Questions tagged [mathematics]

Used for question on application of mathematics in finance - from interest calculation to mathematical description of random processes.

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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^{\...
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0answers
94 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 ...
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2answers
270 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 ...
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1answer
356 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 ...
2
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1answer
447 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 ...
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1answer
54 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 ...
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1answer
122 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 ...
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1answer
158 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
100 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(...
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1answer
68 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 $...
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1answer
330 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
257 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 ...
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1answer
85 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{...
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1answer
3k 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 ...
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1answer
1k 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+\...
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1answer
266 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}...
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1answer
174 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 ...
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1answer
124 views

Formula for conditional expectation. Related to the Fundamental Theorems of Asset Pricing

Let $\lambda$ be a probability measure on $\Omega$ (finite), with filtration $\{\mathcal{F}_t\}$. Define $\nu(X) = \lambda\left(X\frac{d\nu}{d\lambda}\right)$, where $\frac{d\nu}{d\lambda}$ is a ...
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1answer
48 views

Verifying value of claim as an expectation

Background: We have so far taken the bond B to be deterministic for simplicity, but some reflection shows that this is not in any way necessary. Everything works out the same way with a stochastic ...
0
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1answer
59 views

Do we have arbitrage if the probability measures are less than zero

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}...
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1answer
116 views

Value of a perfect hedge

Background Information: The price of a portfolio at time $t$ ($t = 0 ,1$) is $$V_t(\pi) = \phi S_t + \psi B_t$$ The portfolio $\pi$ is a perfect hedge for the claim $X$ if $V_1(\pi) = X$ a.s. as ...
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3answers
177 views

Suppose i want to track S&P500 index using 15 stocks, how do i adjust their weights?

I am given 15 stocks (which is listed in NYSE), and want to track/replicate the S&P500 index. So i am currently have the information about the stock price, and given some capital to invest in (all ...
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0answers
129 views

Can someone try this Boundary Condition for the Black-Scholes PDE out for me?

I have a bit of a favor to ask and if anyone could help me out with this I'd really appreciate it. At the moment I'm trying to use the triangle wave formula as the payoff for the Black-Scholes PDE i.e....
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1answer
48 views

How to understand the following limits when kapa limits to Zero

The equation is quite simple, however it is not very obvious to me to have the following relationship: $$\begin{equation} \frac{1-exp(-\kappa(T-t))}{\kappa}\rightarrow(T-t) \quad \rm{when\space} \...
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1answer
68 views

Implied Expected Stock Return from European Option Prices

We can calculate the expected stock return (under the measure $Q$) from at-the-money ($K=S_t$) option prices as: $$E\left(\frac{S_T-S_t}{S_t}\right)=\frac{e^{rT}}{S_t}(C_t-P_t)$$ The result is ...
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2answers
1k views

Preparation for interview: influx of power of the moon

I am preparing myself for an interview for a quantitative analyst position and one of the sample questions asked in previous examinations was: "Suppose the moon were to disintegrate, and fall to ...
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0answers
222 views

Arrow-Debreu Equilibrium Pricing

I have this problem in asset pricing that I don't know how to solve. Here it is: Consider an economy with a complete set of Securities and $N$ states of the world Tomorrow. Assume that there are two ...
4
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2answers
970 views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\...
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1answer
157 views

Payoff of option

Consider the payoff $g(S_T)$ shown the figure: I believe the payoff represented as a linear combination of the payoffs of some options with different strike and same maturity $T$ is $$g(S_T) = (2K -...
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2answers
545 views

Perpetual American options

Formulate and solve the free boundary problem for the perpetual American options with the following payoffs. a.) $(S - K)_{+} + a$ where $a > 0$ b.) $(K - S)_{+} + a$ where $a > 0$ ...
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0answers
273 views

Two-period binomial model for American option

Consider a two-period binomial model for a risk asset with each period equal to a year and take $S_0 = 1$, $u = 1.5$, and $l = 0.6$. The interest rate for both periods is $R = .1$. a.) Price an ...
2
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1answer
106 views

Trinomial model converges to Black-Scholes weakly

Consider risk-neutral trinomial model with $N$ periods presented by $$S_{(k+1)\delta}H_{k+1}, \ \ \text{for} \ \ k=0,\ldots,N-1$$ where $\delta:=\frac{T}{N}$ and $\{H_k\}_{1}^{N}$ is a sequence of i....
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1answer
52 views

Symmetric probability and subjective return

Let $\{Z_k\}_{k=1}^{N}$ be a sequence of i.i.d. random variables with the following distribution $$Z_k = \begin{cases} \alpha &\text{with probability} \ \hat{\pi}\\ -\beta &\text{with ...
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1answer
264 views

Pricing of Black-Scholes with dividend

Consider the payoff $g(S_T)$ shown in the figure below. Consider Black-Scholes model for the price of a risky asset with $T = 1$, $r = .04$, and $\sigma = .02$ and dividends are paid quarterly with ...
2
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1answer
893 views

Black-Scholes Equation with dividend

Consider a European option with payoff $$g(S_T) = S_T^{-5}e^{10S_T}$$ Assume that the interest rate is $r = .1$ and the underlying asset satisfies $S_0 = 2, \sigma = .2$, an pays dividend at ...
1
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1answer
310 views

Find the solutions of the ODE from SDE

Consider the SDE $$dS_t = rS_t dt + \sigma S_t dB_t \ \ \ \text{where} \ r \ \text{and} \ \sigma \ \text{are constants}$$ a.) Find the ODE for the function $V(x)$ such that $e^{-rt}V(S_t)$ is ...
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0answers
104 views

Find the PDE for a function that makes it a martingale

Given the SDE, find the PDE for the function $V(t,x)$ such that $V(t,S_t)$ is a martingale. $dS_t = \kappa(m - S_t)dt + \sigma\sqrt{S_t}dB_t$ where $\kappa$,$m$, and $\sigma$ are constants. ...
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2answers
430 views

The Relation Between the Ricci flow and the Black-Scholes-Merton Equation

Grisha Perelman once wrote that The Ricci-flow equation, a type of heat equation, is a distant relative of the Black-Scholes equation that bond traders around the world use to price stock and ...
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1answer
168 views

Replicating option strategies

I was curious if there was any references to replicating option strategies i.e. bull spread, bear spread, butterfly, strangle, straddle, etc...? Also what is the insight into replicating of these ...
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3answers
478 views

Linear combination of payoffs of bull and bear spreads

Write the following payoffs as linear combination of call options with different strikes and possibly some cash and give the closed form formula for them. Attempted solution: The payoff for the bear ...
2
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1answer
482 views

Linear combination of Payoffs using Black-Scholes

Write the payoffs in Figure 3.8 as linear combination of call options and derive a closed form formula for the Black-Scholes price, the Delta, and the Gamma of them. All the Greeks of the option are ...
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1answer
126 views

two-period binomial model, with price that is path-dependent

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.03$ and $l = 0.98$. How do you price a look-back option with payoff($\max_{t=0,1,2}...
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2answers
833 views

Two-period binomial model with dividends

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.15$ and $l = 0.95$. The interest rate is $R = .05$. a.) If the asset pays 10% of its ...
4
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1answer
1k views

How to price and find a replicating portfolio for a call spreads using a two-period binomial model?

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.03$ and $l = 0.98$. a.) If the interest rate for both periods is $R = .01$, find the ...
2
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1answer
93 views

Use no dominance to show that the price of the call option satisfies the inequality

Assumption 2.1 - If the payoff $P$ of a financial instrument is non negative, then the price $p$ of the financial instrument is non negative. Assume $C$ is just the price of the call option, and $C^...
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1answer
62 views

Convexity in Markovian contingent claim

Background information: I believe we can use Jensen's Inequality here Show that if the payoff function $V(S_T)$ is a convex function on $S_T$, then the Markovian European contingent claim with ...
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2answers
89 views

Black-Scholes and Markovian contingent claim

Background information: Proposition 4.1 - For a European Markovian contingent claim, the Black-Scholes price satisfies $$\Theta(\tau,S) = -\frac{\sigma^2 S^2}{2}\Gamma(\tau,S) - rS\Delta(\tau,S) + rV(...
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1answer
83 views

European Markovian option

Background information: Consider a European contingent claim with payoff $V(S_T)$, where $V: \mathbb{R}_+\rightarrow \mathbb{R}$ is a function which assigns a value to the payoff based on the price of ...
0
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1answer
224 views

Arrow-Debreu Model and Risk-Neutral Probabilities

Consider one period Arrow-Debreu model with $N = 2$ and $M = 4$ shown in Figure 3.5 and take $R = 0$. a.) Show that any risk neutral probability $\hat{\pi} = (\hat{\pi}_1, \hat{\pi}_2, \hat{\pi}_3, \...
1
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3answers
387 views

Put-Call Parity Application

In the binomial model, how that the Delta of a call option $\Delta^{call}$ and the Delta of a put option $\Delta^{put}$ with the same maturity and strike satisfy $$\Delta^{call}_t - \Delta^{put}_t = ...