# Tag Info

23

Hah! There is no such thing as the “rigorous mathematical underpinning” of high frequency trading - because HFT, like all trading, is not primarily a mathematical endeavour. It’s true that many people who work in HFT have a mathematical background, but that’s because the tools of applied math and statistics are useful when analysing the large amounts of ...

11

I am one of the two authors of the paper. The continuity in time of the path of the underlying suggests that at every trading time, the strategy is self-financing. In fact, if the underlying random process had continuous sample paths of bounded variation, then the binary trading strategy is actually self-financing. In contrast, when these continuous sample ...

10

Pull-to-par says that the bond's price will gradually converge toward par (100% of face value) when yield is unchanged. This process is also known as accretion for a bond trading at a discount (since its price gradually goes higher toward par) and amortization for a bond trading at a premium (since its price gradually declines toward par). Pull-to-par says ...

10

If your working modelling assumptions are such that the dynamics of the log price process $\ln(S_t)$ is space homogeneous, you have that the price of a European vanilla option is itself a space-homogeneous function of degree one. You can then appeal to Euler theorem to get the relationship you need. More specifically, define the price at time $t$ of the ...

10

What they gave you is Newton's formula. If you have a function $f(x)$ then you can find the value $x_0$ such that $f(x_0) = 0$ by this method. It uses the derivative $f'$ which in your case is the vega. Your function is: $$f(x) = BS(x) - M$$ where $BS$ is the theoretical price with volatility $x$ and $M$ is the marketprice. Then $f'(x)$ is the ...

9

It is not financial mathematics in general, but a scientific approach that is beneficial: quantitative views and open objective tools make transactions more transparent. It decreases information asymmetry and thus decrease transaction costs in general (bid-ask spread, prices range, volatility, etc). thanks to (good) models, the consistency between ...

9

Pull-to-par just says that a bond's (clean) price will converge towards its face value as the bonds approaches maturity. There is nothing really interesting about pull-to-par - a bond's (clean) price has to converge to its face value, because a bond with just a few days to maturity is essentially a short-term cash deposit. Look at it this way - the price of ...

9

I would argue, taking a note from John von Neumman, that quantitative finance lacks rigorous underpinnings. Von Neumann warned in 1953 that many things that look like proofs in economics and finance depended on problems that were yet to be solved in mathematics, and where economists were assuming solutions into existence. As the problems were solved in math,...

8

Generally speaking, let us consider a problem where you have a series of simple payoffs $f_{K_i}(S_T)$ of strike $K_i$, $i \in I$, that depend on the value of $S_T$ at time $T$, as well as a more complex, laddered payoff $P_L(T)$ which pays a quantity $g_i(S_T)$ on regions of the form $\{K_i \leq S_T < K_{i+1}\}$ $-$ regions are delimited by the strikes ...

8

Long gamma is being long realized volatility. Long vega is being long implied volatility. Long gamma positions benefit when realized volatility goes up or the actual underlying has volatility. Long vega positions benefit when the price of volatility goes up. Being long plain vanilla options, one is long both gamma and long vega. However, this is not so ...

7

Vega (denoted by $\nu$ in what follows) is the first order sensitivity of the option price with respect to volatility $\sigma$. Gamma (denoted by $\Gamma$ in what follows), is the second order sensitivity of the option price with respect to the underlying spot price $S$. Because for a semi-martingale $(S_t)_{t \geq 0}$ there is a direct link between the ...

6

Let $\tau = T-t$. Then \begin{align*} S_T = S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, Z}, \end{align*} where $Z$ is a standard normal random variable, independent of $\mathcal{F}_t$. Moreover, \begin{align*} E\left(S_T 1_{\{S_T >K\}}\mid \mathcal{F}_t \right) &= E\left(S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, ...

6

A general hedging strategy Let assume that $S_1(t)$ and $S_2(t)$ are the price processes of your 2 stocks and that they follow a Geometric Brownian Motion (GBM): $$\forall \, i \in \{1,2\}, dS_i(t) =\mu_iS_i(t)dt + \sigma_iS_i(t)dW_i(t)$$ We assume both stocks have an instant correlation of $\rho$: $$dW_1(t)dW_2(t)=\rho dt$$ Let also $V(t)$ be the value ...

6

There is an interesting article "How Derivatives and Risk Models Really Work: Sociological Pricing and the Role of Co-Ordination" by R. Rebonato answering your question. In section "3.8 Conferences and Journals" the author formulate his version of the question as follows: It is worth mentioning one last aspect ... of the ‘institutional ecology’ in ...

5

Of course estimating expected returns is the very core of portfolio management. Finding a useful covariance matrix too. To find both fills a book. So I first thought about closing the question. But it is a chance to discuss today's approaches. A nice approach that is very up-to-date where mementum investing seems very fashionable is the following: Momentum ...

5

Because vanilla derivatives with European exercise depend only on total variance , not on it's dynamics in time. If you have a simpler model (like interpolation of these total variances from your volatility surface) you don't have as much of unobservable parameters stochastic volatility models have. Having more parameters (which many times would need to be ...

5

Let define $\mathbb{Q}$ and $\mathbb{P}$ two equivalent probabilities on a filtered space $(\Omega,(\mathcal{F}_t)_{t\geq 0})$ Let define $Z_T=\frac{d\mathbb{Q}}{d\mathbb{P}}$ restricted to $\mathcal{F}_T$ measurable events. It means that for $X_T$ being $\mathcal{F}_T$ measurable we have: \mathbb{E}^{\mathbb{Q}}[X_T] = \mathbb{E}^{\mathbb{P}}\left[...

5

Aside from the independence requirement for the increments, that is, the independence of $X_{s+t}-X_s$ and $\mathcal{F}_s$, you can check whether the increment $X_{s+t}-X_s$ has the distribution of $N(0, t)$. In fact, note that \begin{align*} X_{s+t}-X_s &= (\sqrt{s+t}-\sqrt{s}) Z\\ &\sim N\left(0,\, (\sqrt{s+t}-\sqrt{s})^2\right), \end{align*} which ...

5

This thread will inevitably close because it doesn't meet community guidelines, but I respect your passion in this field and my best suggestion for you is that if you're trying to emulate a MFE education, go look up the course listings of any reputable MFE program, and then look into the sites for those (past) classes and see the recommended readings and ...

5

These are a natural and easiest (most tractable mathematically) choice. A utility function is defined up to a positive affine transformation: economically there is no difference between the utility functions $U(x)$ and $\tilde{U}(x)=Au(x)+B$. Hence, a measure of risk aversion that remains constant w.r.t. affine transformations would be useful. How does one ...

5

It is actually rather simple. Lets start with the fixed rate market. A can borrow at 5% while B can borrow at 7%. Simply said, A has a comparative advantage of 2% in the fixed rate market. In the floating rate market, A borrows at LIBOR + 1% while B borrows at LIBOR + 2.5%. From here, I'm guessing you already know that A has the comparative advantage as ...

5

Intuitively, cadlag expresses the fact that we know a jump has occurred after the fact, but we never have advance knowledge that the jump is about to occur (i.e no knowledge of the starting point for the jump or that a jump is "under way"). Each jump is a surprise, after which we believe there will be no jumps at least for a little while. I hear it in the ...

4

Theoretically speaking (as it's done in financial textbooks at b-school level), variance and covariance are calculated on historical performance of asset classes, forward looking returns are CAPM calculated returns. ARIMA. Practically speaking, ARIMA is useless for predicting long term returns (or portfolio management if you wish). Why? A short answer is ...

4

Trinomial trees give incomplete markets so there is a range of possible risk neutral prices. So you have to find the possible probabilities that make the tree risk-neutral and see what prices you get. You have the correct expressions. Now just have to parametrize the set of solutions. It is one-dimensional and all the probabilities are positive so you need ...

4

First, we have $P(t)+S(t)=C(t)+B(t,T)\cdot K$, Then, $\frac{\partial P(t)}{\partial S(t)} + \frac{\partial S(t)}{\partial S(t)} = \Delta^{\text{put}}_{t}+1$ and $\frac{\partial C(t)}{\partial S(t)} + \frac{\partial [B(t,T)\cdot K]}{\partial S(t)} = \Delta^{\text{call}}_{t}+0$. Finaly, $\Delta^{\text{call}}_{t}-\Delta^{\text{put}}_{t}=1$. This relationship ...

4

Another take on the question which uses stochastic calculus [Digression] Assume deterministic and constant rates without loss of generality. Also assume the absence of arbitrage opportunities and market completeness Let $B_t$ denote the time-$t$ value of a risk-free money market account in which 1 unit of currency $C$ has been invested at $t=0$: \begin{...

4

I'd like to answer this question with one addition: Benefits outside the finance domain. Financial mathematics, if you mean not only mathematical models but also computational finance in this encapsulation, is the ultimate numeric field to improve one's skills in statistics, math and computation (coding, algorithms etc.). The trick to finance is its rules ...

4

As a start: I am sure some asset managers don't know too much mathematical finance and do a good job. They exist. On the other hand as a mathematician I see mathematics (and classicial mathematical finance - math.fin.) still in various areas and sometimes it is not that easy to draw the line. Let's look at examples: If you think of stocks then mathematical ...

4

The question above looks somewhat confused. Where's the error term? A recipe for a standard calculation It's customary to work with monthly returns. For each portfolio $i$, calculate monthly excess returns $R^x_{i,t} = R_{i,t} - R^f_t$ where $R^f_t$ denotes the 1-month risk free rate. Calculate or download the monthly excess return of the market \$R^m_t - ...

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