21

Life Without a Risk-Neutral Measure How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$ is the stochastic discount factor (SDF). This equation holds under very weak assumptions (law of one price) and uses real-world probabilities. So, we take the ...


20

I give you a brief outline about some key properties of Lévy processes. Lévy processes have stationary and independent increments but do not necessarily have continuous sample paths. In fact, Brownian motion is the only Levy process with continuous sample paths. Some Lévy processes (e.g. Poisson process) have single, rare but large jumps (finite activity) ...


11

By construction, the Itô integral, $I_t=\int_0^t X_s\text{d}W_s$, is a martingale if $\int_0^t \mathbb{E}[X_s^2]\text{d}s<\infty$. The martingale property, $\mathbb{E}_s[I_t]=I_s$ implies $\mathbb{E}[I_t]=I_0=0$. Because $W_s\overset{d}{=}\sqrt{s}Z$, where $Z\sim N(0,1)$, we indeed have \begin{align*} \int_0^t\mathbb{E}\left[\frac{1}{(1+W_s^2)^2}\right]\...


11

You need to rotate them so we can find some orthogonal axes. A simple way to think about this is by remembering that we can decompose the second of two brownian motions into a sum of the first brownian and an independent component, using the expression \begin{align} W_{t,2} = \rho_{12} W_{t,1} + \sqrt{1-\rho_{12}^2} \tilde{W}_{t,2} \end{align} where $\tilde{...


11

Besides @StackG's splendid answer, I would like to offer an answer that is based on the notion that the multivariate Brownian motion is of course multivariate normally distributed, and on its moment generating function. We know that $$ \mathbb{E}\left(W_{i,t}W_{j,t}\right)=\rho_{i,j}t $$ i.e. an $N$-dimensional vector $X$ of correlated Brownian motions has ...


8

The means are equal Suppose $f$ is analytic so that we can give it a Taylor series that works everywhere such that $f(x) = \sum a_n x^n$, and then let us let this be bounded too. To show that the expectations are the same consider taking the expectation of $U_t$ \begin{equation} \mathbb{E}(U_t) = \mathbb{E}\left(\int_0^t f(W_s) \,\mathrm{d}s\right)\\ \end{...


7

Because of volatility drag. In very simple terms, assume three periods, $t=t_0, t_1, t_2$, and a process which starts with value $100$ at $t_0$ and which can either go up or go down with same probability when transitioning from one period to the next. Let us assume the same expected change for each period, for example $1\%$, but with different up and down ...


7

First and foremost if $u=e^{\gamma (s-t)}$ then $\frac{du}{ds} = \gamma e^{\gamma(s-t)} \iff du=\gamma e^{\gamma(s-t)} \,ds$. Now, in the Ornstein–Uhlenbeck process $X_t$ is a Wiener process and satisfies $X_0 = 0 \: \: \text{a.s.}$ (see this and this). Then, the first term in the integration by parts formula specified above gives you: $$uv\vert^t_0 = e^{\...


7

This is impossible unless you are very intelligent with good memory-retention skills and already mathemathically proficient in the field of analysis and statistics (and no, a single course in basic probability theory does not suffice). And even if you are, two weeks is an extremely short amount of time. However, assuming these criteria are satisfied, I would ...


6

The theory behind the actual reasoning is a bit complicated than the coverage in Hull's, but staying within the simple reasoning, the difference comes down to the following: The Brownian increments over the interval $dt$ are normally distributed with mean zero and variance $dt$, so in terms of distribution, you can express the increments in terms of a ...


6

Intro: Great answer given by KeSchn above. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no arbitrage" and "replication / hedging" arguments. The way I would like to explain this view is via the following three-step construction: (i) First, I ...


6

Let me try to answer, this topic is much deeper than my answer 1. Why are these models like this unpopular? First, these models produce marginal distributions that does not fit the market, which means they cannot reproduce vanilla option prices traded in the market SV models, e.g. Heston model, may fit to a few vanilla prices, they cannot fit the entire ...


6

Question 1) The Itô integral of a deterministic function is Gaussian, see here or here, i.e. $$\int_0^T f(u)\mathrm{d}W_u \sim N\left( 0,\int_0^T f(u)^2\mathrm{d}u\right).$$ The answer is thus zero. We of course need to require that $\int_0^T f(u)^2\mathrm{d}u<\infty$. Question 2) The simple version of Itô's isometry reads as $$\mathbb{E}\left[\left(\...


6

Let $$g_s = \int_0^s f_u du$$ By Ito-Leibniz product rule: $$ d(W_sg_s) = W_sdg_s+ g_sdW_s +d[g,W]_s $$ Assuming $f_s$ is deterministic, $d[g,W]_s = 0$ and we get: $$ d(W_sg_s) = W_sdg_s + g_sdW_s $$ In integral form, this is: $$ W_tg_t = \int_0^t W_sdg_s + \int_0^t g_sdW_s $$ Getting back to $f$: $$ W_t \int_0^t f_u du = \int_0^t W_sf_sds + \int_0^t \...


6

It is covered very nicely in Iain Clark's Foreign Exchange Option Pricing, A Practitioner’s Guide (pages 98-104). The book also contains references to the relevant literature including Feller's original paper.


5

Cant talk specifically to stock pricing models but in foreign exchange the list in order of use goes: Geometric Brownian motion with time dependent vol and drift Local Volatility, either SABR or some other parametric or cubic-spline+Dupire Heston's stochastic volatility model Stochastic-Local hybrid volatility models, usually some from of parametric local ...


5

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 simplest time continuous stock price model. However, then the stock price is normally distributed and can be negative. Not a great property! So, Samuelson ...


5

It is a vast topic so my answer wont do justice, but staying within the topic of twice continuously differential settings, Ito's lemma can be applied to generalised functions (derivatives defined in the distribution sense)- examples of such functions are the Heaviside function, dirac delta etc. The particular application you referenced goes by the name ...


5

We can use Stochastic Integration by Parts to show this. Taking the corollary from the link above \begin{align} X_t Y_t = X_0 Y_0 + \int_0^t X_s dY_s + \int_0 ^t Y_{s-} dX_s \end{align} We set $X_t$ and $Y_t$ equal to the following: \begin{align} X_t &\to \int_0^t f(u) du\\ Y_t &\to W_t \end{align} then \begin{align} W_t \int_0^t f(u) du &= W_0 \...


5

Here is a simple example that might be useful. Basically finding parameters for a given section. Some of the parameters might be assumed at start instead of calibrated. import QuantLib as ql import matplotlib.pyplot as plt import numpy as np from scipy.optimize import minimize strikes = [105, 106, 107, 108, 109, 110, 111, 112] fwd = 120.44 expiryTime = 17/...


5

Have a look at page 311 in the original paper from Carr, Geman, Madan and Yor (2002). The paramters are for the names of the authors. They explain the role of each parameter there. Note that $C>0$, $G\geq0$, $M\geq0$ and $Y<2$. These parameters play an important role in capturing various aspects of the stochastic process under study. The parameter $C$ ...


4

There are a few recovery mechanisms, for example, recovery of par (i.e., the notional), recovery of treasury (i.e., the recovery value is a constant fraction of the equivalent default-free bond), and recovery of market value (i.e., a fraction of its pre-default market value). Here, your formula, which is also called the Lando formula, assumes the recovery of ...


4

I believe the other answers are nearly exhaustive; but here's a bit of intuition I'd like to add: Think of the decision (= equilibrium price) of a market as: Decision = f(probabilities, risk aversion) where probabilities are the chances of various events happening, and risk aversion is the taste preference of the market. Now it turns out that the 'iso-curve' ...


4

Is GBM still viable for modeling asset prices in a very short time period ? I assume by asset price you mean the mid-price of the asset. Ito diffusions are unable to capture stylized facts of market microstructure: in particular, you can't get volatility clustering, negligible autocorrelation in return series, the signature plots or the Epps effect. We can ...


4

Let's use the following expression (derived in Quantuple's answer in your link), which will help us tidy up the product using Ito's Isometry \begin{align} \int^t_0 W^2_s ds = 2 \int^t_0 (t-s)W_s dW_s + {\frac {t^2} 2} \end{align} Now looking at the expectation \begin{align} {\mathbb E}\Bigl[ \int^t_0 W^3_s dW_s \cdot \int^t_0 W^2_s ds \Bigr] &= {\mathbb ...


4

$X_j$ can be either 1 or -1 with 50% probability each. So this step is just applying the expectation to both possible cases. See definition of the Expectation... \begin{align} {\mathbb E}\bigl[ X \bigr] = \sum_i i \cdot p(x = i) \end{align} It's the sum over all possibilities of the probability of getting that value (both ${\frac 1 2}$ in your case) ...


4

(Bloomberg and Reuters News are fond is reporting that some name is trading at some such CDS spread, "which implies N% probability of default". They neglect to mention what recovery assumption they used, and that this is risk-neutral probability, not physical.) For corporate names, Ed Altman published the well-known paper on Z-score. His basic idea ...


4

An exponential Lévy process is typically modelled via $$ S_t = S_0\exp\left(\left(r-q+\omega\right)t+X_t\right),$$ where $X_t$ is a Lévy process with $X_0=0$. A Lévy process includes three model features: a linear drift, diffusive shocks and jumps (which may be large and rare or small and frequent). The number $\omega$ is called martingale correction or ...


4

Drifts under $\mathbb{Q}$ and $\mathbb{P}$ Some good answers already. Let me just repeat for clarity: under the risk neutral measure $\mathbb{Q}$, the drift of all assets has to equal to the rate at which the Numeraire appreciates, i.e. typically this is the risk-free rate $r$ of the money market. The reason for this is the "no-arbitrage" argument: ...


3

I think the question also brings up a common confusion with notation. I think it is incredibly unfortunate to use notation such as $dW(t)$ (unless it's part of a stochastic integral), and I get upset when I see it being used in textbooks. The definition of Brownian Motion is implicit and goes like this: (i) $W(t=0) = 0$ (ii) $W(t)$ is (almost surely) ...


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