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Book/reference to practice stochastic calculus and PDE for interviews

You may like: Probability and Stochastic Calculus Quant Interview Questions by Ivan Matić, Radoš Radoičić, Dan Stefanica 150 Most Frequently Asked Questions on Quant Interviews, Second Edition by Dan ...
Dimitri Vulis's user avatar
7 votes

The solution to arithmetic brownian motion

To solve the SDE you should use the so called variation of constant method. Define a process $Y_t=e^{-\mu t}X_t$, so that using Itô we obtain: $$dY_t=-\mu Y_t dt+ e^{-\mu t}X_t=e^{-\mu t} \sigma dW_t $...
NSZ's user avatar
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6 votes
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Idea of using logarithm for solving SDE in Black-Scholes model

Black and Scholes (1973) were not the first ones to use the geometric Brownian motion as a model for stock prices. For example, Samuelson did it before them. It all started with a Brownian motion as ...
Kevin's user avatar
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5 votes
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Riccati Equation in spot rate model

As you noted, this is a Riccati type ODE and it can thus be simplified using the standard transformations for this class - see e.g. Wikipedia. We start by defining \begin{equation} C(t, T) = \frac{1}{...
LocalVolatility's user avatar
5 votes
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Pricing the Passport option

You maximize the terms in $q$ in the PDE because this is a consequence of the Bellman principle of optimality in dynamic programming. The intuition is that the global optimal strategy $\{q_t\}_{0 \leq ...
Antoine Conze's user avatar
5 votes

How to solve this PDE using Feynman-Kac?

Martingale Approach As you noted, you need to solve \begin{eqnarray} F(0) & = & e^{-r T} \mathbb{E} \left[ \left( X_T - K \right)^2 \right]\\ & = & e^{-r T} \left( \mathbb{E} \left[ ...
LocalVolatility's user avatar
5 votes

Transformation of local volatility model

Consider a function $f(X_t)$. Ito's lemma gives: $$df(X_t)=\text{time terms}+f'(X_t)\sigma(X_t)dW_t$$ Now any $f$ satisfying: $$f'(X_t)\sigma(X_t)=\text{constant}$$ gives a constant volatility for $f(...
fes's user avatar
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4 votes

PDE for Pricing Interest Rate Derivatives

Note that \begin{align*} M(r_t, t) &\equiv Q(r_t, t) e^{-\int_0^t r_u du} \\ &=E\left(e^{-\int_0^T r_u du} h(r_T, T) \mid \mathscr{F}_t \right) \end{align*} is a martingale. Moreover, \begin{...
Gordon's user avatar
  • 21.1k
4 votes

Evaluating the SDE $dX_t = t\,dS_t$

Using Itô's Lemma, notice that: $$d(tS_t)=tdS_t+S_tdt=dX_t+S_tdt$$ Hence: $$X_t=tS_t-\int S_udu$$ Using independence of Brownian increments, $E(S_udW_u)=E(S_u)E(dW_u)=0$, and the chain rule for the ...
Daneel Olivaw's user avatar
3 votes
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Proof verification : risk free rate

You have: $$r(t):=\theta+(r_0-\theta)e^{-kt}\tag{1}$$ Then: $$r^\prime(t)=-k(r_0-\theta)e^{-kt}\tag{2}$$ which is clearly equal to: $$k(\theta-r(t))\tag{3}$$ based on $(1)$.
Daneel Olivaw's user avatar
3 votes
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What are the advantages and limitations of predicting future stock prices using stochastic differential equations?

The SDE you are describing is called the Geometric Brownian Motion. In the end its just a model, which underlies certain assumptions, which are usually not met in the real world scenarios. There are ...
Question Anxiety's user avatar
3 votes
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Differential of time over Browninan motion

Edits have been made, but the original question asked about $\frac{\mathrm{d}t}{W_t}$. The random variable does exist In your [original] question you ask about $\frac{\mathrm{d}t}{W_t}$, although I ...
oliversm's user avatar
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3 votes

Black Scholes to Heat Equation

Let $n=\sigma^2(T-t)$ $$dn=-\sigma^2dt$$ $\frac{\partial {V}}{\partial {t}} = \frac{\partial {V}}{\partial {n}}\frac{d {n}}{d {t}}$ $\frac{\partial {V}}{\partial {t}} = -\sigma^2\frac{\partial {V}}{\...
ryc's user avatar
  • 401
3 votes

What are the advantages and limitations of predicting future stock prices using stochastic differential equations?

Take the analogy of equations modelling something in physics. Just because you write down an equation, it does not mean it has to be connected to anything in reality. It only do so to the extent you ...
Jesper Tidblom's user avatar
3 votes
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Market Making Strategies Found by Hamilton-Jacobi-Bellman Equation

At the terminal time $T$, the terminal condition is $g(T, q) = -\alpha q^2$, this implies, $$ \begin{aligned} g(T, q) &= \frac{1}{\kappa} \log{\omega(T, q)} = -\alpha q^2\\ \Rightarrow \omega(T,q)...
Danny's user avatar
  • 514
3 votes

Is there an intuitive explanation for the Feynman-Kac-Theorem?

Let's approach this answer in two steps. First, I find it quite intuitive, that for a given stochastic PDE there exists a deterministic PDE that evolves the density to a later time. This equation is ...
davidhigh's user avatar
  • 348
3 votes

How to understand the market price of risk

I think you misunderstood the underlying idea of the risk-neutrality and the market price of risk. The basic idea is to price the option with a portfolio consisting of the underlying asset $S$ and ...
Wiles01's user avatar
  • 267
3 votes
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The PDE of caplet and floors

It must be a typo for the equation in the book. That is, the equation for a caplet is of the form \begin{align*} \frac{\partial V}{\partial t} + LV - r_t V +\max(r_t-r^*, 0) = 0, \end{align*} which ...
Gordon's user avatar
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3 votes

Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$

This is a corollary of Feynman-Kac theorem. For self-containedness, I re-produce the proof as follows. Assume that there exists a $C^{1,2}$-function $F=F(t,x)$ defined on $[0,T]\times\mathbb{R}$ that ...
Danny Pak-Keung Chan's user avatar
2 votes

How to apply the chain rule for partial derivatives to transformations?

As you state in your comment, you only have trouble with the second partial derivative w.r.t. the spot. So you understand how the first partial derivative is obtained \begin{equation} \frac{\partial ...
LocalVolatility's user avatar
2 votes

The PDE of the probability hitting the barrier before T

May be I have overlooked something, but I believe that \begin{align*} Q(t, S) = \mathbb{P}\left(\tau_{B} \le T \mid \mathcal{F}_t\right). \end{align*} Then $\{Q(t, S), \, 0<t < T\}$ is a ...
Gordon's user avatar
  • 21.1k
2 votes

why futures contract has no value

The mathematical analysis above is correct, but to understand WHY we say that "a futures has no value" it is helpful to understand how a Futures Exchange works. When you enter into a position (for ...
nbbo2's user avatar
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2 votes
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How to price the American style Asian option with recent N day average

In convertible bond pricing there is something similar called a "soft call" with similar properties so you might want to search for literature on them. The main difference is that soft calls are an ...
Brian B's user avatar
  • 14.9k
2 votes

What are the advantages and limitations of predicting future stock prices using stochastic differential equations?

The GBM model is liked by practitioners for the modelling of stock prices for the following reasons: (i) The solution is log-normal, so the stock price distribution varies between zero and infinity: ...
Jan Stuller's user avatar
  • 6,118
2 votes
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Approximating Sharpe and Sortino ratios from Exponential moving averages

For anybody still following this: I figured out that the equations and my code work fine; the problem was that I had to scale the returns before doing the risk calculations to avoid float32 precision ...
Alex Pilafian's user avatar
2 votes
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Using the risk neutral version of the First Fundamental Theorem of Asset Pricing to derive a partial differential equation

In the answer to a related question of yours it was shown that under the risk-neutral measure $\mathbb Q$ the process $$ S_te^{-rt}=S_0e^{-\frac{\sigma^2t}{2}+\sigma W^{\mathbb Q}_t} $$ is a ...
Kurt G.'s user avatar
  • 2,033
2 votes

Heston Riccati equation

Partial Answer In [1] on p. 290-291 you find a discussion of the Cox-Ingersoll-Ross model in which the the Riccati equation $$ \textstyle n_t(t,T)-\frac{1}{2}\sigma^2\, n^2(t,T)-b\,n(t,T)+1=0\,,\quad ...
Kurt G.'s user avatar
  • 2,033
2 votes
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Heston Riccati equation

For the proof, it suffices to follow this procedure with $$\begin{align} q_0(t) &= \frac{1}{2}u(1-u)\\ q_1(t) &= \frac{1}{2}\kappa-\rho\sigma u\\ q_2(t) &= -\frac{1}{2}\sigma^2\\ \end{...
NN2's user avatar
  • 1,008

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