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


Agenda





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


Agenda





‘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


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


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


Implied Volatility in options

Sources: http://www.wikipedia.org/ http://www.cboeoptionshub.com/

http://www.wikipedia.org/

http://www.cboeoptionshub.com/


Agenda





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!


Agenda





Stylized Facts

Volatility Clustering and Persistence

Thick Tail Distributions

Negative Correlation between Price and Volatility

Mean Reversion

These are not captured by Black Scholes


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.


Agenda





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]]


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),


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


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


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


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.


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


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


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


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


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


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


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


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.


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


Agenda






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)


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

Date post:	21-Jan-2017
Category:	Economy & Finance
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Local and Stochastic volatility

Economy & Finance