Download - Local and Stochastic volatility
Local and Stochastic Volatility Models
Swati Mital11/11/2015
05/01/2023Swati Mital
Agenda
Black Scholes Pricing Model Holes in the Black-Scholes Model Importance of Volatility Estimation Stylized Facts about Volatility Volatility Models
• Local Volatility Models• Stochastic Volatility Models
Stochastic Volatility Jump Diffusion
05/01/2023Swati Mital
Agenda
Black Scholes Pricing Model Holes in the Black-Scholes Model Importance of Volatility Estimation Stylized Facts about Volatility Volatility Models
• Local Volatility Models• Stochastic Volatility Models
Stochastic Volatility Jump Diffusion
05/01/2023Swati Mital
Review of the Black-Scholes Merton Option Pricing Model
Black-Scholes PDE for an option price
In the Black-Scholes model,
the constant is the (spot) volatility of
As an aside, we can express as
05/01/2023Swati Mital
Agenda
Black Scholes Pricing Model Holes in the Black-Scholes Model Importance of Volatility Estimation Stylized Facts about Volatility Volatility Models
• Local Volatility Models• Stochastic Volatility Models
Stochastic Volatility Jump Diffusion
05/01/2023Swati Mital
‘The Holes in Black-Scholes’
When we calculate option values using the Black-Scholes model, and compare them with option prices, there is usually a difference… The main input that may be wrong is volatility… Different people will make different volatility estimates… We know some specific problems with the formula… A stock’s volatility changes in unexplainable ways, but it also changes in ways related to the price of the stock. A decline in stock price implies a substantial increase in volatility, while an increase in the stock price implies a substantial decline in volatility… a stock may have jumps. Robert Merton shows that taking jumps into account will tend to increase the relative values of out-of-the money options, and in-the-money options and will decrease the relative value of at-the-money options… Finally, the fact that a stock’s volatility changes means that what seems like a close-to-riskless hedge is not. Suppose a call option moves $0.50 for a $1 move in the underlying stock, and you set up a position that is short two option contracts and long one round lot of stock. This position will be fairly well protected against stock price changes in the short run. But if the stock’s volatility increases you will lose. The option will go up even if the stock price stays where it is.
Fischer Black ’89Reference: http://www.risk.net/digital_assets/5955/The_holes_in_Black-Scholes.pdf
‘The Holes in Black-Scholes’
When we calculate option values using the Black-Scholes model, and compare them with option prices, there is usually a difference… The main input that may be wrong is volatility… Different people will make different volatility estimates… We know some specific problems with the formula… A stock’s volatility changes in unexplainable ways, but it also changes in ways related to the price of the stock. A decline in stock price implies a substantial increase in volatility, while an increase in the stock price implies a substantial decline in volatility… a stock may have jumps. Robert Merton shows that taking jumps into account will tend to increase the relative values of out-of-the money options, and in-the-money options and will decrease the relative value of at-the-money options… Finally, the fact that a stock’s volatility changes means that what seems like a close-to-riskless hedge is not. Suppose a call option moves $0.50 for a $1 move in the underlying stock, and you set up a position that is short two option contracts and long one round lot of stock. This position will be fairly well protected against stock price changes in the short run. But if the stock’s volatility increases you will lose. The option will go up even if the stock price stays where it is.
Fischer Black ’89Reference: http://www.risk.net/digital_assets/5955/The_holes_in_Black-Scholes.pdf
01/05/2023Swati Mital
The main input that may be wrong is volatility…
Different people will make different volatility
estimates… We know some specific
problems with the formula… A stock’s volatility changes
in unexplainable ways
05/01/2023Swati Mital
Implied Volatility
BS-pricing formula (for calls and puts) implies, as a function of , an inverse function. For each price (in its range), there exists a unique , which, when put into the BS formula, yields that price.
Given a market price , the implied volatility is the unique volatility, that solves
The function is called a volatility surface. If BS model was correct, would be constant.Remark: Implied volatility is based on current market prices whereas realized volatility is
based on past observations
05/01/2023Swati Mital
Implied Volatility in options
Sources: http://www.wikipedia.org/ http://www.cboeoptionshub.com/
05/01/2023Swati Mital
Agenda
Black Scholes Pricing Model Holes in the Black-Scholes Model Importance of Volatility Estimation Stylized Facts about Volatility Volatility Models
• Local Volatility Models• Stochastic Volatility Models
Stochastic Volatility Jump Diffusion
05/01/2023Swati Mital
Importance of Volatility Estimation
Pricing using IV from a similar exchange-traded option. Is that sensible? What about hedging?
Suppose the trader uses the proceeds from the option sale to form a hedge portfolio with initial value and then uses the hedge at ( is in cash).
Let be the tracking error. Then it can be shown that Hedging strategy makes a profit if the estimated volatility dominates the true volatility!
05/01/2023Swati Mital
Agenda
Black Scholes Pricing Model Holes in the Black-Scholes Model Importance of Volatility Estimation Stylized Facts about Volatility Volatility Models
• Local Volatility Models• Stochastic Volatility Models
Stochastic Volatility Jump Diffusion
05/01/2023Swati Mital
Stylized Facts
Volatility Clustering and Persistence
Thick Tail Distributions
Negative Correlation between Price and Volatility
Mean Reversion
These are not captured by Black Scholes
05/01/2023Swati Mital
Some InterpretationsWe regard the observed prices of a given class of options as
correct, i.e. these prices cannot be arbitraged. «No Arbitrage Pricing»
We find a model that is sufficiently general that it can be calibrated to reproduce all the observed prices for our particular class of options.
Traded call options prices are correct in a no-arbitrage model. After we have calibrated our underlying model against these prices we can then use it to price more complicated contracts. E.g. Barrier Options, Lookbacks, etc.
05/01/2023Swati Mital
Agenda
Black Scholes Pricing Model Holes in the Black-Scholes Model Importance of Volatility Estimation Stylized Facts about Volatility Volatility Models
• Local Volatility Models• Stochastic Volatility Models
Stochastic Volatility Jump Diffusion
05/01/2023Swati Mital
Dupire Local Volatility Model
the volatility becomes a function of time and Stock price
The Black Scholes equation now becomes
If there are continuous prices across expiry and strikes then a unique local volatility exists. [Ref: Gyongy [2]]
05/01/2023Swati Mital
Results for Dupire Local Volatility Model
Forward Kolmogorov Equation (also known as Fokker-Planck) shows that for an SDE
the transition probability satisfies the PDE
Let be the price of a call option at time then by Feynmann-Kac equation, we get, for
If we assume that at time we know all market prices for calls of all strikes. Then we
can compute partial derivatives and This gives us (using result from Breeden-Litzenberger),
05/01/2023Swati Mital
Derivation of Dupire Local Volatility ModelThe transition density satisfies the Kolmogorov forward
equation,
Differentiating the F-K’s equation on previous slides gives us
Solving this equation forward in time we get a unique solution
05/01/2023Swati Mital
Dupire Formula in terms of Implied Volatility
Market quotes implied volatility across expiry and strikes
Transforming the equation to use implied volatility and moneyness
Relation between implied local volatility and local
volatility
05/01/2023Swati Mital
Dupire Local Volatility Implementation
What do we need?• Smooth, Interpolated IV surface • No arbitrage across strikes and expiry • Numerically stable techniques for partial
derivativesInterpolation across Strikes
• SABR, Spline based InterpolationStochastic Alpha Beta Rho
• SABR admits no arbitrage• represents overall level of ATM volatility• represents skewness, • represents shape of skew • measures convexity (stochasticity of • Analytical Formula by Hagan [3] provides
05/01/2023Swati Mital
Dupire Local Volatility Implementation (contd.)
Given market implied volatilities for row of maturities and strikes we fit the SABR parameters for each maturity separately.
Reduce the difference between market and model (SABR) implied volatilities by adjusting for
We then interpolate across time to stitch the volatilities into a smooth surface. Condition for no arbitrage across time = We can use polynomial interpolation across time.
Compute partial derivatives with respect to strike analytically to generate local volatility surface.
05/01/2023Swati Mital
Pros and Cons of Dupire ModelAdvantages of Dupire
LVM• Excellent fit to the market
prices (or equivalently market implied volatility surface)
• Calibration is fast and exact
• Can be treated as a “code block” for transformation of observed implied volatilities to local volatilities
Disadvantages of Dupire LVM−Requires continuous,
smooth implied volatility surface
−Differentiation can be numerically unstable
−Guarantee of no arbitrage −Do not capture dynamics
of volatility
05/01/2023Swati Mital
Stochastic Volatility ModelStochastic Asset Price and Stochastic Volatility
Market is incomplete as there is one traded asset and two driving Brownian Motions W and B. We can hedge randomness of asset but what about volatility? No unique risk free measure Q!!!
Heston took and volatility follows CIR process
Feller condition for
05/01/2023Swati Mital
Heston Stochastic Volatility Model (Closed Form)
To make markets complete, we introduce a second traded asset (e.g. variance swap) hold units of it and perform Vega Hedging.
We proceed in the same way as Black Scholes derivation and create a portfolio
And to obtain risk neutrality we remove the risk and risk, we get
Fundamental PDE for Heston
05/01/2023Swati Mital
Risk Neutral to Real World Transformation
We have two premiums under the physical measure in the Heston ModelEquity Risk Premium since investors are adverse to negative movements in
equity prices.Volatility Risk Premium since they are adverse to positive movements in
volatility.
Equity premium is compensation for risk in . In Heston, these compensations are proportional to and per unit of
Risk premium for stochastic volatility
05/01/2023Swati Mital
Summary of RN and RW Heston Model
Risk neutral dynamics in Heston Model
Real world dynamics in Heston Model
Risk neutral dynamics has lower mean reversion but higher long-run volatility
05/01/2023Swati Mital
Heston Stochastic Model Implementation
Further simplify by introducing because then the coefficients do not contain price of the underlying. Heston solved the PDE using Characteristic Functions.
Monte Carlo Approach (Euler Discretization)- Volatility and price path are discretized using constant - Can lead to discretization errors
05/01/2023Swati Mital
Heston Stochastic Volatility Model Parameters
Heston Parameters• affects the skewness of the distribution (leverage affect)• spread the right tail and squeeze left tail of asset return dist.• spread the left tail and squeeze right tail of asset return dist.
• affects the kurtosis of the distribution, high means heavy tails
• affects the degree of “volatility clustering”• is the long run variance• is the initial variance and it affects the height of the smile
curve
05/01/2023Swati Mital
Calibration of Heston ModelMinimize the least squared error between market price for a given range of
expiries and strikes and the model price.
Local Optimizer like Levenberg-Marquardt• Depends on the selection of seed values (initial guess)• Determines optimal direction of search• Possibility of finding local minima• Only searches small amount of the search space
Find best fit to ATM volatilities because they are the most liquid.
Global optimization approaches better but takes too long.
05/01/2023Swati Mital
Pros and Cons of Heston Model
Advantages of Heston SVM• Non Lognormal probability
distribution in price dynamics
• Volatility is mean reverting
• Takes into account leverage effect between equity returns and volatility
Disadvantages of Heston SVM−Calibration is often difficult
due to number of parameters to fit
−Expensive to do global optimization.
−Prices are sensitive to parameters
05/01/2023Swati Mital
Agenda
Black Scholes Pricing Model Holes in the Black-Scholes Model Importance of Volatility Estimation Stylized Facts about Volatility Volatility Models
• Local Volatility Models• Stochastic Volatility Models
Stochastic Volatility Jump Diffusion
05/01/2023Swati Mital
Stochastic Volatility Jump Diffusion
The stochastic volatility model captures some stylized facts:• Volatility clustering• Mean Reversion• Heavier tails
Fails to capture random fluctuations (for e.g. shock in the market causing crash)
05/01/2023Swati Mital
References
1. Dupire B, 1994 Pricing with a smile2. Gyongy I, 1986 Mimicking the One-Dimensional Marginal Distribution of Processes
Having and Ito Differential3. Heston SL, 1993 A closed-form solution for options with stochastic volatility with
applications to bond and currency options 4. Hagan P, Kumar D, Lesniewski A and D Woodward 2002, Managing Smile Risk5. Monoyios, Michael 2007 Stochastic Volatility , University of Oxford6. Ruf, Johannes 2015 Local and Stochastic Volatility, Oxford-Man Institute of
Quantitative Finance