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7

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 ...


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}\...


6

Yes the option premium is $13. Premium just means how much you have to pay for the option The premium is paid upfront, so whenever the buyer enters the contract/buys the option European options can only be exercised at maturity. An American option is the type of option that can be exercised before maturity at any time. A Bermudan option can be exercised at a ...


5

Completely depends on the asset class. For currencies (including GBP/USD) the spot market is an order of magnitude more liquid than forwards, futures or options. However, some currencies with trading restrictions have a non-deliverable forward contract which can be much more liquid than the spot market for offshore investors (e.g. INR, KRW, TWD). ...


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 ...


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) ...


3

We assume that the price at time $t$ of a zero-coupon bond, with maturity $u$ and unit face value, is of the form \begin{align*} f(u-t, r_t, x_t) = E\left(e^{-\int_t^u r_s ds}\mid \mathcal{F}_t\right). \end{align*} Note that \begin{align*} M(t, r_t, x_t) &\equiv f(u-t, r_t, x_t) e^{-\int_0^t r_s ds} \\ &=E\left(e^{-\int_0^u r_s ds} \mid \mathcal{F}_t ...


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

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....


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 ...


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 ...


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:...


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 ...


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

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 ...


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, ...


1

The reason vega is used like this in quoting a spread is two fold. First, vega gives the change in price with respect to a change in volatility. So when you obtain a bid ask volatility you can multiply by vega to get the bid ask in dollars. It is as if you are pricing two different options, one with a lower vol and one with a higher vol. The second reason ...


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. ...


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