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Please answer following questions. Total. are 18 questions very short and I extend for 3 days to complete. You can access book and google it.

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Phillip Monin MATH 605 Introduction to Financial Mathematics VERSION 1.0: 2020-02-01 19 : 4 5 : 3 1 Z Copyright © 2020 Phillip Monin Contents 1 Options 2 Modeling Asset Prices 3 Brownian Motion 4 Stochastic Integration 5 Ito’s Formula 6 Introduction to Black-Scholes-Merton 7 Introduction to Partial Differential Equations 8 The Black-Scholes-Merton Formula 9 Consequences of Black-Scholes 10 The Greeks 11 Special Topics Index 7 127 17 25 35 47 97 107 87 59 83 69 Introduction This book is a collection of lecture notes specifically made for the students in M605: Introduction to Financial Mathematics at Georgetown University, which I have taught annually since 2018. It is very much a work in progress and will continue to evolve. The impetus for these notes was the unique nature the students in the graduate mathematics and statistics program at Georgetown. These students are extraordinary, and are phenomenally gifted in mathematics. They benefit from a serious introduction to stochastic processes and quantitative finance, but have little background in the field of finance itself. I have yet to find a book that combines graduate mathematics with introductory finance in a way that accounts for the exposure to probability and computational methods that the students taking M605 obtain in their prerequisites and is flexible enough to allow the course to steer itself naturally; hence these notes. These lecture notes are sketches and are not meant to be the sole source for the topics they cover. The material is borrowed from a variety of sources, including books by Wilmott et al. and Steele. I have tried to cite the borrowed material properly along the way. I am responsible for all errors. Again, these notes are specifically tailored to M605, and they should not be distributed beyond the students in that class. I want to thank the students of M605 for inspiring me week in and week out with their intellectual curiosity, energy, and diligence. While they deserve far more, these notes are dedicated to them. 1 Options This course will approach the topic of financial mathematics and financial modeling by considering the Black-Scholes-Merton model, one of the most elegant and widely known financial models. The Black-Scholes-Merton model provides a theoretical framework for pricing options. Options belong to a class of financial assets called derivatives. A derivative is a financial instrument that derives its value from that of another financial instrument, called an underlying asset. Derivatives can be used for a wide variety of purposes: • hedging: reduce overall risk by taking offsetting positions • investment: take calculated risks based on fundamentals or reasoned views • speculation: take outsize risks akin to gambling Derivatives are widespread and essential to modern finance. As with many things in finance and in life, there are several benefits and drawbacks of derivatives. Benefits associated with derivatives: Derivatives provide a means for risk sharing, that is, the slicing and dicing of risks across events and time, so that (in theory) the market participants most willing and best able to bear these risks actually do so. In this way, derivatives help lead to an efficient use of capital, which promotes economic growth. Drawbacks associated with derivatives: Derivatives can be complex, depend nonlinearly on the underlying, and encourage reckless risk-taking, potentially leading to problems or catastrophes. In 2002, Warren Buffett called derivatives “financial weapons of mass destruction.” 8 1.1 math 605 introduction to financial mathematics Options An option gives its holder the right, but not the obligation, to buy or sell an underlying financial asset at a pre-specified price over a pre-specified period of time or at a pre-specified date. An option is a derivative because its value derives from the value of another financial asset, the underlying. There are several types of options. We will focus on some of the more elementary types in this course. Options terminology: • The pre-specified price is called the strike price or an exercise price • The end date of the contract is called the expiry, the expiration date, or the maturity • The holder, or buyer, of the option is said to exercise the option if and when he or she exercises his or her right to buy or sell the underlying according to the terms of the options contract. • The other party in the contract is called the writer of the option. In contrast to the option holder, the option writer has a potential obligation: he or she may be required to buy the asset from, or sell the asset to, the option holder. This requires compensation; the buyer of the option must pay the option writer. This is called the option’s price or premium. We will consider the following questions at length: What is the value of the option, i.e. the right to buy or sell an asset in the future? How can the writer of the option minimize his or her risk? • An American option allows the holder to exercise the option at any time before the expiration date. A European option allows exercise only at the expiration date. • Contracts that provide the option to buy the asset at a pre-specified price are called call options (“call away the asset”) and contracts that provide the option to sell the asset at a pre-specified price are called put options (“put the asset to someone else”). Payoff diagrams Today is time t, and a stock has current price St . Consider a call option with strike K and maturity T. The payoff is ( CT = ST − K, 0, ST ≥ K = (ST − K )+ ST < K (1.1) options where ( x+ ≡ x, 0, x≥0 x<0 (1.2) Figure 1.1: Payoff diagram for a long call option. Similarly, for a put option, ( K − ST , PT = 0, K ≥ ST = (K − ST )+ K < ST (1.3) Link to option chain for Apple: https://finance.yahoo.com/quote/AAPL/options/ 1.1.1 Example: Obtaining Leveraged Exposure with a Call Option First, a primer on financial leverage. Leverage refers to using borrowed money to buy something. For example, you want to buy a $500,000 house. To avoid having to purchase mortgage insurance, you want to put down 20%, or $100,000, and borrow the $400,000 balance in the form of a mortgage loan to buy the house. You have used borrowed money to purchase an asset, and are said to be 5-to-1 leveraged or have a leverage ratio of 5-to-1. One of the most important aspects about leverage is that it magnifies returns. Suppose the house appreciates 5% to $525,000. For simplicity, suppose the mortgage rate is 0%. What is your equity in the 9 10 math 605 introduction to financial mathematics Figure 1.2: Payoff diagram for a long put option. house? What is the return on your initial capital of $100,000? If you sell the house for $525,000 and pay off the $400,000 mortgage, then you are left with $125,000 on net in equity, which is a 25% return on your initial investment of $100,000. So the asset appreciates 5% but your equity in it goes up 25% because you are 5-to-1 leveraged. However, it should be easy to see that this also works in reverse: if the house instead depreciates 5% to $475,000, then your equity in it will be $75,000, which is a -25% return on your capital. Leverage can also refer to a situation where you may not explicitly borrow money to buy an asset, but the nature of the asset, usually a derivative, permits exposure “as if” you had borrowed money. That is, one position in some asset is equivalent to some other position in other assets wherein money was borrowed. This is called embedded leverage, and will be a general theme of the analysis with options. Today is January 10th. We’d like to value a certain call option. This call option contract stipulates that on November 18th the holder has the right, but not the obligation, to purchase one share of XYZ for $2.00 per share. (Options are most often written in lots of 100 shares, but to keep things simple we usually only consider an option for one share.) What is the price of this option? Suppose that on November 17th the share price of XYZ is $2.20. The holder of the option can purchase XYZ for only $2.00, when in the market it is selling for $2.20. What does the holder do? options She exercises the option, buying the share for $2.00 and immediately sells it in the market for $2.20, making a cool profit of $0.20 per share. On the other hand, if the share price of XYZ is only $1.80, then it would not be sensible to exercise the option. Why? Exercising the option means that she can buy it for 2.00, but it’s cheaper in the market. Suppose a simplified world where XYZ only takes the values $1.80 and $2.20 on November 18th, and does so with equal probability. Then the expected profit for the option is 0.5 ∗ 0 + 0.5 ∗ $0.20 = $0.10 Ignoring interest, it seems reasonable that the value of the option might be close to $0.10. Things aren’t this simple really, as this course will show, but assume for the moment that the price was indeed $0.10. Then the net profit if XYZ raises to $2.20 is $0.20 - $0.10 = $0.10, a 100% increase. However, the net loss if XYZ declines to $1.80 is $0 - $0.10 = - $0.10, a 100% decrease. A 100% increase or decrease is huge, especially when you consider what would’ve happened if the holder just bought the stock. If he or she bought the stock at $2.00 and sold it at $2.20, then a profit of $0.20 on an initial investment of $2.00 yields a 10% gain; if XYZ dropped to $1.80, then a loss of $0.20 yields a 10% loss. What have we learned? • Option prices can respond in exaggerated ways to price changes in the underlying. This is variously called embedded leverage, economic leverage, gearing, and represents a use for options as a way to get higher amounts of exposure to the underlying. Buy stock, get ±10%. Buy option get ±100%. 10x or 10-to-1 leverage. • Dependence of the option’s price on today’s price of the underlying. The greater (lesser) the share price of XYZ on November 18th, the greater (lesser) the profit. While we do not know what XYZ will be in the future, we do know it now, and it seems reasonable that the higher the share price is now, the higher it will be in the future. Thus the price of the option today depends on today’s share price. • Dependence of the option’s price on the strike. The dependence of the value of the option on the strike price is clear. Why? Same asset but different strikes yield different payoffs. • Dependence of the option’s price on the time to maturity. On November 17th, one day before the option expires, there is little 11 12 math 605 introduction to financial mathematics time left for the underlying’s price to change by much. However, today, on January 10th, there is a lot of time for the underlying’s price to change. Thus it seems that the option price should depend on the time to maturity of the option. • Dependence of the option’s price on the volatility. We will see later that the option price depends on a property of the “randomness” of the asset price called the volatility. The volatility is intuitively a measure of how jagged the graph of the asset price is against time. This clearly affects the distribution of the asset price at expiration, and thus should affect the option price. • Dependence of the option’s price on interest rates. Finally, the option price should depend on interest rates. The option is paid for up-front and any potential payoff doesn’t come until later. The price should thus reflect income that would’ve otherwise been made by investing the premium in the bank. 1.1.2 Example: Hedging Downside Risk with a Put Option Today is January 10th. We’d like to understand how to value a certain put option. This put option contract stipulates that on November 18th the holder has the right, but not the obligation, to sell one share of XYZ for $3.00 per share. Suppose that on November 17th the share price of XYZ is $2.70. The holder of the option can sell XYZ for $3.00, when in the market the price is $2.70. What does the holder do? She exercises the option, selling the share for $3.00 and immediately buys it in the market for $2.70, making a cool profit of $0.30 per share. On the other hand, if the share price of XYZ is higher at $3.30, then it would not be sensible to exercise the option. Why? Buy at 3.30 sell at 3, 0.30 loss. Suppose a simplified world where XYZ only takes the values $2.70 and $3.30 on November 18th, and does so with equal probability. Then the expected profit for the option is 0.5 ∗ 0 + 0.5 ∗ $0.30 = $0.15 Ignoring interest, it seems reasonable that the value of the option might be close to $0.15. Again, things aren’t this simple in reality, but assume for the moment that the price was indeed $0.15. Then the net profit if XYZ drops to $2.70 is $0.30 - $0.15 = $0.15, a 100% increase. However, the net loss if XYZ increases to $3.30 is $0 - $0.15 = - $0.15, a 100% decrease. So, like the call example, a put option lets its holder obtain leveraged exposure to the underlying. options 13 In a departure from the last example, let’s use the parameters of this example to consider a popular use of options: hedging. We will consider the use of a put option to hedge downside risk. The construction is called a protective put and involves the following portfolio: own a share (ie long) of XYZ plus long a put option. Suppose I bought the share of XYZ at the same time as I entered the put contract, when the price of XYZ was $3.00 per share. Now, if the price of XYZ increases to $3.30 per share, then my total position is worth (we ignore the option premium for now to illustrate the hedge): ($3.30 − $3.00) + 0 = $0.30 This reflects the price appreciation in the value of the share of XYZ and the fact that when the price goes up I will decline to exercise the option and it will expire worthless. Suppose instead that the price of XYZ falls to $2.70 per share. Now I have lost $0.30 on my position in the stock, but because I hold a put option with a strike at $3.00, I can exercise it and partially offset my loss, for a combined position worth: ($2.70 − $3.00) + $0.30 = $0 Figure 1.3: Payoff diagram for a protective put. 14 math 605 introduction to financial mathematics Algebraically, a protective put has payoff ( K, K > S + S + (K − S) = S, S > K 1.2 (1.4) Exercises Figure 1.4: Payoff diagram for a long call option. The payoff for a long call option, i.e. for an investor who buys a call option, with strike K at expiration is given by CT = (ST − K )+ and that the corresponding payoff diagram is given by Figure 1.4. It is useful to observe that the x-axis in the payoff diagram is the value of the stock at expiration, ST , and the y-axis is the payoff of the option, CT . Also note that, for simplicity, we have ignored the option premium paid by the buyer of the call option at initiation. Please ignore option premia paid throughout this exercise. Similarly, the payoff for a long put option, i.e. for an investor who buys a put option, with strike K at expiration is given by PT = (K − ST )+ and the corresponding payoff diagram is given by Figure 1.5. Exercise 1.1 Write a mathematical expression for the payoff of a short call option with strike K, i.e. for an investor who sells (writes) a call option, at expiration. Draw the corresponding payoff diagram. options Figure 1.5: Payoff diagram for a long put option. Exercise 1.2 Write a mathematical expression for the payoff of a short put option with strike K, i.e. for an investor who sells (writes) a put option, at expiration. Draw the corresponding payoff diagram. Exercise 1.3 Suppose I buy a call option with strike K and sell a put option with the same strike on the same underlying stock and with the same expiration date. What is the payoff MT at expiration for this portfolio? (Hint: It is a simple function of S and K.) Draw the corresponding payoff diagram. Exercise 1.4 Use the definition of x + to simplify (ST − K )+ − (K − ST )+ . This is the payoff of a position composed of a long call option and a short put option. What do you notice? Exercise 1.5 Suppose the share price of Amazon (AMZN) is $1200. Consider two call options on AMZN with the same expiration, one with strike $1250 and one with strike $1300. Which option is likely to be more expensive, i.e. have a higher premium? Why? 15 16 math 605 introduction to financial mathematics Exercise 1.6 Suppose you buy a call with strike price K1 and sell a call on the same stock with the same expiration date but with a strike price of K2 , where K2 > K1 . Write an expression for the payoff of this options portfolio and draw the payoff diagram (ignoring the premiums on the two options). Is the initial cost of this portfolio positive or negative? 2 Modeling Asset Prices The Black-Scholes-Merton (BSM) model is our entree into the world of financial modeling. The model produces the theoretical price of a European option given certain dynamics for the option’s underlying, i.e. the stock. Like all models, the BSM model makes certain assumptions, and one of these assumptions is that the dynamics of the stock are taken as exogenous to the model and are given by a geometric Brownian motion (GBM). In this section we pursue a heuristic derivation of a GBM. Along the way we get glimpses of Brownian motion, stochastic processes, stochastic integrals, and Ito’s lemma, all of which we will come back to with more rigor in a subsequent section. As mentioned above, the dynamics, or data-generating process, for the stock in the BSM model are taken as given by a GBM. This means that in the BSM model we do not intend to ascertain where the price of the stock comes from, or how it evolves over time. Instead we want a statistical model for how a stock price might evolve. So it makes sense to look at the time series of stock prices for inspiration. Figures 2.1 and 2.2 depict time series graphs of the Dow Jones Industrial Average and Amazon’s (AMZN) stock price during selected times. There seems to be a certain jaggedness to the graphs. Daily changes seem to be random, and tend to be relatively small most of the time and occasionally big. If we try to use the tools and functions of the calculus of real variables or ordinary differential equations to produce a similar statistical object then it seems we will fail. We need some new mathematical techniques. Postpone for a second further discussion of what these new mathematical techniques are and consider the field or domain that we’re in. We’re trying to model financial asset prices. What does financial theory have to say about this? One of the central topics and debates in finance is the efficient markets hypothesis. In brief, the EMH says that the history of the stock and the associated company is fully re- 18 math 605 introduction to financial mathematics 15000 14000 13000 12000 11000 10000 9000 004n 2005 l 2005n 2006 l 2006n 2007 l 2007 2 l Ju Ja Ju Ja Ju Ja Ju Date 12000 11500 11000 10500 10000 9500 9000 99 1999 1999 2000 2000 2000 2000 9 1 Apr Jul Oct Jan Apr Jul Oct Date 3000 2800 2600 2400 2200 2000 1800 1600 1400 86 1987 1987 1988 1988 1989 1989 1990 1990 9 1 Jul Jan Jul Jan Jul Jan Jul Jan Jul Date Figure 2.1: Time series graphs of the Dow Jones Industrial Average modeling asset prices 1200 1000 800 600 400 200 0 Figure 2.2: Time series graphs of Amazon’s (AMZN) stock price 201 3 201 4 5 6 201 201 Date 7 201 90 80 70 60 50 40 30 20 10 002000200020002000200020002000200020002000 0 2 Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Date 19 20 math 605 introduction to financial mathematics flected in its present price, and that its present price does not hold any further information. Moreover, markets and thus prices respond immediately to new information about a company that’s relevant to its valuation. Thus modeling of financial asset price ...
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