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Questions tagged [mathematics]

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16 views

Prove coherency scenario risk measure

The scenario risk measure is defined as follows: $max\{L_i(x) : x \in X\}$, Monotonicity, translation invariance and positive homogeneity follow trivially, but i'm wondering how to prove ...
0
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1answer
55 views

Why do most interest rate formulas, and indeed finance in general, add 1 to a rate and then subtract afterwards? [closed]

For example, in the formula that shows the relationship between the nominal and effective interest rate shown below, 1 is first added to in/m and then 1 is subtracted from the result. What is the ...
0
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0answers
18 views

Estimating monthly return based on known yearly return and different monthly dataset

I have two data sets: (1) Data set of monthly equity returns (Index 1) and (2) data set of yearly equity returns (Index 2) I want to estimate the monthly returns in Index 2 using the known yearly ...
1
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1answer
41 views

Approximation of Forward Rates in discrete time

The forward rate from time $t$ to $T$ ($f_{t,T}$) can be approximated by: $$ f_{t,T}= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{{T-t}}}-1 \sim \frac{(1+r_T)^T-(1+r_t)^t}{T-t} $$ Why is ...
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1answer
42 views

standard brownian vs brownian motion

We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons . Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
1
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1answer
101 views

Finding the process of $X/Y$

This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11. If $$dX_t = \alpha X_t dt + \beta X_t dW_t$$ $$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$ with $W$ ...
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66 views

Spread State definition

I am a bit confused about the meaning of the definition 4.0.3 in the following article page 12. Definition 4.0.1 (Strategy) : A strategy is a $n$-vector function $L: \mathbb{R}^{+} \times \mathbb{Z}...
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2answers
183 views

Is there an intuitive explanation for why Kelly gambling ignores odds?

I have just learned about Kelly gambling from Chapter 6 of Cover & Thomas' Introduction to Information Theory. The mathematical setup is that we have a horse race, with horse $i$ winning with ...
0
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0answers
50 views

Computing the Expectation to a Max function

if $X_T$ is log-normally distributed and $k$ is a constant, how do I compute: $$E[\max(X_T-k,0)]$$ I can compute $E[X_T-k]$ and $P(X_T-k>0)$. I was thinking that an approach will be compute to $$E[...
1
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1answer
42 views

4-point Trapezium rule for numerical integration

Background: This is in reference to Mark Joshi's concepts of mathematical finance ch.7 problem 11. Question: We have in the Black-Scholes model: $S_0 = 1, T = 1, \sigma = 0.1, r = 0$. A ...
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0answers
41 views

Anti-thetic sampling and second moment matching

Background: This is in reference to ch 7 problem 10 of Mark Joshi's concepts of mathematical finance. Question: A normal random generator produces the following draws: $$0.68, -0.31, -0.49, -0....
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1answer
88 views

Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 6 [duplicate]

Suppose a stock allows a geometric Brownian motion in a Black-Scholes world. Develop an expression for the price of an option that pays $S^2 - K$ if $S^2 > K$ and zero otherwise. What PDE will this ...
2
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1answer
215 views

Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 4

Let an asset follow a Brownian motion $$dS = \mu dt + \sigma dW$$ with $\mu$ and $\sigma$ constant. The constant interest rate is $r$. What process does $S$ follow in the risk-neutral measure? ...
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1answer
109 views

Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance

If $$dX_t = \mu(t,X_t)dt + \sigma(X_t)dW_t$$ with $\sigma$ positive, show there exists a function $f$ such that $$d\left(f(X_t)\right) = v(t,X_t)dt + V dW_t$$ where $V$ is constant. How unique is $f$...
1
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1answer
114 views

Can someone please verify or disprove this Sharpe Ratio math logic for me

I want to start by stating a problem that I wanted to figure out initially so that this all ties in somehow. I initially wanted to figure out if individual securities in an efficient portfolio all ...
1
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2answers
103 views

I need liquidity metrics of a portfolio (2-5 bonds) that takes into consideration difference in size of bonds and maturity profile

Context: I have bond A from say Apple, Apple also issued different types of bonds , namely B , C, D, E bonds. Bonds A B C D E are all same, except, they were issued at different times, have ...
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2answers
413 views

Mark Joshi, Quant Interview Question problem 2.34; replicating a digital option on a 4-step symmetric binomial tree

Question: Team $A$ and team $B$, in a series of $7$ games, whoever wins $4$ games first wins. You want to bet $100$ that your team wins the series, in which case you receive $200$, or $0$ if they ...
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0answers
56 views

Why do we have to use discretization methods for SDE?

I haven't found the answer for the question above in google. Why can't we just discretize the equation instead of using methods like euler or milstein for the discretization.
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1answer
101 views

How to run optimization to achieve an equal active weight portfolio?

I am trying to build an equal active weight portfolio, while minimizing the total risk. However, my constraint of equal active weight always leads to 0 active weight for everything. I know 0 active ...
0
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1answer
70 views

CDS protection/contingent leg pricing, taking expectation of interest and hazard rates

The Pricing and Risk Management of Credit Default Swaps, with a Focus on the ISDA Model Screenshot: Pricing protection leg of a CDS, by OpenGamma In the screenshot above, I am having trouble ...
1
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1answer
30 views

Calculating the ideal initial capital value to optimize a growth model

I'm trying to work out a method for finding the initial capital value that allows someone to run out of money at the exact time they reach mortality. Currently, I'm graphing the annual total capital ...
6
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1answer
327 views

The Heston Solution For European Option - Jim Gatheral

I have this equation (Eq. (2.4) "The Volatility Surface - A Practitioner's Guide" by Jim Gatheral (Ed. 2006)): $$-\frac{\partial C(v, x, \tau)}{\partial \tau}+\frac{1}{2}v \frac{\partial^2 C(v,x,\tau)}...
1
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1answer
192 views

Likelihood Ratio Method - Delta

I was checking Glasserman(2004) - Monte Carlo for Financial Engineering and got to the likelihood ratio method. I am also looking in my textbook (M. Cerrato: The Mathematics of derivatives securities ...
0
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1answer
311 views

Intuition behind log return of portfolio = weighted sum of log returns

Suppose we have $n$ assets, each of which has weight $w_i$ in the portfolio. The log return of asset $i$ is denoted by $r_i$. What's the intuition why this holds approximately: $$ ln \left( \sum_i ...
1
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1answer
2k views

Swap contract comparative advantage

Corporation $A$ has an excellent credit rating and can borrow at a fixed rate of $5\%$ or a floating rate of LIBOR + $1\%$. Corporation $B$ has a somewhat less excellent credit rating and can borrow ...
1
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1answer
658 views

Pathwise Derivative To Estimate Delta

I am trying to estimate delta using the pathwise derivative method (Broadie and Glasserman (1996)) and I stuck on this part: Here is the other notation defined: Here is my C++ code I have written so ...
4
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1answer
248 views

European call delta derivation

Let's write $S(T) = S_T$ and $S(0) = S_0$. We want to compute $\frac{d}{dS_0}\mathbb{E}[f(S_T)]$. From a previous discussion this is equal to $$\mathbb{E}_{S_0}\left[f(S_T)\frac{g'_{S_0}(S_T)}{g_{S_0}(...
2
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1answer
196 views

Using crude Monte Carlo

Background Information: The crude Monte Carlo algorithm for the arithmetic Asian call option is $$Y = e^{-rT}(\overline{S}_A - K)^{+}$$ and the control is $$C e^{-rT}(\overline{S}_G - K)^{+}$$ The ...
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0answers
66 views

Deriving Cox, Ingersoll and Ross expression for the relationship between forwards and futures, how do they conclude a specific step?

I'm trying to derive a specific relationship about the relationship between forwards and futures from "The relationship between forward and futures prices", written 1981 by Cox, Ingersoll and Ross (...
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0answers
130 views

What jobs in Finance are most math intensive? [closed]

I'm a math major and I've always been really interested in Finance; however, I'm starting to enjoy math more and more and would like to know which jobs in Finance use the most/more advanced math. Also,...
2
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1answer
514 views

Given two risky stocks calculate the rate of return, standard deviation, beta, and risk-free rate

Consider a world where there are only two risky stocks, $A$ and $B$, whose details are listed in the table below: Furthermore, the correlation between the returns of stocks $A$ and $B$ is $\rho_{A ...
0
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1answer
112 views

Using CAPM to derive the following

Background Information: Say there are $s = 1,\ldots,S$ possible future outcomes (states) with known probabilities $\pi_s > 0$, $\sum_{s=1}^{S}\pi_s = 1$. Define the expected payoff as $\mathbb{E}_\...
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2answers
95 views

Given three stocks what is the fraction of each stock's risk is diversified away

Consider an equally weighted portfolio of three stocks, each of which is independently distributed of the others but have the same risk. I.e., $cov(r_i, r_j) = 0$; $\forall i \neq j$, and $\...
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0answers
29 views

Numerical method to extracting a piece of a summation function?

So this is a pension framework. I am trying to code a system and I don't want to have to brute force this answer, but I can't figure out a clean solution. $$Fund = \sum_{i=1}^t [\cfrac{I\cdot e^{\...
2
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0answers
78 views

Is the exponential Shannon entropy sub-additive?

In a recent paper of Salazar et al. (2014), The Diversification Delta: A Different Perspective, forthcoming in the Journal of Portfolio Management , the authors propose to use the exponential Shannon ...
8
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2answers
233 views

Why won't Bjork ever show that the integrability condition is satisfied?

A major technique employed throughout Bjork's "Arbitrage theory in Continuous Time" is that when taking the expectation of a stochastic integral, the result is 0. This is based on a result presented ...
3
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1answer
249 views

Is Black-Scholes complete?

If we have a Black-Scholes model $B_t = \exp{(rt)}$ and $S_t = S_0\exp{(\sigma W_t + \mu t)}$ then is it complete? What if $W_1$ and $W_2$ are independent Brownian motions. Then the two-stage ...
3
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1answer
226 views

Black-Scholes evaluating the squared of the stock price

Consider a Black-Scholes model $S_t = 5\exp{(\sigma W_t + \mu t)}$, $B_t = \exp{(rt)}$, where $W_t$ is Brownian motion with respect to a given measure $\mathbb{P}$. Suppose you hold a forward ...
0
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1answer
51 views

For discrete models, the existence of strong arbitrage is equivalent to a particular self-financing strategy

Background Information: This question is from Lectures on Financial Mathematics: Discrete Asset Pricing. Question: Prove that for discrete models, the existence of a strong arbitrage is also ...
1
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1answer
93 views

All martingale measures price the attainable claim equally

Background Information: This question is from Lectures on Financial Mathematics: Discrete Asset Pricing. Theorem 3.2 First Fundamental Theorem of Asset Pricing - Suppose $\nu$ is any measure such ...
0
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1answer
149 views

Show that there exists a fully invested portfolio such that the covariance between their returns is zero

Background Information: I came across this question in chapter 2 of Active portfolio Management by Grinold and Kahn. It pertains to the efficient frontier which is displayed below: Question: If $...
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1answer
94 views

What is the value this “special” forward contract at maturity?

Background Information: I am not sure this is relevant: Terminal value pricing: If the derivative $X$ equals $f(S_T)$, for some $f$ then in the value of the derivative at time $t$ is equal to $V_t(...
0
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1answer
59 views

If there is an inconsistent pricing strategy then by defintion we have strong arbitrage

Background Information: An Inconsistent pricing strategy is a self financing strategy $\phi$ with $V_T(\phi)= 0$ and $V_0(\phi) \neq 0$ A strong arbitrage is a self-financing strategy $\phi$ with $...
3
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1answer
205 views

How to prove we have a $\mathbb{Q}$-Brownian motion?

Background Information: This question comes from the book Financial Calculus by Baxter and Rennie. WE start with looking at the marginal of $W_T$ under $\mathbb{Q}$. We need to find the likelihood ...
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1answer
195 views

How to solve $dX_t = X_t(\sigma_t dW_t + \mu_t dt)$?

Solve the SDE $$dX_t = X_t(\sigma_t dW_t + \mu_t dt)$$ where $\sigma_t$,$\mu_t$ are deterministic. Attempted solution We have $$dX_t = X_t(\sigma_t dW_t + \mu_t dt)$$ Let $f(x) = \log X$, applying ...
0
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1answer
68 views

Do we have a Brownian motion

Background Information: The process $W = (W_t:t\geq 0)$ is a $\mathbb{P}$-Brownian motion if and only if i) $W_t$ is continuous, and $W_0 = 0$ ii) the value of $W_t$ is distributed, under $\mathbb{...
0
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1answer
2k views

How to Calculate a Negative ROI?

How to properly calculate negative ROIs? I am just wanting to calculate very simple ROIs, which could be very negative, but it doesn't seem to work. Wikipedia defines ROI as this formula: return ...
0
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1answer
426 views

taylor expansion of PnL

I have a question about the following derivation in this pdf (sample chapter from Bergomi - Stochastic Volatility Modeling). He derives the PnL for a delta hedged position as $$PnL = -[P(t+\delta,S+\...
0
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1answer
151 views

Showing the discounted stock is a martingale

Background Information: This question follows from here It is tempting to write $$V_0(X) = \beta\left[\left(\frac{\beta^{-1}S_0 - S_1(d)}{S_1(u) - S_1(d)}\right)X(u) + \left(\frac{S_1(u) - \beta^{-1}...
2
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1answer
146 views

Law of One price and the Inconcistent pricing strategy

Background Information: A market satisfies the Law of One Price if every two self-financing strategies that replicate the same claim have the same initial value. An inconsistent pricing strategy is ...