Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
367
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Ito's lemma in stochastic volatility models [closed]
I couldn't help but notice that in all stochastic volatility models articles I consulted, whenever Ito lema is applied with a process of the sort
$$\frac{d S_t}{S_t} = \sigma_t d W_t $$
With $(\...
- 267
2
votes
0
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343
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Model based PnL explain for FX Options
In FX options the vol surface for a given maturity is usually described by three or five points, I.e. Atm, 25 delta risk reversal and butterfly and 10 delta risk reversal and butterfly. Then models ...
- 253
3
votes
1
answer
270
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When calculating VIX, how to deal with the problem of asymmetry of put and call data?
I'm trying to calculate the VIX index according to the methodology of CBOE. I am looking at commodity options. I found that at some time, like at this minute, there are 13 call options out of the ...
- 31
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0
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179
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Stochastic vs. local volatility model choices for greeks
As a follow-up of another question (which is I feel slightly separate, hence a new question). Assume we want to fit a volatility surface with the goal of calculating good greeks, not prices. We can ...
- 113
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1
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377
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Delta hedging when volatility is stochastic
From my understanding in a BSM world you can make a bet on volatility using options and delta hedging with the underlying.
If you think realized volatility of the underlying will be higher than the ...
- 61
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236
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How to price american barrier with Local-Stochastic Volatility
I have attended a conference where one speaker mentioned that the market standard to price FX and Equity derivatives is now the Local-Stochastic volatility model.
I understand this class of model is a ...
- 113
5
votes
1
answer
224
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Independence vs correlation in stochastic vol models
I am struggling a bit with some basic stuff lately:
Consider a SV model
\begin{align}
dS_t &= \sigma_t S_t dW_t \\
d\sigma_t &= b(\sigma_t,t) dZ_t
\end{align}
with $dW_t dZ_t = 0$.
I know that ...
user34971
4
votes
2
answers
641
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Introductory material for getting started with local and stochastic volatility modelling
Are you able to provide some suggestions for resources to get started with non-flat volatility modelling? The models I am interested in are the likes of CEV, Heston, SABR etc.
I have tried looking ...
- 152
1
vote
1
answer
135
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Pricing & hedging vanilla interest rate options with SABR LMM
Are there any advantages of pricing and hedging plain vanilla interest rate options with more complex SABR LMM instead of simpler SABR model? Should one always go with the SABR LMM as a universal ...
- 764
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1
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217
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Why is the LMM with mixture dynamics (Brigo & Mercurio) inconsistent for the pricing of exotics?
I am reading about the LMM with lognormal-mixture dynamics. Consider the following dynamics for the forward rate $F_{i}(t)$ fixing at $T_{i-1}$ and paying at $T_i$:
\begin{align}
dF_{i}(t) = (F_i (t) +...
- 428
4
votes
1
answer
2k
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When to use a Local Vol model vs Stochastic Vol Model?
I'm new to volatility modeling, I'm struggling to understand when to use a Local Vol model and when to use Stochastic Vol Model, Also now we use a hybrid model combining the two models ? Can someone ...
- 616
5
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1
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620
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ATM Implied Volatility and Expected Variance
This answer claims that
$$\sigma^2_{ATM}\approx E^Q\left(\frac{1}{T}\int_0^T\sigma^2_t dt\right)$$
ie implied ATM vol = risk-neutral expectation of integrated variance.
Is there some proof available? ...
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1
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196
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Help needed in replicating FX Implied Vol Surface
I am relatively new to this area and am doing some self studying on SLV model. I am however getting stuck on trying to replicate this implied vol surface (which I will use to calculate the local vol)
...
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150
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Single barrier options in stochastic volatility models
In this note/sketch, I derive among others a closed-form formula for an up and in put (UIP) in stochastic volatility models of the form
$$
dS(t) = \sigma(t) S(t) \left[ \rho dW(t) + \sqrt{1-\rho^2} dZ ...
user34971
1
vote
1
answer
824
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Calibration and pricing with the Stochastic Local Volatility model
I'm reading the stochastic local volatility model literature, e.g., the Heston Stochastic Local Volatility model (https://ir.cwi.nl/pub/22747/22747D.pdf); but I'm a bit unsure about its calibration ...
- 301
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1
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431
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Deriving vol of vol from volatility futures price
From Colin Bennet's trading volatility (pg 117), he says:
"A forward on a volatility future is short vol of vol. This means it is possible to back out the implied vol of vol from the price of ...
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0
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101
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Best Way To Compute the Volatility Risk Premium
I'm trying to come up with a measure for the volatility risk premium (VRP) for a strategy I want to implement, but I'm not entirely sure how to proceed. My situation is as follows.
The underlying is ...
- 41
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0
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282
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SABR LMM for RFR
Is there a research showing a way to use SABR LMM with new RFRs such as SOFR, i.e. pricing exotic path-dependent RFR derivatives with volatility smile and skew?
I'm aware that
Looking Forward to ...
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2
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0
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351
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Pricing a put-option in the Heston Model
Assume the Heston Model with dynamics under the martingale measure $Q$ given by
\begin{align}
dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\
dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{...
- 548
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0
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184
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GARCH option pricing
I have been trying to implement GARCH(1,1) model for pricing call options. Suppose I have calibrated Garch(1,1) model for modelling the conditional volatility using the historical data of an equity ...
2
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0
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165
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Calculating model-free implied volatility [closed]
I am trying to come up with model-free implied volatility as in Britten-Jones, M. and Neuberger, A. (2000) Option Prices, Implied Price Processes, and Stochastic Volatility, Journal of Finance, 55, ...
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2
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886
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Local Vol vs Stoch Vol Option Pricing
This is an interview question:
Imagine you have a double knock-out barrier option: the current spot is 100, the lower barrier is 80, and upper barrier is 120. The barrier is continuous, meaning that ...
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1
answer
149
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Calculating Expectation of Stochastic Volatility
I have a question while reading THE NELSON–SIEGEL MODEL OF THE TERM
STRUCTURE OF OPTION IMPLIED VOLATILITY
AND VOLATILITY COMPONENTS by Guo, Han, and Zhao.
I don't understand why the above equations ...
- 11
3
votes
0
answers
111
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Comparison of Option-Pricing Models (volatility models) vs Product-Mapping
I scoured this forum, looking for some indicative (updated as of year 2021) comparison of volatility/option-pricing models. There were some, but they seem dispersed and lacking in general details...
...
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2
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0
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208
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How do you hedge volatility risk?
Suppose I model an asset $S_1(t)$ under a stochastic volatility model. To price an option on $S_1$, I must assume the existence of an asset $S_2$ that is used to hedge against changes in the ...
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0
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133
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Implied vol bounded if and only if instantaneous vol bounded
I'd like to show that in diffusion models IV is bounded iff instantaneous vol is bounded if there is to be no arbitrage. So, assume a model under the pricing measure of the form
$$
dS_u = \sigma_u S_u ...
user34971
2
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0
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173
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Are Stochastic Differential Equation diffusion terms always invariant under a change of measure?
I'm struggling with learning change of numeraire, and stochastic differential equations. I'm reading the beginning of Brigo and Mercurio's Interest Rate Models- Theory and Practice, and I'm on the ...
- 163
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1
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411
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Rogers Satchell Volatility
I am trying to implement Roger Satchell volatility in Go, but my results do not match reality... I have been at this all day, but cannot find my error. The 30 day Rogers Satchell vol is at 8.75%, but ...
1
vote
1
answer
156
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Stochastic (volatility) models with the elements of fundamental analysis - are there such models and why not?
I read about stochastic volatility models (e.g. https://en.wikipedia.org/wiki/Stochastic_volatility) and those models are quite simple, but the most important feature is that parameters are quite ...
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534
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Forward volatility smile: Local Volatility vs Stochastic volatility
I was reading this great answer: What are the advantages/disadvantages of these approaches to deal with volatility surface?
And I have the following question:
How to show that the forward volatility ...
- 853
3
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1
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632
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Correlation Spot Vol - when is it important?
I know that a local volatility model does not allow to control the correlation between Spot and Vol. I know also that the correlation Spot Vol is important for products like autocalls.
Why is ...
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0
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371
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Are there any public implementations of realized kernels? (preferably in Python)
looking to implement a realized kernel model to forecast realized variance of around ~140 equities and indices in Python given order book data.
I have read "Realised Kernels in Practice: Trades ...
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1
answer
201
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Reduced volatility in local stochastic volatility model
in Local Stochastic Volatility models I always read or hear "first the stochastic volatility model is calibrated to reduced vols and then the local volatility model corrects it" also I head ...
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1
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343
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Question about pricing forward start option with Heston Monte Carlo
I'm trying to price a forward start option with payoff $\Big(\dfrac{S_{T_2}}{S_{T_1}}-1\Big)^+$ with Heston Monte Carlo.
Heston Model:
$$ dS_t = rS_tdt + \sqrt{v_t}S_tdW_t^1$$
$$ dv_t = \kappa(m-v_t) +...
- 121
2
votes
1
answer
212
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HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$
I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway ...
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0
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86
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What is the relationship between the estimated GARCH(1,1) conditional volatility and the true conditional volatility
Suppose that the data has been generated by a GARCH(1,1) model, i.e.
\begin{align}
y_t &= h_t \epsilon_t, \; \epsilon_t \sim N(0,1) \\
h_t &= \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \...
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335
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Why calibrate volatility Models to volatility surfaces rather than underlying's historical price data?
I'm trying to grasp the rationale for calibrating stochastic volatility models (i.e. Heston model) to empirical IV data from market prices. Doesn't this assume that the options are fairly priced and ...
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2
answers
492
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Heston: Variance of Integrated Variance
Consider the standard Heston model\begin{align*}
dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\
dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\
dBdW&=\rho dt.
\end{align*}
Computing $\mathbb{E}\...
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321
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Derivation of Bergomi model
In Stochastic Volatility Modeling, L. Bergomi introduces in Chapter 7 the pricing equation (7.4) :
$$
\frac{dP}{dt}+(r-q)S\frac{dP}{dS}+\frac{\xi^t}{2}S^2\frac{d^2P}{dS^2}+\frac{1}{2}\int_t^Tdu\int_t^...
- 747
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1
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145
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Can you shift a standard libor market model with regard to only at-the-money options?
Suppose I have an LMM defined using the spot measure as in Brigo and Mercurio:
$dF_k(t) = \sigma_k(t)F_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)F_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)...
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1
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503
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Fourier transform of a European put
In book The concepts and practice of mathematical finance, in the context of illustrating the stochastic volatility model, the Fourier transform $\hat{P}(\xi, V, T)$ of a European put $P(x, V, T)$ is ...
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397
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In what cases characteristic function of (log-)price process is known?
Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
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314
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Characteristic function of the Bates model
I have a misunderstanding concerning the derivation of the SVJ model :
Firsty,I understand how to reach the final differential equation from :
\begin{gather}
dS_t = (r - q - \lambda t (e^{m-\frac{\nu}{...
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Is $C(K,S_t)$ a (local) martingale if PCS is broken?
When put-call symmetry holds
$$
P(S_t,K) = C(K,S_t) = \frac{K}{S_t} C \left( S_t, \frac{S_t^2}{K} \right)
$$
where $P$ is the market price of a put option and $C$ is the market price of a call option. ...
user34971
3
votes
1
answer
484
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Bergomi Volatility Model
I was studying on the Bergomi volatility model(using forward variance represented as $\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating ...
- 446
3
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1
answer
305
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Non-constant Volatility of the Volatility in Stochastic Volatility Models
In pricing financial derivatives, we often first assume that the volatility of the stock price is constant.
$$\mathrm{d}S(t) = \alpha S(t) \mathrm{d}t + \sigma S(t) \mathrm{d}W(t)\text{.}$$
The ...
- 437
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0
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450
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Heston Nandi Garch Implementation Problem for Python
I have a coded my own Garch class in order to implement the Heston-Nandi Garch model.
...
- 446
3
votes
0
answers
116
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Fractional Brownian Motion's Covariance Proof
Let's have the non independent Brownian motion such :
$B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$
...
- 446
2
votes
1
answer
169
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Variance swaps and the Log-Moment formula
I was looking at the paper of Raval and Jaquier The Log Moment Formula For Implied Volatility
available here : https://arxiv.org/pdf/2101.08145.pdf
On the page 4 they wrote(with $<logS>_T$ and $&...
- 446
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2
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545
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What is wrong in my Heston model's code
I am trying to code a heston model pricer.However,it seems correct at the beginning but when inserting extreme data I retrieve myself with negative probabilities or negative prices.
There is the code :...
- 446