by where is the market price of -risk and is the volatility of
  • Assuming that all investors are risk-neutral

Example (36.1)

The cost of renting commercial real estate in a certain city is quoted as the amount that would be paid per square foot per year in a new 5-year rental agreement. The current cost is $30 per square foot. The expected growth rate of the cost is 12% per annum, the volatility of the cost is 20% per annum, and its market price of risk is 0.3. A company has the opportunity to pay $1 million now for the option to rent 100,000 square feet at $35 per square foot for a 5-year period starting in 2 years. The risk-free rate is 5% per annum (assumed constant).

How to evaluate the option?

Example (36.1)

Define as the quoted cost per square foot of office space in 2 years. Assume that rent is paid annually in advance. The payoff from the option is

where is an annuity factor given by

The expected payoff in a risk-neutral world is therefore

According to the Black-Scholes formula, the value of the option is

where

and

The expected growth rate in the cost of commercial real estate in a risk-neutral world is , where is the real-world growth rate, is the volatility, and is the market price of risk. In this case, , , and , so that the expected risk-neutral growth rate is 0.06, or 6%, per year. It follows that . Substituting this in the expression above gives the expected payoff in a risk-neutral world as $1.5015 million. Discounting at the risk-free rate the value of the option is million.

This shows that it is worth paying $1 million for the option.

Extension to Many Underlying Variables

• When there are several underlying variables qi we reduce the growth rate of each one by its market price of risk times its volatility and then behave as though the world is risk-neutral

• Note that the variables do not have to be prices of traded securities

Estimating the Market Price of Risk

• Contimuous CAPM:

• According to the previous slides:

• The market price of risk:

Types of Options

• Abandonment

• Expansion

• Contraction

• Option to defer

• Option to extend life

Example (page 795)

• A company has to decide whether to invest $15 million to obtain 6 million units of a commodity at the - rate of 2 million units per year for three years.
• The fixed operating costs are $6 million per year and the variable costs are $17 per unit.
• The spot price of the commodity is $20 per unit and 1, 2, and 3-year futures prices are $22, $23, and $24, respectively.
• The risk-free rate is 10% per annum for all maturities.

The Process for the Commodity Price

• We assume that this is

where and

• We build a tree as in Chapter 32 (Interest Rate Tree) and Chapter 35 (Energy and Commodity Derivatives)

Value of Expansion Option; Fig 36.4 (Company Can Increase Scale of Project by 20% for $2 million)

Node	A	B	C	D	E	F	G	H	I
	0.1667	0.1217	0.1667	0.2217	0.8867	0.1217	0.1667	0.2217	0.0867
	0.6666	0.6566	0.6666	0.6566	0.0266	0.6566	0.6666	0.6566	0.0266
	0.1667	0.2217	0.1667	0.1217	0.0867	0.2217	0.1667	0.1217	0.8867

Appendix: Interest Rate Tree

Interest Rate Trees vs Stock Price Trees

• The variable at each node in an interest rate tree is the -period rate

• Interest rate trees work similarly to stock price trees except that the discount rate used varies from node to node

Two-Step Tree Example

Payoff after 2 years is , ; ; ; Time step=1yr

B:

A:

Alternative Branching Processes in a Trinomial Tree

Procedure for Building Tree

• [1] Assume and
• [2] Draw a trinomial tree for to match the mean and standard deviation of the process for
• [3] Determine one step at a time so that the tree matches the initial term structure

Example

Suppose that , , and . The zero-rate curve is given below.

Building the First Tree for the rate

• Set vertical spacing:

• Change branching when nodes from middle where is smallest integer greater than

• Choose probabilities on branches so that mean change in is and S.D. of change is

The First Tree

Shifting Notes

• Work forward through tree

• Remember the value of a derivative providing a $1 payoff at node at time

• Shift nodes at time by so that the bond is correctly priced

The Final Tree

Formulas for 's and 's

To express the approach more formally, suppose that the have been determined for . The next step is to determine so that the tree correctly prices a zero-coupon bond maturing at . The interest rate at node is ,so that the price of a zero-coupon bond maturing at time is given by

where is the number of nodes on each side of the central node at time .

The solution to this equation is

Once has been determined, the for can be calculated using