# Questions tagged [mathematics]

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### 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 ...
3answers
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### 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 ...
0answers
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### 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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### 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 ...
0answers
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### 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. ...
2answers
579 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 ...
3answers
376 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 ...
1answer
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### 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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### 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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### 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 ...