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 ...
- 20.8k
29
votes
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 ...
- 20.8k
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 \\
&...
- 20.8k
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 ...
- 481
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, ...
- 2,109
20
votes
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}\...
- 20.8k
17
votes
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 ...
- 5,901
17
votes
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'...
- 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 ...
- 3,659
17
votes
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 ...
- 2,233
16
votes
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 ...
- 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, ...
- 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 ...
- 7,705
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 ...
- 15k
14
votes
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 ...
- 761
14
votes
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{...
- 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:
- 2,255
13
votes
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 ...
- 146
13
votes
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 ...
- 20.8k
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 ...
- 6,843
13
votes
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 ...
- 2,255
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 ...
- 14.4k
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 \...
- 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 ...
- 20.8k
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 ...
- 5,901
12
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 ...
- 11.2k
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 ...
- 5,723
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 $\...
- 7,705
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 ...
- 341
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}{\...
- 15.5k
Only top scored, non community-wiki answers of a minimum length are eligible