# Tag Info

Accepted

### Arbitrage opportunity interview question

A similar question for put option has been discussed in this question: Finding Arbitrage in two Puts. Basically, the call option payoff is a convex function of the strike. Then the call option price ...

For a sufficiently smooth function $f$, positive constant $a$, and $x>0$, Note that, \begin{align*} f(x) -f(a) &= \int_a^{x} f'(v) dv \\ &= \int_a^{x} \big[f'(v) -f'(a) + f'(a) \big] dv \\ &...

### Value of Call Option as Volatility goes to Infinity

The value of a call option does not go to infinity as the volatility goes to infinity. It tends to the discounted value of the forward $F=S_0 e^{(r-q)T}$, which when the dividend yield is zero, ...

### What is the importance of alpha, beta, rho in the SABR volatility model?

We created the SABR model because we realized that (a) option values were nonlinear in the volatility, and (b) volatilities are stochastic. This means that if one had an option (or portfolio of ...
Accepted

### Gamma Pnl vs Vega Pnl

For an option with price $C$, the P$\&$L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$, is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\...
Accepted

### How much can be said about the Greeks without picking a model?

Find the topic of model-independent properties of option prices very interesting as well. Here are some results that I am aware of and the respective references in the literature. Some are already ...
Accepted

### Problems with local volatility models (vs stochastic volatility models)

1. What does it mean by the vol surface is the current view of vol? The local volatility model is calibrated to vanillas prices (and equivalently their implied volatilities), which reflect the market'...

### What is the importance of alpha, beta, rho in the SABR volatility model?

Let's relabel this as What (TF) is SABR? Alpha, Beta and Rho are the point of the model. So explaining them is explaining the model. A model of two processes Unlike earlier models in which the ...
Accepted

### What's the most efficient way to store options and time series data for backtesting?

If the only purpose is to backtest with the data, the primary (in some cases, only) access pattern is to seek to a start time and read all of the data serially through to an end time. Then, there is a ...
Accepted

### Barrier option (autocallable) Vega profile

You have a multidimensional problem - there isn't an answer of "this is what the greeks look like" for all cases, because it depends on the various levels of the different parameters. For example, if ...

### Mark Joshi's book - quant interview questions

For large values of the spot S, this payout goes to infinity like the square of S. However, the hedging instruments available are vanilla options, which go like S to the first power. Mathematically, ...

### Which of the three options is the most valuable?

The greater the optionality, the greater the price. Hence, in your case: a European call "gives" you optionality on a single day; a Bermudan call "gives" you optionality on a series of days between ...
Accepted

### Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$

I provide a solution in three steps. The first step carefully outlines how to split up the expectation and what new measures are used. This first step does not require any special model assumption ...
Accepted

### Bergomi: Skew arbitrage

Great question. Let me try to provide some insights and thoughts regarding the points and questions you raised. It may not be a full answer but hopefully it will help connecting the contents in the ...
Accepted

### Path-dependent options valuation

Risk-neutral pricing A time-$T$ payoff is an integrable, $\mathcal{F}_T$-measurable random variable $\xi$. The value process of the discounted payoff is then a $\mathbb{Q}$-martingale, i.e., \begin{...
Accepted

### What are the main flaws behind Ross Recovery Theorem?

This is a loaded question. Ross' recovery theorem has both flaws and insights. The single answer thus far did a great job of addressing the flaws from an economics perspective. No one questions that ...
Accepted

### How to exploit calendar arbitrage?

The answer by @HenriK is certainly correct. However, for justification, technique such as the Jensen inequality is needed. For example, since $x^+$ is a convex function, assuming zero interest and ...
Accepted

### Why does it take so many lines of code to price even the simplest of options with QuantLib

I've been using QuantLib for quite a while. Let me share some experience: QuantLib is a highly sophisticated quantitative framework. It can do much and much more than a simple pricing of European ...
Accepted

Basically, the author is saying that the delta of an option, $dC/dS = \frac{\partial C}{\partial S} + \frac{\partial C}{\partial v}\frac{\partial v}{\partial S}$, where the \frac{\partial C}{\... 13 votes Accepted ### Least Squares Monte Carlo To compute the price of an American option or a callable instrument in general, at each potential exercise date, one is required to compare its continuation value (discounted risk-neutral expectation ... 13 votes Accepted ### Options Market Making Used Implied Volatility Surface Your question is twofold How a market maker should adjust its quotes on a vol surface with respect to his inventory? How to adjust the vol surface when a new trade is observed on the markets? Let me ... 12 votes Accepted ### Link between Vega and Gamma Under the Black-Scholes model, \begin{align*} Gamma &= \frac{N'(d_1)}{S \sigma \sqrt{T-t}}\\ Vega &= SN'(d_1) \sqrt{T-t}. \end{align*} Then, it is easy to see that \begin{align*} Vega = S^2 \... 12 votes ### How to derive the price of a square-or-nothing call option? I provided an answer, based on an elementary approach, to an exactly same question yesterday. However, that question has disappeared, even though I like to keep a record for what I wrote. I would ... 12 votes Accepted ### Breeden-Litzenberger formula for risk-neutral densities I assume that for approximating the second derivative of the call priceC (K,T) $at the bounds of the strike domain (see first 2 "if" cases of the last for loop of your code) you tried to set up ... 12 votes Accepted ### Prove that the butterfly condition is always greater than zero You generally can't simply subtract two inequalities as you did in your attempt. Here are two approaches to solve your problem: No-Arbitrage Argument Assume that the initial value of the Butterfly ... 12 votes Accepted ### Why is realized volatility typically lower than implied volatility? Consider what happens when IV is lower than realised vol. The person long the IV would make money. So there would ideally be no one selling IV if it's lower than realised vol on an average. Next if ... 12 votes Accepted ### Why do institutional Traders prefer Short Selling instead of Buying Puts? In my opinion, professionals mainly trade options if they want to trade the volatility. I believe there is a mathematical proof that shows that if the realized underlying volatility between the option ... 12 votes ### Mark Joshi's book - quant interview questions I suspect this is because, conditional on being in-the-money, the payoff of your option is convex in stock price$-$whereas for a vanilla call, the payoff is linear. As a consequence, the delta$\...
With the underlying asset price $S_t$ following a geometric Brownian motion with drift $\mu$ (risk-neutral or otherwise) , we have at time $t = T$, S_T = S_0e^{(\mu- \frac{\sigma^2}{2})T}e^{\sigma ...
I'm not completely certain from your question, but I'm going to assume you have a basket of $n$ stocks with prices $S_0(t)$ to $S_n(t)$, and you want to price an option with payoff at $C(\tau)$ at ...