Questions tagged [stochastic-volatility]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
60 views

Mid-curve swaption pricing - how to get the spread vol?

I believe I understand the following (from the accepted answer to the Quantitative Finance question called "volatility of a mid curve option"): A swaption in which the underlying swap ...
  • 1
2 votes
0 answers
120 views

Implied volatility skew decay over expiry

I seem to remember the implied volatility skew of European options decreases as the expiry increases. It is true for the Heston model under some approximation. What are the good references that prove ...
  • 2,511
0 votes
1 answer
49 views

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 $(\...
  • 237
2 votes
0 answers
191 views

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
167 views

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 ...
0 votes
0 answers
82 views

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 ...
1 vote
1 answer
174 views

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 ...
0 votes
0 answers
112 views

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
0 votes
0 answers
48 views

Milstein Discretization of Heston Model

Given the following representation of the Heston Model: $$d\left(\begin{array}{l}S_{t} \\ V_{t}\end{array}\right)=\left(\begin{array}{c}\mu S_{t} \\ \nu-\varrho V_{t}\end{array}\right) d t+\left(\...
5 votes
1 answer
208 views

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 ...
4 votes
2 answers
531 views

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 ...
1 vote
1 answer
79 views

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 ...
  • 545
7 votes
1 answer
168 views

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
3 votes
1 answer
311 views

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 ...
  • 466
4 votes
1 answer
354 views

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? ...
  • 696
-1 votes
1 answer
111 views

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) ...
3 votes
0 answers
101 views

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 ...
1 vote
1 answer
250 views

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 ...
  • 57
1 vote
1 answer
121 views

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 ...
1 vote
0 answers
61 views

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
3 votes
0 answers
114 views

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 ...
  • 545
2 votes
0 answers
135 views

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_{...
  • 355
0 votes
0 answers
87 views

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 votes
0 answers
114 views

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, ...
  • 31
2 votes
2 answers
310 views

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 ...
1 vote
1 answer
125 views

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
4 votes
0 answers
76 views

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... ...
  • 622
2 votes
0 answers
193 views

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 ...
7 votes
0 answers
123 views

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 ...
2 votes
0 answers
132 views

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 ...
0 votes
0 answers
46 views

Sabr model asset classes

For the SABR stochastic volatility model, can it be used for any asset classes that exhibit a volatility smile (not skew) or is it just restricted to interest rate derivatives? Thanks
1 vote
1 answer
242 views

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
135 views

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 ...
  • 193
0 votes
0 answers
167 views

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 ...
  • 805
3 votes
1 answer
152 views

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 ...
  • 805
2 votes
0 answers
200 views

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 ...
3 votes
1 answer
134 views

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 ...
  • 125
2 votes
1 answer
208 views

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
126 views

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 ...
3 votes
0 answers
75 views

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 + \...
  • 2,336
4 votes
0 answers
172 views

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 ...
7 votes
2 answers
403 views

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}\...
  • 696
2 votes
0 answers
181 views

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^...
  • 540
2 votes
1 answer
100 views

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)...
  • 113
1 vote
1 answer
438 views

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 ...
  • 356
0 votes
1 answer
195 views

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 ...
  • 43
4 votes
0 answers
206 views

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}{...
  • 446
2 votes
0 answers
103 views

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. ...
3 votes
1 answer
253 views

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 votes
1 answer
154 views

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

1
2 3 4 5
7