Advanced Calculus for Financial Engineering, Baruch refresher assignments - weiyialanchen/advanced-calculus. A Probability Primer for Mathematical Finance, sioteketerhost.ga Kosygina. 4. Differential Equations with Numerical Methods for Financial Engineering, by Dan Stefanica. Solutions Manual - A Primer for the Mathematics of Financial Engineering - Ebook download as PDF File .pdf), Text File .txt) or read book online. Solutions .
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Explaining the magic of Greeks computations (Download); Primer For The Mathematics Of Financial Engineering Pdf a math primer. Reviews for “A Primer for the Mathematics of Financial Engineering”, First Edition: ``One of Get your Kindle here, or download a FREE Kindle Reading App. 𝗥𝗲𝗾𝘂𝗲𝘀𝘁 𝗣𝗗𝗙 on ResearchGate | A Primer for the Mathematics of Financial Engineering / D. Stefanica. | Contenido: 1) Preliminares de matemáticas;.
Dollar duration, Dollar convexity, DV01; the effect of parallel shifts in the yield curve to changes in bond yields; bond portfolio immunization; arbitraging the Put-Call parity; percentage vs. New or expanded sections: Financial applications selected: Put-Call parity, bond mathematics, numerical computation of bond yields, Black-Scholes model, numerical estimation for Greeks, implied volatility, yield curves bootstrapping. Mathematical topics selected: This book covers linear algebra concepts for financial engineering applications from a numerical point of view.
The book contains many such applications, as well as pseudocodes, numerical examples, and questions often asked in interviews for quantitative positions. FE Press, A ten questions selection , with solutions, can be downloaded here.
This book builds the solid mathematical foundation required to understand the quantitative models used financial engineering and can be used as a reference book or as a self-study book.
Top 3 QuantNet bestselling book in , , , and Sample Sections: Table of Contents Download 3. Explaining the magic of Greeks computations Download 5. In particular. What is the new value of the portfolio? Use risk-neutral valuation to write the value of the put as an integral over a finite interval. Using risk-neutral valuation. K] into 4 intervals. We report the Midpoint Rule and Simpson's Rule approximations to 2. This is due to the fact that the Black-Scholes value of the put.
If the underlying asset follows a lognormal distribution. To compute a numerical approximation of the integral 2. The approximation error of these approximations is on the order of Using numerical integration.
Create a butterfly spread by going long a call and a call. Intervals Midpoint Rule 4 5. If we consider that convergence is achieved when the error is less than The payoff V T of the butterfly spread at maturity is V T.
This was to be expected given the quadratic convergence of the Midpoint Rule and the fourth order convergence of Simpson's Rule. Since interest rates are zero. Using 2. We conclude that. Chapter 3 Probability concepts. You throw two fair dice. If the sum of the dice is k.
Find the smallest value of w k thats makes the game worth playing. Since the dice are assumed to be fair and the tosses are assumed to be independent of each other. Consider the probability space S of all possible outcomes of throwing of the two dice. The value X of your winning or losses is the random variable X: From 3. Let X be the number of times you must flip the coin until it lands heads. A coin lands heads with probability p and tails with probability 1 — p.
What are E[X] and var X? If the first coin toss is heads which happens with probability p. If the first coin toss is tails which happens with probability 1 — p. P 1—p 3. The coin will first land heads in the k-th toss. Ek2 1. The probability space S is the set of all different paths that the stock could follow three consecutive time intervals.
The value ST of the stock at time T is a random variable defined on S. Let a E R be an arbitrary real number. For our problem, it follows that. Problem 6: Input for the Black-Scholes formula: How do you hedge a short position in such a call option? The price of the put option is 0. A call with strike 0 will always be exercised, since it gives the right to download one unit of the underlying asset at zero cost.
This can be seen by building a portfolio with a long position on the call option and a short position of e-qTshares, or by using risk-neutral pricing: A short position in the call option is hedged statically by downloading one share of the underlying asset.
The sensitivity of the vega of a portfolio with respect to volatility and to the price of the underlying asset are often important to estimate,. These two Greeks are called volga and vanna and are defined as follows:. The name volga is the short for "volatility gamma". Also, vanna can be interpreted as the rate of change of the Delta with respect to the volatility of the underlying asset, i.
For at-the-money options. Use the Put-Call parity. Note that ad. Ke-r T-t N d2. N d2 T-t Ke-r T-t N d2 with respect to K.
T—t —. Show that the price of a plain vanilla European call option is a convex function of the strike of the option. We note that the positive value of e C is nonetheless small.
This can be seen by plotting the Delta of a call option as a function of spot price. What can you infer about the hedging of ATM options with different maturities? By differentiating 3. We note that Gamma decreases as the maturity of the options increases. Compute the Gamma of ATM call options with maturities of fifteen days. The cost of Delta-hedging ATM options. If you have a long position in either put or call options you are essentially "long volatility".
For simplicity. This could be understood as follows: Show that the value P of the corresponding put option must satisfy the following no-arbitrage condition: Assume that interest rates are constant and equal to r. Show that. One way to prove these bounds on the prices of European options is by using the Put-Call parity.
The value P of the put at time 0 cannot be more than Ke-rT. Consider a portfolio made of a short position in one call option with strike K and maturity T and a long position in e-qT units of the underlying asset. The value C of the call at time 0 cannot be more than Se-qT.
If the dividends received on the long asset position are invested continuously in downloading more units of the underlying asset. To establish the bounds 3. This portfolio will be Delta. How do you make the portfolio Deltaneutral and Gamma-neutral? Take positions of size x1and x2. A portfolio containing derivative securities on only one asset has Delta and Gamma The solution of this linear system is What is the value of your position the option and shares position?
You download six months ATM Call options on a nondividend-paying asset with spot price What Delta-hedging position do you need to take? A long call position is Delta-hedged by a short position in the underlying asset. You are long call options with strike 90 and three months to maturity. To understand how well balanced the hedged portfolio H is. For the Delta-hedged portfolio. Compute the expected value and variance of the Poisson distribution.
You hold a portfolio made of a long position in put options with strike price 25 and maturity of six months. What is the expected number of tosses in order to get k heads in a row for a biased coin with probability of getting heads equal to p? What is new value of your portfolio. Show that the values of a plain vanilla put option and of a plain vanilla call option with the same maturity and strike.
Calculate the mean and variance of the uniform distribution on the interval [a. What is the expected number of coin tosses of a fair coin in order to get two heads in a row? What if the coin is biased and the probability of getting heads is p?
The outcomes of the first two tosses are as follows: If p is the probability of the coin toss resulting in heads. If E[X] denotes the expected number of tosses in order to get two heads in a row.
You can trade in the underlying asset. What trades do you make to obtain a A—. The probability density function of the uniform distribution U on t We conclude that the expected number of tosses in order to get n heads in a row for a biased coin with probability of getting heads equal to p is P P" 1-P ' If the coin were unbiased.
What is the expected number of tosses in order to get n heads in a row for a biased coin with probability of getting heads equal to p? The probability that the first n throws are all heads is pm. Let X be a normally distributed random variable with mean p.
We conclude that E[ IX! E[X] 2. While this would provide the correct result. Problem 5: From the Put-Call parity.
Ke-rT 3. To obtain a Delta—neutral portfolio. The Black—Scholes value of the put option is P T. Here and in the rest of the problem.
The Delta—neutral portfolio will be made of a long position in put options a long position in shares of the underlying stock. You hold a portfolio with A To make the portfolio Delta—neutral.
You can make the portfolio F— and vega. F— and vega. What trades do you make to obtain a A-. By trading in the underlying asset. Recall that if X1 and X2 are independent normal random variables with mean and variance pi and a?. Assume that the normal random variables X1. Xn are independent. This happens. Xn of mean it and variance a2 are uncorrelated. Chapter 4 Lognormal random variables. If S t is the price of the asset at time t.
E E cicicov xi. Problem 4: Solution 1: Solution 2: Similarly, we obtain that. The results of the previous two exercises can be used to calibrate a binomial tree model to a lognormally distributed process. In other words, we are looking for u, d, and p such that. Since there are two constraints and three unknowns, the solution will not be unique. To obtain 4. Show that the series EkD. To show that the series EL -1kt is divergent.
This can be seen as follows: For example, 4. It is easy to see that oo. Consider a put option with strike 55 and maturity 4 months on a non-dividend paying asset with spot price 60 which follows a lognormal.
Assume that the risk-free rate is constant equal to 0.
Show that the Delta of the option is always greater than 0. For most cases, the Delta of an at-the-money call option is close to 0. For an at-the-money call on a non-dividend paying asset, i. Use risk-neutral pricing to price a supershare. Assume that the underlying asset pays no dividends. Recall that 0.
From 4. K e-rT N d2. Assume that the risk free rate is constant and equal to r. If the price of an asset follows a normal process. Ke' dx 2 fgr d dx. Using risk-neutral pricing. N -d e-rT a ff. The limit of this sequence is -yR. If you play American 1 roulette times. Use risk—neutral pricing to find the value of an option on a nondividend—paying asset with lognormal distribution if the payoff of the option at maturity is equal to max S T a — K.
European roulette.. Find a binomial tree parametrization for a risk—neutral probability of going up or down equal to 1. Recall from 4. The sequence is therefore convergent to a limit between 0 and 1. If the asset has lognormal distribution. If you play American roulette times. Since every bet is independent of any other bet.
Recall that an American roulette has 18 red slots. Use risk—neutral pricing to find the value of an option on a non—dividend—paying asset with lognormal distribution if the payoff of the option at maturity is equal to max S T a — K. Ke-rT N a.. By completing the square for the argument of the exponential function under the integral sign we obtain that 1 r Using 4.
We solve 4. Find a binomial tree parametrization for a risk—neutral probability of going up equal to Z. From 5. Use the Taylor series expansion of the function e2' to find the value of e0. ATM approximation of Black—Scholes formulas. Chapter 5 Taylor's formula and Taylor series. Find the Taylor series expansion of the functions ln 1 — x2 and 1 1 — x2 around the point 0. It is then enough to compute xo. By taking absolute values in 5.
In the Cox-Ross-Rubinstein parametrization for a binomial tree. We conclude that lim If x. We will show that the Taylor expansion for 5.
Recall the following Taylor approximations: From the Black—Scholes formula. It is interesting to note that the approximate formulas T C -. Compute the approximate value an ATM put option and estimate the relative approximate error Papprox. Compare this error with the relative approximate error 5.
Problem 8: T 1 Papprox PBS 4. This is. We obtain that Bnew. D the approximate value given by formula 5. Denote by Bnew. A five year bond worth has duration 1. We expect the precision of the approximation formula for ATM options to decrease as the maturity of the option increases.
D —B and. Use both the formula AB — DAy. D Bnew. C — Bnew. Ay Bnew. Supplemental Exercises 1. Find the linear and quadratic Taylor approximations of e9 x around the point 0.
Compute the relative approximation error to the Black—Scholes value of the option of the approximate value S T 7. The goal of this exercise is to compute fo i ln 1 — x ln x dx. Assume that the interest rates are constant at 4. The quadratic Taylor approximations of e-x and 1. Recall from 5. Prove that x 1 '2 1 The left inequality of 5. The goal of this exercise is to compute fo1 ln 1 — x ln x dx.
Compute the relative approximation error of the approximation , T 1. The approximate option values and the corresponding approximation errors are given below:. Error Papprox PBS 1 4. Problem 1: The options are on the same underlying asset and have the same maturities. A security that pays 1 in a certain state and 0 in any other state is called an Arrow-Debreu security, and its price is called the Arrow-Debreu price of that state.
The value of a call option as a function of the strike of the option is infinitely many times differentiable for any fixed point in time except at maturity. Therefore the expression from 6. Let S K be a fixed value of the spot price of the underlying asset.
From 6. A position in bull spreads as above. Find a second order finite difference approximation for f' a using f a. To obtain a finite difference approximation for f' a in terms of f a. By solving 6. We multiply 6. This type of approximation is needed. For symmetry reasons. Find a central finite difference approximation for the fourth derivative of f at a. The goal of this exercise is to emphasize the importance of symmetry in finite difference approximations.
We solve for f 4 a and obtain the following second order finite difference approximation: What is the order of this finite difference approximation? We investigate what happens if symmetry is not required. We eliminate the terms containing f" a by multiplying 6. In other words.. By solving for f' a. By eliminating from 6. By solving for 7 a.
The finite difference approximation 6. We write the cubic Taylor approximation Show that this finite difference discretization is convergent. This is due to the fact that. Induction step: Initial condition: Consider the following first order ODE: We look for yo. Consider the following second order ODE: The following second order finite difference discretization of 6. Note that we actually showed that the finite difference discretization is first order convergent.
Find A and b. Yn Solution: By writing the ODE 6. Of Ac F. IA dS 0. Use finite differences to find approximate values for the Greeks. Find the following forward difference approximations for vega. Consider a six months plain vanilla European call option with strike 18 on a non—dividend—paying underlying asset with spot price Denote by C S.
AEI 0. The corresponding forward and central difference approximation errors for the Delta and Gamma of the option are: Although direct computation can be used to show this result. The corresponding relative approximation errors are Ivega. It is easy to see that of at qSe-q T-t -rKe-r T F and 8 the values. Show that f S.. V27 T. The value at time t of a forward contract struck at K and maturing at time T. If the asset pays no dividends. T denotes the value of a plain vanilla call option with strike K and maturity T on an underlying asset with spot price S following a lognormal distribution.
Show that the central difference approximation for f' x around the point 0 is a fourth order approximation. T x2 where. AI and Ire — rl..
How do you explain this? Note that these approximation errors stop improving.. From the definition of f S. T -2C K.. To do so. For the sake of completeness. Since g is continuous. Since the function g: Vr Recall that the "magic" of Greek computations is due to the following result: The table below records the values of r c. Lemma 3. Note that these approximation errors stop improving.
AI and I Fc. AI are recorded in the table below: When computing the finite difference approximation Ac. The central finite difference approximations A. To explain this phenomenon. CBs S. This value is computed using a numerical approximation of N di that is accurate within 7.
Ke-rT N d2. Cexact S 0. Since the central finite difference is a second order approximation. ABSI deteriorates as alN6 becomes large. The only estimate we can find using 6.
Using 6. The central finite difference approximations Pcand the approximation errors lc. While the approximation error IA. Cexact S. Almay be better in practice.
The reason for this is similar to the one explained above for the finite difference approximations of A. The finite difference approximations of F became more precise while dS decreased to Barrier options. Optimality of early exercise. From 7.
Chapter 7 Multivariable calculus: Note that lims. The integral 7. Compute f fD xxy y dxdy. The Jacobian of the change of variable x. Let D be the domain bounded by the x—axis. Let V S. Use the change of variables to polar coordinates to show that the area of a circle of radius R is 7rR2. R] x [0. R Solution: We use the polar coordinates change of variables x. Assume that the function V S. From the first equation.
Solve for a and b the following system of equations: The PDE 7. It can be shown that V S. Using Chain Rule. An Asian call option pays the maximum between the spot price S T of the underlying asset at maturity T and the average price of the underlying asset over the entire life of the option. T can be solved numerically. Similarity solutions of the type 7. The price of a non-dividend-paying asset is lognormally distributed.
The values of the out options are summarized in the table below: This feature was observed for the options priced here: Assume that the spot price is For which of these options is the intrinsic value max K — S.
Show that the premiumsof the Black-Scholes value of a European call option over its intrinsic value max S. CBS S.. This is to be expected. Note that f S is a continuous function. Use the formula B 2a c V S. Show that the value of a down-and-out call with barrier B less than the strike K of the call.
It is easy to see that f S CBs S. We rewrite formula 7. VT Similarly. Let u. What does the boundary condition V B. This is a version of Gronwall's inequality. Compute I where fox dxdy. Which number is larger. Assume that options with all strikes K exist. Find an implicit differential equation satisfied by uznip K. For the same maturity. Define the implied volatility az. The values of Asian options with continuously sampled geometric average satisfy the PDE 7.
One way to see that American calls on non—dividend—paying assets are never optimal to exercise is to note that the Black—Scholes value of the European call is always greater than the intrinsic premium S — K. The goal of this problem is to compute the rate of change of the implied volatility as a function of the strike of the options.
Show that this argument does not work for dividend—paying assets. E x— Solution: It is easy to see that dxdy ax ay ax ay as at at as dsdt 1 2t-vi 2. By taking the natural logarithm. II f 2 f2e dxdy. V x Solution: Define the function w: Using the Product Rule. Consider the following change of variables: K for S large enough. One way to see that American calls on non-dividend-paying assets are never optimal to exercise is to note that the Black-Scholes value of the European call is always greater than the intrinsic premium S.
Using chain rule. We conclude that CBS S. Note that CBs S. Show that this argument does not work for dividend-paying assets. We want to show that. Find an implicit differential equation satisfied by climp K. We conclude from 7. Recall that CBS S. We conclude that the implied differential equation 7. Note that C K is assumed to be known for all K.
Using Chain Rule and 7. N— dimensional Newton's method. Chapter 8 Lagrange multipliers. We reformulate the problem as a constrained optimization problem.
The Lagrangian associated to this problem is F x. Implied volatility. Find the maximum and minimum of the function f xi. We now find the critical points of F x. Note that f 2. Note that f We conclude that the point The expected values. This system has two solutions: Assume that you can trade four assets and that it is also possible to short the assets. This allows us to conclude directly that the point 2. We do not require the weights wi to be positive.
Recall that the expected value and the variance of the rate of return of a portfolio made of the four assets given above are..