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31 votes
Accepted

Variance replication using options

Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance ...
Gordon's user avatar
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29 votes
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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 ...
Gordon's user avatar
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26 votes

Carr-Madan Formula

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 \\ &...
Gordon's user avatar
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23 votes

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 ...
Patrick S Hagan's user avatar
22 votes

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, ...
Dom's user avatar
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20 votes
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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}\...
Gordon's user avatar
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17 votes
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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 ...
LocalVolatility's user avatar
17 votes
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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'...
byouness's user avatar
  • 2,160
17 votes

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 ...
Phil H's user avatar
  • 3,659
17 votes
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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 ...
databento's user avatar
  • 2,233
16 votes
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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 ...
will's user avatar
  • 2,521
16 votes

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, ...
Peter A's user avatar
  • 494
14 votes

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 ...
Daneel Olivaw's user avatar
14 votes
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 ...
Kevin's user avatar
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14 votes
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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 ...
SI7's user avatar
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14 votes
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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{...
Kevin's user avatar
  • 15k
13 votes

How to replicate a digital call option

A digital call option (cash-or-nothing) can be replicated with two call options with different Strike. When we make the delta infinitely small and assume we have arbitrary strike prices. We get:
ABCD's user avatar
  • 2,255
13 votes
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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 ...
peter carr's user avatar
13 votes
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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 ...
Gordon's user avatar
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13 votes

Why the expected return rate of a stock has nothing to do with its option price?

Because you can hedge. Once you have delta hedged, the pay-off is symmetric about up and down moves so drift doesn't matter. Also the delta-hedged call and the delta hedged put have to have the same ...
Mark Joshi's user avatar
  • 6,843
13 votes
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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 ...
ABCD's user avatar
  • 2,255
13 votes
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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 ...
Quantuple's user avatar
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12 votes
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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 \...
Gordon's user avatar
  • 20.8k
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 ...
Gordon's user avatar
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12 votes
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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 ...
LocalVolatility's user avatar
12 votes
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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 ...
lehalle's user avatar
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12 votes
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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 ...
Jan Stuller's user avatar
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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 $\...
Daneel Olivaw's user avatar
11 votes

The meaning of Ornstein-Uhlenbeck parameters

$\theta$ is the "mean" for this process. If $X_t > \theta \implies (\theta - X_t) < 0 $, which means that the drift for the process is negative and tends towards $\theta$. The opposite case can ...
eltigrechino's user avatar
11 votes
Accepted

Skew and shadow delta

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}{\...
dm63's user avatar
  • 15.5k

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