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### 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 ...
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### 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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### 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 $f$...
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### 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 ...
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### 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 ...
540 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 ...
91 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...