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3

Gonzalez-Perez (2015) Model-free volatility indexes in the financial literature: A review makes some remarks on this topic in section 2.2. Andersen, Bondarenko & Gonzalez-Perez (2013) identify a new error source in VIX that generates a significant number of jumps in the volatility index unconnected with the underlying volatility process and that ...


2

Approximating implied volatility of European options can be done in a few ways--this is just one. Below is a python implementation that uses Newton Raphson. You can use the implied_volatility function to find the approximate implied volatility. You can then check it by plugging the output from that back into the option_price function. import numpy as np ...


1

Expected volatility in the underlying price over the life of the option is a major component of the BSM option pricing model. When you calculate the volatility based on the current market price, you're figuring out what the market thinks the volatility would be, that's why it's called implied volatility. So to answer your question, you can either assume a ...


1

You can simply apply formula (3.4) in Brigo and Mercurio's book (page 56). There is a simple put-call parity for the prices of European-style options written on zero-coupon bonds, i.e. \begin{align*} \mathbf{ZBP}(t,T,S,X) = \mathbf{ZBC}(t,T,S,X) -P(t,S)+XP(t,T). \end{align*} The formula is kind of identical to the standard equity put call parity where you ...


2

I think the variance of the instantaneous shifts in the spread is meant: $V \left[ dX \right]=V \left[ dS_1-dS_2 \right]$ And the individual variances (in the conditional and local sense) are: $V \left[ dS_1 \right]= \sigma_1^2 S_1^2dt$ $V \left[ dS_2 \right]= \sigma_2^2 S_2^2dt$ And the covariance term is, assuming the two Brownians are correlated:...


2

I am not sure what you are trying to do, but I think you are trying to use the Modified Euler Method to find the option value. If the Delta at $S=52$ is $0.7041836$ the Delta at $S=53$ can be approximated as $0.7041836+(53-52)0.04429147=0.74847507$ The Delta to be used in the modified Euler method (or Heun Method) is half-way between these i.e. $(0....


8

Using our good friend Taylor, we know that \begin{align*} C(S+\Delta_S)\approx C(S)+\Delta_C\Delta_S+\frac{1}{2}\Gamma_C(\Delta_S)^2, \end{align*} where $\Delta_C$ and $\Gamma_C$ are the call's sensitivities and $\Delta_S$ a small change in the price of the underlying asset. In your example, $\Delta_S=1$ and thus, \begin{align*} C(52+1) &\approx 5.057387 ...


0

$E_0[Y_{\lambda,t}] = 1\,\, \forall t$, hence $Y_t$ is a martingale. Hint: Look at the arithmetic moments section of this wiki page on lognormal distribution


0

We define the process $Y_t=Y(t,S_t)$ as follows: $$Y_t=\left(\frac{S_t}{S_0}\right)^\lambda \exp\left\{-\left(r\lambda-\lambda(1-\lambda)\frac{\sigma^2}{2}\right)t\right\}$$ Let: $$\alpha=\lambda\left(r-(1-\lambda)\frac{\sigma^2}{2}\right)$$ Then by Itô's Lemma: $$\text{d}Y_t=-\alpha Y_t\text{d}t+\frac{\lambda}{S_t}Y_t\text{d}S_t+\frac{1}{2}\frac{\lambda(\...


1

Under the condition $r=\frac{\sigma^2}{2}$, it is true that $S_t = S_0e^{\sigma W_t}$. Since \begin{align*} E\Big( S_T - \min_{0 \le t \le T} S_t\Big) = E\big( S_T\big) - E\Big(\min_{0 \le t \le T} S_t\Big), \end{align*} what you need is the expectation $E\big(\min_{0 \le t \le T} S_t\big)$. Note that \begin{align*} \min_{0 \le t \le T} S_t = S_0e^{\sigma \...


1

The theta for puts and calls at the same strike should be the same, so it seems the SPX theta is somehow wrong. Edit: thanks @maxim, I see now what the issue is. I think the difference is coming from the fact that the options on the e-mini futures are using the Black formula where the futures price is held constant when calculating the theta. However ...


3

By applying the Ito's lemma on $X_t^2$, you find easily the process $(X_t^2)$ satisfying $$d(X_t^2) = 2X_tdX_t +\frac{1}{2} 2 <dX_t,dX_t> = X_t^2 ( \sigma^2 dt +2\sigma dW_t)$$ so $(X_t^2)$ is a geometric Brownian motion with drift $\mu = \sigma^2$ $$\frac{dX_t^2}{X_t^2} = \sigma^2 dt +2\sigma dW_t$$ we can deduce that ( $r = 0$) $$V_0 = E^Q(X_T^2) ...


6

The stock price process $(X_t)$ is a geometric Brownian motion with drift $\mu=0$. Thus, $$X_t=X_0\exp\left(-\frac{1}{2}\sigma^2t+\sigma W_t\right).$$ Assume you have constant interest rates $r_t\equiv r$ and are interested in a European-style claim, then, using risk-neutral pricing, the time zero price of a claim paying $H=X_T^2$ equals $$ V_0 = e^{-rT}\...


2

It's best to think of the sum of $da_t A_t$ and $da_t dA_t$: $$ da_t A_t + da_t dA_t = da_t(A_t + dA_t) $$ which is the cost of rebalancing at the new price $A_{t'} = A_t + dA_t$. You don't rebalance the portfolio at $t$ but at $t'$. And the self-financing condition means you need to finance this cost by rebalancing other assets (including possibly the money ...


0

Say you start with 1000 units worth 100 equals value 100,000. You buy 100 more but the price falls to 90, giving you 1100*90 equals 99,000 value. Change in Value = dP * U = -10 * 1000 = -10,000 P * dU = 100 * 100 = 10,000 dP * dU = -10 * 100 = -1,000 The first two, the change in value of existing units, and the value brought in with new purchases ...


0

Going back to the original question, there is no static replication. This is clear from the first answer above which states that a call spread with an infitesimal difference between their strikes is needed. To arrive at an approximate replication, we need the probability of the underlying fixing in the USD 5 interval between the calls to be small, hence: ...


1

Below is the link to curated topics related to programming in quant finance. https://github.com/wilsonfreitas/awesome-quant (this contains all programming languages(python, R, C++ etc) and there resources in quant finance). Apart from quantopian.com as mentioned above you can try quantiacs(https://www.quantiacs.com/) (which is actually a quant finance-algo ...


3

I'd highly recommend the http://www.quantopian.com platform and the correspoding forum there, e.g. the trading strategies thread. You could pick a topic and also use a concrete example in job interview.


2

To add a bit to Will Gu's answer: Compute $\mathbb{E} \left[ \left. S_T \right| S_T > K \right]$ using the fact that $S_T$ is lognormally distributed with mean $ln(S_0) + (r - \sigma^2/2)T$ and variance $\sigma^2 T$. Then find the pdf of the lognormal distribution on, e.g., Wikipedia, and compute the expectation integral. You may find the following ...


4

When you delta hedge a currency option, you are hedging in both currencies. In your example, since you have a short position of $600,000 as your hedge, this would be against JPY; therefore you would be long 600,000 * 112 = 67.2MM JPY at the same time. Regarding your second question, you would still hedge the delta using the underlying of the option--in ...


1

The market will quote Call and Put options prices within a bid-ask spread. In order to imply the volatility, one may choose to use the bid, the ask, or the mid. Although the mid is a better idea in general, there is no right choice. The point is that there is always a spread in the implied volatility. Now, the Put-Call parity only holds within the a spread. ...


1

FX differs from other asset classes in that certain amount of market manupulation by cebtral banks is the norm. For almost any currency, if its exchange rate versus other currencies moves outside a certain band, the central banks will try to intervene, usually by just buying the currency in the market. The bank's goal is not to make money by speculation, ...


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