JUMP DIFFUSION MODELS

Karina Mignone

Option Pricing under Jump Diffusion

Previous knowledge

Diffusion: A process that has diffusion has random motion causing random distribution of data.

Option: In finance, an option is a contract between a buyer and a seller that gives the buyer of the option the right, but not the obligation, to buy or to sell a specified asset on or before the option's expiration time, at an agreed price (the strike price). This is often done in the stock market.

Call option: an option to buy. Put option: an option to sell. European option: An option that can be

exercised only at expiry date. American option: An option that can be

exercised at any date between the issue date and the expiry date.

Stochastic process: a random process; non-deterministic.

Stochastic drift: The average change over time in a random process.

Brownian motion A zero-mean continuous-time stochastic

process with independent increments (also known as a Wiener process).*

It is the scaling limit of a random walk in one dimension

According to the model the returns on a certain stock in successive, equal periods of time are independent and equally distributed.

*definition from www.mathworks.com

According to the geometric Brownian motion model the future price of financial stocks has a lognormal probability distribution and their future value can therefore be estimated with a certain level of confidence.

Brownian motion however assumes a constant expected rate of return and volatility and does not consider discontinuity

Background

First model Bachelier’s Gaussian model for stock returns

60 years later Osborne proposed the use of Brownian motion to ensure price levels did not achieve negative values which as in Bachelier’s model

Then, Brownian motion was known as the only process with stationary and independent increments that has continuous paths

Brownian motion is a main tool for financial modeling however fits data poorly

It underestimates the likelihood of a large movement in the underlying

Instead need a model that conforms to empirical data: a stochastic model that is skewed (to the left) and leptokurtic (more peaked at its mean and with greater probability mass in its tails)

Different Option Pricing methods Press proposed a model where the

natural logarithm of the stock price is assumed to follow a distribution that is a Poisson mixture of normal distributions and a Brownian motion, in the following way

Where Y1…Yk is a sequence of mutually independent random variables normally distributed, Nt is a Poisson process and Wt is Brownian motion

t

N

k kot WYSS t 1lnln

Black-Scholes model (extension of Brownian motion)

[The model develops partial differential equations whose solution, the Black–Scholes formula. Close approximation to real observed prices, however assumes the stock price follows a geometric Brownian motion with constant drift and volatility.]

Cox, Ross and Rubinstein binomial model [the model uses a "discrete-time"

(lattice based) model of the varying price over time of the underlying. Binomial model provides a discrete time approximation to the continuous process underlying the Black-Scholes model]

Merton (1976) and Tucker and Pond (1988) provide a more thorough discussion of mixed-jump processes. Mixed-jump processes are formed by combining a continuous diffusion process and a discrete-jump process and may capture local and nonlocal asset price dynamics.

Merton derived an option pricing formula as the underlying stock returns are generated by a mixture of both continuous and the jump processes.

The two basic building blocks of every jump-diffusion model are the Brownian motion (the diffusion part) and the Poisson process (the jump part).

Jump Diffusion

Assume the underlying follows Brownian motion, plus jumps governed by Poisson distribution

(The size of the jump can be given by a distribution of our choice)

Poisson process is used for modeling systematic jumps caused by surprise effects

[The arrival times follow a Poisson distribution] =mean arrival rate of event during time

interval dt.

First define Poison process:

where =Poisson arrival intensity. The event is a jump of size u, which can

itself be a random variable Thus there is a probability dt of a jump in

the timestep dt. Qt is the number of events that occur by

time t.

𝑑𝑞={ 1 , h𝑤𝑖𝑡 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 λ 𝑑𝑡0 , h𝑤𝑖𝑡 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 1−λ 𝑑𝑡 }Event

occurring

Event will not occur

2 important properties of Poisson process

1. Probability of at least one event occurring in is

2. Probability of two or more events occurring in is o( ). In other words, we do not see 2 or more events happening at the same time.

0,0)(][ tastoteventoneleastatP

t

t

t

Stochastic differential equation for an option with jumps

From Black-Scholes Model :

Represents the jump

Another way of writing it: if no jump occurs if one jump occurs

The jump size follows a log-normal distribution ,

where m is the average jump size, v is the volatility of jump size and N(0, 1) is the standard normal distribution.

𝑛 𝑚𝑒𝑥𝑝(− 𝑣2

2+𝑣𝑁 (0,1))

From a risk management perspective, jumps allow to quantify and take into account the risk of strong stock price movements over short time intervals, which appears non-existent in the simple diffusion framework.

It can be shown that for all derivatives with convex payoff (which includes regular call and put options) the price always increases when jumps are present—regardless of the average jump direction.

This increase in price can be interpreted as compensation for the extra risk taken by the option writer due to the presence of jumps, since this risk cannot be eliminated by hedging.

Example 1

when there are no jumps, the jump diffusion model reduces to the Black–Scholes model, in which returns follow a normal distribution

Example 2

the effect of jumps can be observed clearly by "turning down" the volatility of the diffusive component to zero

Summary

for the purpose of studying option pricing, a jump diffusion model, in which the price of the underlying asset is modelled by two parts: a continuous part driven by Brownian

motion, and a jump part following a Poisson process

Represents a good model for option pricing.

Any questions?

References

http://www.hoadley.net/options/bs.htm, date accessed 22/5/11 http://www.few.vu.nl/en/Images/werkstuk-dmouj_tcm39-91341.pdf, date accessed 22/5/11 http://what-when-how.com/finance/jump-diffusion-model-finance/, date accessed 19/5/11 http://www.datasimfinancial.com/UserFiles/articles/PIDE.pdf, date accessed 12/5/11 http://www.math.nyu.edu/~benartzi/Slides5.2.pdf, date accessed 24/5/11 http://janroman.dhis.org/stud/AFI%20Jump%20diffusionPPT.pdf, date accessed 22/5/11 http://www.ems.bbk.ac.uk/for_students/msc_finance/TOF2ctpt_emec043p/slides4.pdf, date

accessed 24/5/11 http://demonstrations.wolfram.com/DistributionOfReturnsFromMertonsJumpDiffusionModel/

, date accessed 24/5/11 http://docs.google.com/viewer?a=v&q=cache:yOkPsHM3WggJ:www.ems.bbk.ac.uk/

for_students/msc_finance/TOF2ctpt_emec043p/lecture4.pdf+jump+diffusion+lecture&hl=en&gl=au&pid=bl&srcid=ADGEESg0c1YzS7Eb9cecGB-mEdSGYIDnhP8MuVGFrgY0VH45loi95rhG5ESBXw1rC5cTyze4nW3ww13cEUaUXgrOYqNlvJZX6ogQXPettJsfTu5BIZ6Fy9hV2ot3fnSaWiE5ap8udYDU&sig=AHIEtbQQK9V8I7GFyAdwk6ShxwMjJq6yBQ, date accessed 25/5/11

http://www.google.com.au/search?hl=en&q=Brownian+motion&tbs=dfn:1&tbo=u&sa=X&ei=MrzdTbP1GMvnrAff9Nj2CQ&ved=0CB0QkQ4&biw=1131&bih=687, 26/5/11

Demonstration

http://demonstrations.wolfram.com/DistributionOfReturnsFromMertonsJumpDiffusionModel/

Download from here http://demonstrations.wolfram.com/Mert

onsJumpDiffusionModel/

If v(x,t) is smooth enough, then V(S(t),t) is also a jump diffusion, providing a partial integrodifferential equation (PIDE):