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;.
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Advanced Calculus for Financial Engineering, Baruch refresher assignments - weiyialanchen/advanced-calculus. A Probability Primer for Mathematical Finance, lesforgesdessalles.info 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 .
At time 0. The third bond pays coupons in 2. Therefore r 0. Note that 1. How do you synthesize this derivative security i. We reformulate the problem as a constrained optimization problem.
Since f x is a second order polynomial. Problem 9: Formula 1. This exercise shows that the function u x. By direct computation and using the Product Rule. In fact. All options are on the same underlying asset and have maturity T. Draw the payoff diagram at maturity of the portfolio. Consider a portfolio with the following positions: The payoff at maturity of a butterfly spread is always nonnegative..
This is a butterfly spread. For our particular example. V47rt 6 4t 1. Depending on the values of the spot S T of the underlying asset at maturity. A butterfly spread is an options portfolio made of a long position in one call option with strike K1 a long position in a call option with strike K3. Note that being long Portfolio 1 and short Portfolio 2 is equivalent to being long a butterfly spread. Draw the payoff diagram at maturity of a bull spread with a long position in a call with strike 30 and short a call with strike Which of the following two portfolios would you rather hold: Depending on the value of the spot price S T.
All options are on the same asset and have the same maturity. For an arbitrage opportunity to be present. Is there an arbitrage opportunity present? If yes.
The value of the portfolio at time T is detailed below: Call options with strikes Denote by x1. More precisely. Depending on the value S T of the underlying asset at maturity. X1 Ki. C3 0 are such that we can find xi. The constraints 1. These constraints are satisfied. The easiest way to find values of xi. The Put-Call parity is not satisfied. At time 0. The short will be closed at maturity T by buying shares on the market and returning them to the borrower.
The initial value of the portfolio is zero. Buying one option with strike The riskless profit at maturity will be the future value at time T of the mispricing from the Put-Call parity. The following values are given: Assume that the dividend payments are financed by shorting. This will generate the following cash amount: Show that the Put-Call parity is not satisfied and explain how would you take advantage of this arbitrage opportunity.
Note that by shorting the shares you are responsible for paying the accrued dividends. Assume that the risk free rate is equal to 0. This value represents the risk-free profit made by exploiting the discrepancy from the Put-Call parity. From the bid and ask prices. Which strategy is better.
To act on the expected stock price appreciation. It is easy to see that. The index is at Is there an arbitrage opportunity.. The breakeven point of the two strategies. Problem The realized gain is the interest accrued on the cash resulting from the short position minus How do you synthesize this derivative security i.
Compute I xnex dx. Compute J xn ln x dx. A derivative security pays a cash amount c if the spot price of the underlying asset at maturity is between K1and K2. Compute f 1n x 71 dx. Create a portfolio with the following payoff at time T: Use plain vanilla options with maturity T as well as cash positions and positions in the asset itself. C2 and C3 such that no—arbitrage exists corresponding to a portfolio made of positions in the three options.
Find necessary and sufficient no—arbitrage conditions for Cbid Cask. Find necessary and sufficient conditions on the prices C1. Denote by Cbid and Cask. Assume that the risk—free interest rates are constant equal to r. Call options on the same underlying asset and with the same maturity. If n —1. C2 and C3.
Compute J eln x dx. Problem 2: Compute f xnex dx. By using integration by parts. Compute f 1n x n dx. OC x—. Note that 1. R R is a continuous function which is uniformly bounded2. Since g x is a continuous function. From 1. V y E [ 6. V [ 6. Since V T is discontinuous. A derivative security pays a cash amount c if the spot price of the underlying asset at maturity is between Ki. We conclude. Using plain vanilla options. We approximate the payoff V T of the derivative security by the following payoff V.
It is easy to see that the payoff VE T is the same as in 1. This is equivalent to a long position in two units of the underlying asset. Denote by C20 and P20 i and by C40 and P First of all. This is equivalent to a long position in two units of the underlying asset and a short position in three calls with strike We check that the payoff of this portfolio at maturity.
If the asset does not pay dividends and if interest rates are zero. To synthesize a short position in three calls with strike The first replicating portfolio will be made of the following positions: Portfolio 1: Long the K1-option. An arbitrage exists if and only if a no-cost portfolio can be set up with non-negative payoff at maturity regardless of the price of the underlying asset at maturity. Consider a portfolio made of positions in the three options with value 0 at inception.
The second replicating portfolio will be made of the following positions: Note that any piecewise linear payoff of a single asset can be synthesized. C3 Cs 1. C2 and C3 such that no-arbitrage exists corresponding to a portfolio made of positions in the three options.
For no-arbitrage to occur.. K1 — x2. Long the Ki—option. Portfolio 3: Using the value of x3 from 1. K1 Cl. Assume that the risk-free interest rates are constant equal to r. We conclude that Find necessary and sufficient no-arbitrage conditions for Cbid.
It is easy to see that 1. Cbid — Pask were greater than the value Se-qT. Ke-rT of the forward contract. Recall the Put-Call parity C. The inequality 1. Compute the integral of the function f x. Numerical integration.
We first identify the integration domain D. Interest rates. Chapter 2 Improper integrals. We include a mathematically rigorous arguments for. Since 1 1 1 lim 1. In a similar intuitive way. By definition. Making these intuitive arguments precise is somewhat more subtle. From 2. We can then use 2.
By integration by parts. Conclude that Simpson's rule converges. Compute an approximate value of fi. The approximate values of the integral found using the Midpoint.
Define the function g: Intervals 4 8 16 32 64 Simpson's Rule 0. Intervals Midpoint Rule 4 0. Compute g' x. V exP Problem 6: Let h x be a continuous function such that exists.
Note that f a t. A similar result can be derived for improper integrals. Recall that. The continuously compounded 6-month. The data below refers to the pseudocode from Table 2.
For a 2-year semiannual coupon bond. Problem 7: What is the par yield for a 2-year semiannual coupon bond? Par yield is the coupon rate C that makes the value of the bond equal to its face value. For the zero rates given in this problem. The formulas for the price. The quarterly bond will pay a cash flow of 1. The price. What are the price. Compute the price. Check your answer mathematically. Discount factors: If the coupon rate goes up.
The duration of the bond is the time weighted average of the cash flows. This is due to the fact that the earlier cash flows equal to the coupon payments become a higher fraction of the payment made at maturity.
To compute the duration and convexity of the bond. If the coupon rate increases. The yield can be computed. If the coupon rate of a bond goes up.
We obtain that the yield of the bond is 0. Give a financial argument.. The duration of a zero—coupon bond is equal to the maturity of the bond. For small changes Ay in the yield. Recall that the percentage change in the price of the bond can be approximated by the duration of the bond multiplied by the parallel shift in the yield curve.
AB The new value of the bond is B. Find an approximate price of the bond if the yield decreases by fifty basis points. Establish the following relationship between duration and convexity: A five year bond with duration 32 years is worth By how much would the price of a ten year zero-coupon bond change if the yield increases by ten basis points? One percentage point is equal to basis points. Assume that the continuously compounded instantaneous rate curve r t is given by 0.
The prices of three call options with strikes Create a butterfly spread by going long a 45—call and a 55—call. Find the value of the put option with six decimal digits accuracy using the Midpoint Rule and using Simpson's Rule.
When would the butterfly spread be profitable? Use risk—neutral valuation to write the value of the put as an integral over a finite interval. What positions could you take in these bonds to immunize your portfolio i.
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 buy 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 buying 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 buying 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 buy 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 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.
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