Questions tagged [derivatives]

A financial contract whose payoff is linked to the evolution of an underlying security.

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1answer
206 views

FX Options price vs implied vol

From the screenshot below, what is the difference between the option price by strike in the table versus the implied volatilities by delta in the chart at the bottom? https://www.investing.com/...
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6answers
10k views

How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings. However, the behavior of ...
6
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4answers
713 views

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

Develop a formula for the price of a derivative paying $$\max(S_T(S_T-K))$$ in the Black Scholes model. Apparently the trick to this question is to compute the expectation under the stock measure. So,...
11
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2answers
4k views

How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?

I have $\frac{dS_t}{S_t} = rdt + \sigma dW_t$ as usual under the money-market numéraire and I need to price options with payoffs $$(S_T f(S_T))^+$$ How do I express the stock dynamics using the ...
10
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3answers
5k views

Why discounted derivative price is a martingale?

Usually after showing that discounted stock price process is martingale under the risk-neutral measure, most authors say that this implies that the discounted derivative price process is a martingale ...
7
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2answers
414 views

What are the books in which to study the basics of the derivative financial instruments?

Books similar to Options, Futures, and Other Derivatives by John C. Hull. I need another academic book that explains the basics of quantitative finance derivatives (forward, futures, options)
14
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1answer
1k views

Derivation of VIX Formula

I've read a lot of derivations about VIX formula. I can say it is -adjusted- fair strike of variance swap. But I can't see how it goes from variance swap rate to VIX formula. In particular I can't see ...
8
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1answer
936 views

Theoretical limits for contango and backwardation

What do you think would be the theoretical limit for contango? What about backwardation? This was asked in an interview. I am still not so sure about the answer.
3
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1answer
261 views

Cap option on Libor

We denote discount factor $D(t),$ zero coupon bond $B(t,T),$ $E_t[X] = E[X|\mathcal{F}(t)]$ and $T$-forward measure $E_t^{T}[\ ]....
3
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1answer
182 views

Cox-Ingersoll-Ross: Monte Carlo Simulation

I am trying to build a Monte Carlo simulation in Excel (yes, far from optimal) for valuation of a callable bond. I have some experience with MC simulation on path dependent derivatives with stocks as ...
1
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1answer
141 views

Trying to understand brazil derivatives market

I am trying to get a better understanding of brazil's market, specially derivs. I know they have certain instruments such as "Convertibility" (based on the yields spread between onshore and ...
11
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2answers
4k views

Why is the SABR volatility model not good at pricing a constant maturity swap (CMS)?

I have heard that the SABR volatility model was not good at pricing a constant maturity swap (CMS). How is that?
4
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2answers
2k views

Delta Hedging with fixed Implied Volatility to get rid of vega?

I'm wondering if i should use a floating IV or a fixed IV to delta hedge my options every day. I've read this post but would like different information : Delta Hedging with fixed Implied Volatility ...
12
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2answers
5k views

Why a self-financing replicating portfolio should always exist?

According to my understanding the derivation of the Black-Scholes PDE is based on the assumption that the price of the option should change in time in such a way that it should be possible to ...
7
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5answers
605 views

Heston Model Integration Oscillations

Is there a way to reduce oscillations for the numerical integration when evaluating the Heston model. I am pricing a series of 5000 options scattered over the Heston model parameter space and I find ...
10
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2answers
442 views

Differences between main classes of interest pricing derivatives models

There seems to be 3 main classes of interest rate pricing models: 1) Short rate models, 2) Heath Jarrow models and 3) Libor Market Model. My book doesnt seem to explain why we need all these different ...
8
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1answer
372 views

Replicating a portfolio with a certain payoff function

Assume there are two stocks $S_1$ with price $p_1(t)$ and $S_2$ with price $p_2(t)$ where $t$ indicates time. Assume, there is a hypothetical derivative $D$, which is such that, price of $D$ at a time ...
4
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4answers
400 views

Using a Constant as a Numeraire

Please provide steps to justify the below. 1) Can we use a constant as a numeraire? Related Question: Scaling Stock Price and Strike etc. by a Constant The rest of standard Geometric Brownian ...
5
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3answers
757 views

How to hedge a derivative that pays the reciprocal of the stock price?

1) Suppose S is the stock price, how to hedge a derivative that pays $1/S_t$ at time $t$? 2) Suppose there will be a dividend of amount $d$ between $t$ and $T$, how to hedge a derivative that pays $...
2
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1answer
3k views

How to understand the market price of risk

Consider the stochastic vol: $$dS = \mu Sdt + \sigma SdW_1$$ $$d\sigma = p(\sigma,S,t)dt + q(\sigma,S,t)dW_2$$ $$dW_1dW_2 = \rho dt$$ We want to obtain the price of option $V(\sigma,S,t),$ we use the ...
1
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2answers
8k views

Long/Short Vega and Option Positions

Why do you get long vega when you buy an option and short vega when you sell an option? I would have thought that for both buying and selling options the vega would change according to whether the ...
1
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2answers
359 views

Volatility smile shows individual Put and Call IV or combination

IV is calculated per strike AND option type basis(for example WTI 50 CALL its x and WTI 50 PUT its y). The question is when its shown in "Smile" its just shown on strike basis, so does that mean lower ...
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1answer
4k views

Why does increased stock borrow costs decrease a stock's forward price?

The author in this article -- http://streetwiseprofessor.com/?p=7294 -- states that an increase in stock borrowing costs decreases a stock's forward price: In the absence of manipulation, the ...
6
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2answers
291 views

Radon Nikodym derivative when changing numeraires

I note from Wikipedia that if $Q$ and $Q^N$ are two measures corresponding to numeraires $M$ and $N$, then the Radon Nikodym derivative is given by: $$\frac{dQ^N}{dQ} = \frac{M(0)}{M(T)}\frac{N(T)}{N(...
5
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1answer
295 views

The positivity of the market price of risk

Does the market price of risk, be it of stochastic volatility, interest rate or equity return, have to be positive? What is the rationale if it does?
2
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2answers
705 views

The dice game and derivatives trading

I happened to a interview question: Give a equal dice, you will gain the money which is the number you roll, then how much will you pay for the game. Naturely, the answer is 3.5. But the interview ...
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2answers
256 views

Structuring and Customization

It seems complex derivatives in particular exotic options are not available at any retail broker. Can a regular retail trader get access to these instruments? Maybe through prop firms or banks? ...
1
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1answer
281 views

Rebasing of Cap Volatilities

I recently found this article where towards the end the author describes a method to rebase cap volatilities. Their method works like this: for a fixed strike assume that you are given the implied ...
1
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0answers
95 views

Scaling Stock Price and Strike etc. by a Constant

Please provide steps to justify the below. 1) If the stock prices, strike and other price related parameters are scaled by the same constant, will the derivative price scale accordingly? I would ...
3
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1answer
125 views

What is the Radon-Nikodym derivative in the Heston model?

It is clear to me that $$ \frac{dQ}{dP} = e^{-\lambda W_T-\frac{\lambda^2}{2}T}$$ is the Radon-Nikodym derivative that defines the change of measure in the framework described by Black and Sholes. But ...
2
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3answers
211 views

Why don't we take the differential to the Delta in the Delta hedge-portfolio

For option $V(S,t)$ with underlying asset $S$, we have a hedge portfolio $$\Pi = V(S,t) - \Delta(S,t)S$$ I always confuse here, when we take the differential of $\Pi$ $$d\Pi = dV -\Delta dS$$ why ...
2
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1answer
345 views

What is the filtration described?

What is the filtration $(\mathfrak{F}_t)$ encircled below? Is it $(\mathfrak{F}_t) = (\sigma(W_t)) = (\sigma(\tilde{W_t})), t \in [0,T]$? Or is it $(\mathfrak{F}_t) = (\sigma(\hat{W_t})), t \in [0,T]...
2
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1answer
432 views

Cap price as bond options

I am currently struggling with model calibration of the Hull-White (or Vasicek) model to Caps and Floors. My main problem is that I am confused about the notation. In Brigo & Mercurio (2006, p. ...
1
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1answer
301 views

Upper bound option price in volatility dimension

All, I have a theoretical question about the value of an option when spot price goes to infinity as a function of volatility going to infinity. I know that for a call option: The option value ...
0
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1answer
193 views

Reference for why a derivative is a derivative and not say an insurance contract

I recently spoke to an options trader that tried to demonstrate option pricing by considering a random walk of balls dropping down a lattice so the underlying stochastic process is a simple random ...
0
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1answer
98 views

Extensive list of financial derivatives and what method is used to value them

What I'm imagining is a long list of different types of financial instruments traded on the market along with the model(s) that is industry standard for valuing it. Something like: European equity ...
0
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2answers
2k views

Proof Black Scholes Theta

I saw the following proof of theta in a paper I read, and I thought it looked pretty neat. Unfortunately I don't understand the step that they do. This is what they do: Now, I don't get how they go ...
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1answer
1k views

Calculating arbitrage- S&P 500 stocks vs S&P 500 Index future?

How exactly would I go about investigating whether the S&P 500 stocks were currently over-valued compared with the price of the S&P 500 Index futures contract? Is it just a case of taking each ...