P-1. Preface. Thank you for choosing ACTEX.

Transcription

1 Preface P- Preface Thank you for choosing ACTEX ince Exam MFE was introduced in May 007, there have been quite a few changes to its syllabus and its learning objectives To cope with these changes, ACTEX decided to launch a brand new study manual, which adopts a completely different pedagogical approach The most significant difference is that this edition is fully self-contained, by which we mean that, with this manual, you do not even have to read the required text Derivatives Markets by Robert L McDonald By reading this manual, you should be able to understand the concepts and techniques you need for the exam ufficient practice problems are also provided in this manual As such, there is no need to go through the textbook s end-of-chapter problems, which are either too trivial simple substitutions or too computationally intensive Excel may be required Note also that the textbook s end-of-chapter problems are not at all similar in difficulty and in forma to the questions released by the ociety of Actuaries oa We do not want to overwhelm students with verbose explanations Whenever possible, concepts and techniques are demonstrated with examples and integrated into the practice problems Another distinguishing feature of this manual is that it covers the exam materials in a different order than it occurs in Derivatives Markets There are a few reasons for using an alternative ordering: ome topics are repeated quite a few times in the textbook, making students difficult to fully understand them For example, estimation of volatility is discussed four times in Derivatives Markets ections 4, 5, 85, 3! In sharp contrast, our study manual presents this topic fully in one single section Module 3, Lesson 4 The focus of the textbook is somewhat different from what oa expects from the candidates According to the oa, the purpose of the exam is to develop candidates knowledge of the theoretical basis Nevertheless, the first half of the textbook is almost entirely devoted to applications Therefore, we believe that reading the textbook or following the textbook s ordering is not the best use of your precious time Perhaps you have passed some oa exams by memorizing formulas However, from the released exam questions, you can easily tell that it is difficult, if not impossible, to pass Exam MFE/3F simply by memorizing all formulas in the textbook In this connection, in this writing, we help you really learn the materials By having the reasoning skills, you will discover that there is not really much to memorize To help you better prepare for the exam, we intentionally write the practice problems and the mock exams in a similar format as the released exam and sample questions This will enable you to, for example, retrieve information more quickly in the real exam Further, we have integrated Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

2 P- Preface the sample and previous exam questions provided by the oa into the study manual into our examples, our practice problems, and our mock exams This seems to be a better way to learn how to solve those questions, and of course, you will need no extra time to review those questions Our recommended procedure for use of this study manual is as follows: Read the lessons in order Immediately after reading a lesson, complete the practice problems we provide for that lesson Make sure that you understand every single practice problem 3 After studying all 5 lessons, work on the mock exams If you find a possible error in this manual, please let us know at the Customer Feedback link on the ACTEX homepage wwwactexmadrivercom Any confirmed errata will be posted on the ACTEX website under the Errata & Updates link Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

3 Preface P-3 A Note on Rounding and Using the Normal Table To achieve the desired accuracy, we recommend that you store values in intermediate steps in your calculator If you prefer not to, please keep at least six decimal places In this study guide, normal probability values and z-values are based on the standard normal distribution table, which is provided on page T0-3 When using the standard normal distribution table, do not interpolate Use the nearest z-value in the table to find the probability Example: uppose that you are to find PrZ < 0759, where Z denotes a standard normal random variable Because the z-value in the table nearest to 0759 is 076, your answer is PrZ < 076 = Use the nearest probability value in the table to find the z-value Example: uppose that you are to find z such that PrZ < z = 07 Because the probability value in the table nearest to 07 is 06985, your answer is 05 Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

4 P-4 Preface yllabus Reference Module 0 and Module Our tudy Manual The Required Text Module 0: Review 0 4, 5 p3 only , 9 Module : Risk-Neutral Valuation in Discrete-time Lesson : Introduction to Binomial Trees 0 up to the middle of p38 0 from the middle of p38 to the middle of p from p30 to the middle of p3 Lesson : Multiperiod Binomial Trees 0, from the middle of p3, 3 from the middle of p358 to the middle of p359 Lesson 3: Options on Other Assets 3 05 up to p p33, up to the middle, 9 formula 94 only from the middle of p33 to the middle of p334 Lesson 4: Pricing with True Probabilities 4 up to the middle of p347 4 second half of p347 to p350 Lesson 5: tate Prices 5 Appendix B 5 ample questions #7 53 Appendix B Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

5 Preface P-5 Module Module : Risk-Neutral Valuation in Continuous-time Lesson : Brownian Motions 3 up to the beginning of p353 0 up to the first two lines of p from the bottom of p653 to the bottom of p up to the middle of p656 Lesson : tochastic Calculus cattered in 0 and excluding multivariate Ito s lemma 3 Mainly scattered in 0 and 03, Example p65 to p653 Lesson 3: Modeling tock Price Dynamics 3 8, 8 3 0, up to the end of p from the bottom of p60 to formula from the middle of p353 to p354 Lesson 4: The harpe Ratio and the Black-choles Equation 4 p , p68, but we generalize the approach here 43 from p683 to the middle of p , 3 except the backward equation, 07 except finding the lease rate and valuing a claim on a Q b Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

6 P-6 Preface Module 3 Module 3: The Black-choles Formula Lesson : Introduction to the Black-choles Formula 3 up to the middle of p706 p706 ordinary options and gap options, up to 3 p Lesson : Greek Letters and Elasticity 3 3 p38 385, p387 up to middle, p388 Greek measures for portfolio, 34 up to p p386, middle of p starting from p389 to the end of the section 3 34 Appendix 3B Lesson 3: Risk Management Techniques 33 3, 33 up to the next-to-last paragraph on p p47 to p the bottom of p49 to the end of the section, p the bottom of p433 to the end of the section Lesson 4: Estimation of Volatilities and Expected Rates of Appreciation 34 5, 3, 4, 3 up to the middle of p746, Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

7 Preface P-7 Module 4 Module 4: Further Topics on Option Pricing Lesson : Exotic Options I Exercise except p455 and example 4 Lesson : Exotic Options II Exercise , p706 Ordinary options and gap options Lesson 3: imulation 43 9, Lesson 4: General Properties of Options from p99 to the first lines on p p93 94 European versus American options and maximum and minimum option prices p97 time to expiration Lesson 5: Early Exercise for American Options p94 to the third paragraph of p96, p96 Early exercise for puts p455 and Example 4 Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

8 P-8 Preface Module 5 Module 5: Interest Rate Models Lesson : Binomial Interest Rate Trees 5 cattered in Chapter up to Figure 49 on p806 Lesson : The Black Model 5 9 p86 Options on bonds 5 p38 Options on futures Lesson 3: An Equilibrium Equation for Interest Rate Derivatives 53 4 p78 An equilibrium equations for bonds up to p783 the first paragraphs p783 to the middle of p the middle of p783 to the end of section 4 Lesson 4: The Rendleman-Bartter, Vasicek and Cox-Ingersoll-Ross Model 54 4 p785 The Rendleman-Bartter model 54 4 p786 The Vasicek model p787 The Cox-Ingersoll-Ross model and Comparing Vasicek and CIR Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

9 ample pages from Module Lesson This lesson contains 4 pages We are showing pages to 6

10 Module : Risk-neutral valuation in continuous-time Lesson : tochastic calculus M-9 Lesson tochastic Calculus OBJECTIVE To understand stochastic differential equations and diffusion processes To use Itô s lemma to obtain the stochastic differential equation for a function of a diffusion process 3 To solve three types of stochastic differential equations 4 To understand the concept of variations of Brownian motions tochastic Differential Equations Differentials uppose that the rate of change in x depends on the time t and the value of x itself That is, dx dt = f t, x We can interpret this equation by using the concept of differentials We multiply both sides of the equation by dt to obtain dx = ft, xdt, which says that the change in x over a very short time interval [t, t + dt] is given by ft, xdt Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

11 M-0 Module : Risk-neutral valuation in continuous-time Lesson : tochastic calculus To illustrate, we apply the above to a bank account crediting a constant force of interest r uppose that you put $ into the bank account at time 0 and that a is the accumulated value at time t Consider a very short time interval [t, t + dt] Noting that r is the interest rate per annum, the interest rate credited in [t, t + dt] is rdt The dollar amount of interest earned in [t, t + dt] is a rd 3 Thus, the change in the bank account is radt In other words, da da = radt or = ra dt tochastic differential equations In financial markets, not all variables are deterministic What if the change in x is perturbed by a standard Brownian motion? uppose that or in shorthand notation dx = a t, X dt + b t, X dz, dx = adt + bdz The above is called a stochastic differential equation DE and X is said to be a diffusion In this equation, dz is the change in the standard Brownian motion over [t, t + dt], while dx is the change in X over [t, t + dt] Intuitively, you may view them as Zt + d Z and Xt + d X To interpret a DE, you need to know the following ince dz is random, dx, and hence X, are random That is why we have used an uppercase letter for X The distribution of dz = Zt + d Z is N0, d Hence, E[dZ] = 0 and Var[dZ] = dt 3 It follows from independent increments that dz is independent of the history {Zu: 0 u t} In particular, dz and Z are independent 4 Given the value of X, the terms at, X and bt, X are no longer random, and hence E[dX X] = at, Xdt and Var[dX X] = b t, Xdt By virtue of 4, ax, and bx, are called the drift and volatility of the DE If ax, = 0, then X is said to be driftless Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

12 Module : Risk-neutral valuation in continuous-time Lesson : tochastic calculus M- Itô s Lemma uppose that Y is a function of time t and a standard Brownian motion Z Then the change in Y, dy, should be related to t, Z and dz We are interested in obtaining the DE for Y This can be accomplished by using Itô s lemma Itô s Lemma simplified version Let Y = f t, Z Then where [dz] = dt dy = f t t, Zdt + f z t, ZdZ + F O R M U L A f zz t, Z [dz ], Thus, the drift is f t t, Z + f zz t, Z and the volatility is f z t, Z To derive the DE for Y = f t, Z, use the following procedure tep : Recognize the function ft, z This can be done by replacing all Z by z tep : Find the three partial derivatives f t, f z and f zz tep 3: Plug f t, f z and f zz into Itô s lemma Remember that there is a attached to fzz tep 4: Use [dz] = dt and collect like terms to get the drift and volatility Read the following example Example Let Z be a standard Brownian motion Find dy for the following: a Y = Z b Y = tz Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

13 M- Module : Risk-neutral valuation in continuous-time Lesson : tochastic calculus olution a tep : ince Y = ft, Z = Z, by replacing Z by z, we have ft, z = z tep : f t t, z = 0, f z t, z = z, f zz t, z = tep 3: By Itô s lemma, dy = 0 dt + ZdZ + [dz ] tep 4: By using [dz] = dt, we get dy = ZdZ + dt = dt + Z dz b tep : ince Y = ft, Z = tz, by replacing Z by z, we have ft, z = tz tep : f t t, z = z, f z t, z = tz, f zz t, z = t tep 3: By Itô s lemma, dy = Z dt + tzdz + tep 4: By using [dz] = dt, we get dy = Z dt + tzdz + t [dz ] d [ t t t Z t ]d t tz t d Z t = + + [ END ] Now let us find the DEs satisfied by ABMs and GBMs introduced in Lesson of this module Example Let Z be a standard Brownian motion and Y0 be a constant Find dy for the following: a Y = Y0 + μ t + Z b Y = Y0 exp μ t + Z Note: Here we use Y = Y0 exp μ t + Z but not Y = Y0 exp[ μ t + Z ] The reason will become clear in the next section Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

14 Module : Risk-neutral valuation in continuous-time Lesson : tochastic calculus M-3 olution a tep : ince Y = ft, Z = Y0 + μ t + Z, by replacing Z by z, we have ft, z = Y0 + μ t + z tep : f t t, z = μ, f z t, z =, f zz t, z = 0 tep 3: By Itô s lemma, dy = μ dt + dz + 0[dZ ] tep 4: A further simplification is not needed The final answer is b tep : ince Y = ft, Z = Y0 e we have ft, z = Y0 e dy = μ dt + dz μ t+ z μ t+ Z, by replacing Z by z, μ t+ z tep : f t t, z = Y0 μ e, f z t, z = Y0 e f zz t, z = Y0 e μ t + z μ t+ z, tep 3: By Itô s lemma, μ t + Z t μ t + Z dy = Y0 μ e dt + Y0 e dz μ t+ Z + Y 0 e [dz ] tep 4: By using [dz] = dt, we see that the last term can cancel out the latter part resulting from the expansion of the first term Thus, dy = Y0 μe Finally, by using Y0 e μ t+ Z μ t+ Z dt + Y0 e = Y, we get dy = μydt + YdZ μ t+ Z dz [ END ] Now suppose that Y is a function of time t and another function X uppose also that we are given the DE for X: dx = at, Xdt + bt, XdZ To find the DE for Y, we use the general version of Itô s lemma Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

15 M-4 Module : Risk-neutral valuation in continuous-time Lesson : tochastic calculus Itô s Lemma general version Let Y = f t, X Then dy = f t t, Xdt + f x t, XdX + where [dx] = b t, Xdt F O R M U L A f xx t, X [dx ], Note that the simplified version of Itô s lemma is the special case when at, x = 0 and bt, x = To use of Itô s lemma for a function of t and X, follow the procedure below tep : Recognize the function ft, x This can be done by replacing all X by x tep : Find the three partial derivatives f t, f x and f xx tep 3: Plug f t, f x and f xx into Itô s lemma Remember that there is a attached to fxx tep 4: Use the DE of X, [dx] = b t, Xdt and collect like terms to get the drift and volatility Example 3 uppose that dx = μxdt + XdZ Find dy for Y = ln X olution tep : ince Y = ft, X = ln X, by replacing X by x, we have ft, x = ln x tep : f t t, x = 0, f x t, x =, fxx t, x = x x tep 3: By Itô s lemma, dy = 0 dt + dx + [dx ] X X tep 4: We have at, X = μx and bt, X = X, so that [dx] = X dt dy = [μxdt + XdZ] + X X dt = μ dt + dz X [ END ] Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

16 ample pages from Module Lesson 4 This lesson contains 33 pages We are showing pages to 8

17 M-6 Module : Risk-neutral valuation in continuous-time Lesson 4 : The harpe ratio and the Black-choles equation Lesson 4 The harpe Ratio and the Black-choles Equation OBJECTIVE To state the meaning of the Black-choles framework To understand the concept of the harpe ratio and the restriction it imposes on the drift and volatility of a contingent claim 3 To understand the Black-choles equation and its uses 4 To understand the concept of risk-neutral valuation 5 To recognize the relation between the true and the risk-neutral measures 4 The Black-choles Framework In this lesson we study the concept of risk-neutral valuation in continuous-time This lesson is the most theoretical in the exam syllabus Because questions from this lesson tends to be tricky, it is important to fully grasp the concepts taught in this lesson In exam MFE, you would see statements such as Assume the Black-choles framework and uppose that follows the Black-choles model very often By the Black-choles framework, we mean the following: The underlying asset follows a geometric Brownian motion The underlying asset is either nondividend-paying or pays dividends continuously at a level proportional to its price The risk-free interest rate is constant There are no transaction cost or taxes Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

18 Module : Risk-neutral valuation in continuous-time Lesson 4 : The harpe ratio and the Black-choles equation M-63 It is possible to purchase or short-sell any units of the underlying asset The borrowing rate and the lending rate are both equal to the risk-free interest rate There are no arbitrage opportunities The most important assumption is of course the first one: GBM A lot of things can be said on this assumption and you should now understand why we have spent three lessons in particular Lesson 3 on GBMs To recap, recall that the five statements below are equivalent is a GBM with drift α δ and volatility d = α δdt + dz where Z is a standard Brownian motion 3 d[ln ] = α δ dt + dz 4 = 0 e α δ t+ Z 5 ln is normal distributed with mean ln 0 + α δ t and variance t 4 The harpe Ratio Let X be the price of an asset The harpe ratio of X at time t is defined as the ratio of the instantaneous average risk premium to the instantaneous volatility Heuristically, the harpe ratio is a measure of the risk-return trade off We assume the following: The dynamics of X follow dx = mdt + sdz X Warning: We are not using dx = mdt + sdz! Here m = mx, and s = sx, can depend on t and the time-t price of X For simplicity, we suppress the arguments X and t The asset pays dividends continuously at a rate proportional to its price The continuous dividend yield is δ We have δ = 0 if the asset is nondividend-paying The dollar amount of dividend over an infinitesimally short time period t, t + d is Then, X δdt Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

19 M-64 Module : Risk-neutral valuation in continuous-time Lesson 4 : The harpe ratio and the Black-choles equation dx X is the instantaneous return due to capital gains, m dt = dx E X s dx dt = Var X X is the expected instantaneous return due to capital gains, X is the variance of the instantaneous return due to capital gains The total return on X is the sum of capital gains and dividends As a result, the instantaneous total return is m + δ, and the instantaneous risk premium is m + δ r The harpe ratio is defined by the ratio of the instantaneous risk premium to the instantaneous standard deviation F O R M U L A The harpe Ratio of an Asset dx If = mdt + sdz and the continuous dividend yield is δ, then the harpe ratio is X m + δ r φ = s Read the following examples Example 4 Consider the Black-choles model for a stock price Find the harpe ratio of olution The dynamics of is d = α δ dt + dz pays dividends continuously at a constant rate proportional to its price The dividend yield is δ α δ + δ r α r Thus the harpe ratio of is φ = =, which is a constant because α and are both constants [ END ] Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

20 Module : Risk-neutral valuation in continuous-time Lesson 4 : The harpe ratio and the Black-choles equation M-65 Example 4 Consider again the Black-choles model for a stock price Let V, = F P t, T be the time-t prepaid forward price for the delivery of one share of at time T Find the harpe ratio of this contract olution We need to derive the dynamics of V ince V, = F P e δt t, T =, we use Itô s lemma Let fs, = se δt Then f t s, = δse δt, f s s, = e δt, f ss s, = 0 dv, = δ e δt dt + e δt d = δe δt dt + e δt [α δdt + dz] = e δt [α dt + dz] = V, [α dt + dz] dv, o, the dynamics is = α dt + dz In this case, m = α, and s = V, A prepaid forward contract pays no dividend The dividend yield is 0 Thus, the harpe ratio is φ = m + δ r s α r = [ END ] We observe from Example 4 and Example 4 that both the stock and the prepaid forward contract have the same harpe ratio Is it a coincidence? Actually it is not The reason for the equality of harpe ratio is that both the stock and the prepaid forward contract have the same underlying source of risk: it is the same Z that causes the stock price and prepaid forward price to change over time randomly Equality of harpe Ratios F O R M U L A The harpe ratios of two assets driven by the same Brownian motion must be the same In particular, any contingent claim written on a stock that follows a GBM must have a α r harpe ratio of φ = Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

21 M-66 Module : Risk-neutral valuation in continuous-time Lesson 4 : The harpe ratio and the Black-choles equation We need to study the proof of the above theorem, because the proof can be and has been tested in various ways The idea of the proof involves hedging, an important concept that we will explore in more detail in Module 3 This is the only proof in this lesson o please read it Proof: et-up: Two risky assets X and Y, with price processes dx = m X dt + s X dz, X dy = m Y dt + s Y dz Y The continuous dividend yield rates for X and Y are δ X are δ Y, respectively Notice that here s X and s Y can be negative This does not mean that the volatility of the return is negative, but that the return moves in a direction that is opposite to Z Hedging: uppose that we have unit of X at time t Our goal is to purchase / sell appropriate units of Y and cash, so that we have an instantaneously risk-free and costless portfolio uppose we purchase N units of Y and invest W dollars at risk-free interest rate r Then the value of the portfolio at time t is V = X + NY + W To make the portfolio costless, W should be chosen so that W = X NY What would happen after an infinitesimally short period dt? The change in the price of X and Y are dx and dy The amount of dividends generated from unit of X and Y are Xδ X dt and Yδ Y dt The interest earning for dollar is rdt o, dv = dx + Xδ X dt + NdY + Yδ Y d + rwdt = m X X + δ X X + Nm Y Y + Nδ Y Y + rwdt + [s X X + Ns Y Y]dZ To make the portfolio instantaneously risk-free, we pick N so that s X X + Ns Y Y = 0, or equivalently, s X X N = s Y s X X s X As a result, W = X Y X syy = s Y 3 Equating the drift of the hedged portfolio to 0: Y By picking N and W as in, V is instantaneously risk-free and costless What should such a portfolio earn? The return on a risk-free investment of zero dollars must be zero, or otherwise there will be an arbitrage! o, the drift of dv must be 0: Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

22 Module : Risk-neutral valuation in continuous-time Lesson 4 : The harpe ratio and the Black-choles equation M-67 m X X + δ X X + Nm Y Y + Nδ Y Y + rw = 0 After some algebraic simplifications, we get The formula for N can be written as m X + δ s X X r my = + δ s Y Y r [ END ] s X volatility coefficient of dx X N = = 4 Y volatility coefficient of dy s Y This formula makes a lot of sense uppose that s X and s Y are positive If Z increases, then the risky part of X and Y would both be positive In order that they cancel each other, one should sell Y If X is riskier ie, s X is large relative to s Y, then we need more units of Y for a complete cancellation of risk Example 43 Consider two assets X and Y There is a single source of uncertainty which is captured by a standard Brownian motion {Z} The prices of the assets satisfy the stochastic differential equations dx = 007dt + 0dZ and d[lny] = Adt + 06dZ, X where A is a constant You are also given that i X is nondividend-paying; ii Y pays dividends continuously at a rate proportional to its price The dividend yield is 3% iii The continuously compounded risk-free interest rate is 004 Determine A olution We compute the harpe ratio of Y We first derive the dynamics of Y By the equivalence of different representations of a GBM, we have dy 06 = A dt + 06dZ Y + 06 A o, the harpe ratio of Y is A + φ Y = = Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

23 M-68 Module : Risk-neutral valuation in continuous-time Lesson 4 : The harpe ratio and the Black-choles equation The harpe ratio of X is φ X = = A By equating the harpe ratios, = 0 5, we have 06 A = = 0037 [ END ] Example 44 The prices of two stocks are governed by: dx dy = 006dt + 00dZ, = 009dt + kdz X Y where Z is a standard Brownian motion and k is a constant You are given: i The current stock prices are X0 = 5 and Y0 = 50 ii Both stocks pays dividends at a rate proportional to its price The dividend yields of X and Y are δ + 00 and δ, respectively iiithe continuously compounded risk-free interest rate is 4% To construct a zero-investment, risk-free portfolio in which there are exactly 6 shares of X, one needs to trade a certain number of Y and borrow 00 dollars at risk-free rate Find δ olution Equating the harpe ratios, δ δ 004 = 00 k uppose one has share of X, then the hedge portfolio has 005 N = = k50 00k units of Y and W = X0 NY0 = k = 5 + dollars of cash From the question, we know that in a portfolio with 6 shares of X, we nee d to borrow 00 o 00 for a portfolio with one share of X, we have W = = 65 On solving 5 + = 6 5, 6 k we get k = ubstituting the value of k into the equation for δ, we have 375 k By 4 Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

24 Module : Risk-neutral valuation in continuous-time Lesson 4 : The harpe ratio and the Black-choles equation M-69 which gives δ = δ 00 = δ, [ END ] 4 3 The Black-choles Equation In Example 4, we calculated the harpe ratio for a prepaid forward contract What if we apply the same procedure to a derivative on? We consider a derivative whose time-t price when the stock price is is V, tep : By using Itô s lemma, we obtain dv, tep : Then by finding m and s, we obtain the harpe ratio of the derivative It is not hard to see that Vs m = V + α δ V + V, s = V t s ss V But these two formulas are not important and you do not need to remember them now We will revisit the formula for s in Module 3 α r tep 3: By setting the harpe rati o to, we get the Black-choles equation The Black-choles Equation V t V + r δ + s F O R M U L A V s = rv The pricing formula for any derivative must satisfy the Black-choles B equation o, the B equation can be used as a polygraph: if you are presented a formula V, that looks like the time-t price of something, you can tell if it is indeed the price of a certain derivative by checking if it satisfies the B equation To familiar yourself with this funny concept, let us take a look at a toy example Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

25 ample pages from Module 3 Lesson This lesson contains 33 pages We are showing pages to 0

26 M3-0 Module 3 : The Black-choles Formula Lesson : Greek Letters and Elasticity Lesson Greek Letters and Elasticity OBJECTIVE To study Greek letters To calculate the mean return and volatility of a derivative 3 To calculate the elasticity of a derivative In this lesson, we focus on different measures of risk These measures, as we will demonstrate in the next lesson, can help us hedge the risk associated with a portfolio of risky assets 3 Greek Letters: Delta, Gamma and Theta In the Black-choles framework, the price of any derivative security depends on the following six factors: tock Option Environment Current tock Price Time Risk-free Rate Volatility Payoff Feature Dividend Yield For example, for a power contract, we have V, = a exp[r + ar δ + 05aa T ], which is a function of the current stock price, volatility, dividend yield δ, time t, payoff feature a, and the risk-free rate r For a cash-or-nothing European call, Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

27 Module 3 : The Black-choles Formula Lesson : Greeks Letters and Elasticity M3- V, = e rt Nd, d = ln K + r δ T, T t which is a function of the current stock price, volatility, dividend yield δ, time t, payoff feature K, and the risk-free rate r As time proceeds, t increases, changes Both would lead to a change in the price of a derivative One way to quantify the risk of a derivative is to measure how sensitive V, is when or t changes The sensitivities can be estimated by the partial derivatives of V with respect to and t Delta, Gamma and Theta F O R M U L A Δ = V, Γ = V Δ =, θ = V t The textbook gives the following verbal and non-rigorous interpretations of the first three Greek letters: Delta Δ measures the change in the price of a derivative when the stock price increases by $ A large Δ means that the derivative price is very sensitive to small changes in Therefore, a derivative is riskier if it has a larger Δ Gamma Γ measures the change in delta when the stock price increases by $ Theta θ measures the change in the price of a derivative when there is a decrease in the time to expiration T, that is, an increase in t as T is fixed Example 3 Assume the Black-choles framework Compute the time-t delta and gamma for a cash-ornothing call Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

28 Module 3 : The Black-choles Formula Lesson : Greek Letters and Elasticity M3- Actex 0 Johnny Li oa Exam MFE / CA Exam 3F olution We have V, = e rt Nd, where d = t T t T r K + δ ln The time-t delta is t T d N e d d N e t V t T r t T r = = Δ =,, where N z / z e = π is the pdf of N0, The time-t gamma is,, t T d N e t T d N e t T d N e d d N t T e t t T r t T r t T r t T r = + = Δ Γ = where d d / / z zn e z e z z N z z = = = π π [ END ] It is very tedious to compute the time-t theta of the cash-or-nothing call by differentiating V, with respect to t Rather than working directly on the partial derivative, we can calculate theta by using the Black-choles equation, which relates Δ, Γ, and θ as follows F O R M U L A Relation between Delta, Gamma and Theta rv r = Γ Δ + + δ θ

29 Module 3 : The Black-choles Formula Lesson : Greeks Letters and Elasticity M3-3 The following table shows the formulas for Δ, Γ, and θ of European calls and puts Greek Call Put Δ = V V Γ = V θ = t e δt Nd e δt N d δ T e N d same as Γ of call T t δ T t δ T r T e N d call θ δe N d rke N d r T δ T T t + rke δe You must remember the formulas for Δ for Exam MFE However, the formulas for Γ and θ are optional If you are asked to compute Γ, you can simply differentiate the formula for Δ In the unlikely event that you are asked to compute θ, you should first compute V, Δ and Γ, and then use the Black-choles equation to solve for θ Example 3 Assume the Black-choles framework You are given that: i A stock has a current price of 5 ii The stock pays dividends continuously at a rate that is proportional to its price The dividend yield is 4% iiithe volatility of the stock is less than 03 iv A 3-month at-the-money European put option on has a delta of v The continuously compounded risk-free interest rate is 8% Compute the price of the put option olution Let the current time point be t = 0 The delta of the put is e δt Nd = Nd = 04360e 004/4 = d = 05 5 ln = = 0 = 0 or 04 rejected Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

30 M3-4 Module 3 : The Black-choles Formula Lesson : Greek Letters and Elasticity As a result, d = d T = , and Nd = 0480 = The price of the put is 5e 008/ e 004/ = [ END ] There are a few special relations between the Greeks for calls and puts These relations are derived from put-call parity: c, p, = e δt Ke rt Differentiating both sides of the put-call parity equation with respect to, we get δt call delta put delta = e Differentiating both sides of the put-call parity equation with respect to twice, we get call gamma put gamma = 0 Differentiating both sides of the put-call parity equation with respect to t, we get call theta put theta = δe δt rke rt Example 33 Assume the Black-choles framework You are given that: i A nondividend-paying stock has a current price of 0 and a volatility of 40% ii A T-year K-strike European put option on has a price of 4954 and a theta of ii A T-year K-strike European call option written on has a delta of and a gamma of Find r, the continuously compounded risk-free interest rate olution Let the current time point be t = 0 ince the stock is nondivdend-paying, δ = 0, and call delta put delta = e δt = put delta = = 0550 Moreover, since call gamma = put gamma, put gamma is We now have the price, delta, gamma and theta of the put The Black-choles equation says that r = r 4954, which gives r = 005 [ END ] Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

31 Module 3 : The Black-choles Formula Lesson : Greeks Letters and Elasticity M3-5 We now study the properties of the three Greeks for European calls and puts Properties of Δ Δ Δ δt e longer T longer T K K shorter T δt e shorter T long call long put For calls, Δ is positive and bounded by 0 and e δt For puts, Δ is negative and bounded by e δt and 0 If an option is deeply OTM call: low, put: high, Δ would be close to 0 Explanation: When an option is very OTM, it is unlikely that it will be exercised and thus V 0 In this case Δ would be close to 0 since it is not very sensitive to when changes by a small amount, V is still very close to 0 If the option is deeply ITM, Δ approaches e δt for calls and e δt for puts Explanation: When a call is deeply ITM, we expect that the final payoff from the call would be T K, and hence V e δt Ke rt, which means Δ e δt The explanation for deeply ITM puts is similar Properties of Γ Γ longer T shorter T K long call / put Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

32 M3-6 Module 3 : The Black-choles Formula Lesson : Greek Letters and Elasticity Calls and puts with the same strike and time to expiration have the same value of Γ For long positions of calls and puts, Γ must be positive Recall that a function is said to be convex if its second derivative is always non-negative European calls and puts are hence called convex derivatives ince Δ does not change much it reaches either 0 or ± e δt when a call / put is deeply OTM or ITM, Γ is close to 0 when is very low or very high Properties of θ θ θ K longer T longer T K shorter T shorter T long call long put The value of θ can be positive or negative It is usually negative because call and put prices tend to drop as time passes One exception is a deeply in-the-money European put on a nondividend-paying stock When a put is very ITM, we expect that the final payoff from the put would be K T, and hence V Ke rt This means θ rke rt > 0 Another exception is a deeply in-the-money European call on a currency with a very high interest rate The theta of a European call on a nondividend-paying stock is always negative The theta of a deeply OTM option is close to zero, while the theta of an at-the-money option is large and negative The Delta-Gamma-Theta Approximation Apart from quantifying the risk of a derivative, delta, gamma and theta can also be used to approximate the price of a derivative when t or changes by a small amount uppose that at time t, the price of the derivative is V, If the stock price suddenly changes to + ε, how would the price of the derivative change? By Taylor s theorem, we have Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

33 Module 3 : The Black-choles Formula Lesson : Greeks Letters and Elasticity M3-7 V + ε, V, + V, ε + V, ε This leads to the following result Delta-Gamma Approximation V + ε, V, + Δ, ε + Γ, ε F O R M U L A If we drop the gamma term, the resulting formula V + ε, V, + Δ, ε is called a delta approximation Example 34 [MFE 07 May #9] Assume that the Black-choles framework holds The price of a nondividend-paying stock is $3000 The price of a put option on this stock is $400 You are given Δ = 08 and Γ = 00 Using the delta-gamma approximation, determine the price of the put option if the stock price changes to $350 A $340 B $350 C $360 D $370 E $380 olution We have V, = 4, = 30, + ε = 35, and hence ε = +5 Using a delta-gamma approximation, we have V + ε, = 3695 o the answer is D [ END ] It is unusual that jumps suddenly A more comprehensive description is that the stock price changes from to t + h when time proceeds from t to t + h We can model this mathematically with a delta-gamma-theta approximation, which is derived from the multivariate version of Taylor s theorem Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

34 M3-8 Module 3 : The Black-choles Formula Lesson : Greek Letters and Elasticity Delta-Gamma-Theta Approximation F O R M U L A Vt + h, t + h V, + Δε + Γ ε + θ h where ε = t + h, and the three Greeks are evaluated at and t Delta, Gamma and Theta of a Portfolio of Derivatives uppose that an investor forms a portfolio with n derivatives written on the same underlying stock The investor takes a position of w i units of the ith derivative, whose price is denoted by V i If w i > 0, then it is a long position; and vice versa The value of the portfolio is given by Hence, the delta of the portfolio is P = n P = i= n i= w V i i Vi wi = The above says that the delta of the portfolio is the sum of the deltas of the individual portfolio components This property also applies to the gamma and theta of the portfolio n i= w Δ i i Example 35 uppose that is a nondividend-paying stock which has a current price of 30 Assume the Black-choles framework, and that the volatility of the stock is % The continuously compounded risk-free interest rate is 5% Compute the time-0 price and delta of the following two derivatives: a a straddle that has a time-05 payoff of ; b a bull spread that has a time-05 payoff of max[0, min05, 30 8]: Payoff Actex 0 Johnny Li oa Exam MFE / CA Exam 3F

35 Module 3 : The Black-choles Formula Lesson : Greeks Letters and Elasticity M3-9 olution a The payoff of the straddle can be decomposed into [05 30] + + [30 05] + As a result, the straddle is just a 30-strike call plus a 30-strike put To find the price of the straddle, we calculate the following: 30 0 ln d 30 = = 0686, d = = Nd = 05675, Nd = 0539 The call price is e 005/ = 5034 The put price is 30e 005/ = 3057 The price of the straddle is = 6338 We then calculate the deltas for the calls and puts: The call delta is Nd = The put delta is = 0435 Thus, the delta of the straddle is = 0350 b The payoff of the bull spread can be decomposed into [05 8] + [05 30] + As a result, the spread is just a 8-strike call minus a 30-strike call To find the price of 8-strike call, 30 0 ln d 8 = = 07958, d = = Nd = 0788, Nd = The 8-strike call price is e 005/ = The price of the bull spread is = 65 Note that the delta of the 8-strike call is Nd = 0788 Thus, the delta of the bull spread is = 006 [ END ] Actex 0 Johnny Li oa Exam MFE / CA Exam 3F