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Merton’s JumpDiffusion Model
David Bonnemort, Yunhye Chu, Cory Steffen, Carl Tams
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Outline
Background
The Problem
Research
Summary & future direction
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Background Terms
Option: (Call/Put) is a derivative written on a security thatgives the owner the right to buy or sell the security at apredetermined price on a specified date in the future
Premium: price of the option Strike: the price at which the owner of an option has the
right to buy or sell the option Maturity: the date on which the option may be exercised Volatility: standard deviation of the change in value of an
asset over time In general, the more volatile the asset, the more a derivative
contract is worth
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Background
Example: Call OptionSuppose the strike = $100
Payoff If stock price reaches $110 at expiration, then the
buyer makes a profit of $10
If the stock is below $100 at expiration, then it hasno value
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Background
Build Optimal PortfolioDepends on goal: risky v. conservative
Options By itself it is VERY RISKY
When coupled with stock appropriately the portfoliocan become LESS RISKY
This is called ‘Hedging’
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Background
Want to know how a financial product willreact in the market If stock goes up option moves accordingly.
Design option to react how you want it to (ie. callsand put)
This is important because understanding thesereactions will help people optimize their portfolios i.e. make more money by knowing how to allocate
their assets within their investments
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Background
Black-Scholes FormulaTo calculate the fair-market value for any option,
the formula uses multiple variables.
S = current stock price,
K = strike price,
r = risk-free interest rate,
T = time to expiration,
_ = volatility of the stock.
t = current time
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Background
Black-Scholes Formula
where N(x) is the cumulative normal distributionfunction
And d1 and d2 are defined as:€
C s,t,σ,k( )= SN d1( )−Ke−r(T− t )N d2( )
€
d1
€
d1 =logS K + r( +σ 2 2)(T − t)
σ T − t
d2 = d 1̀ −σ T − t
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Background
Assumptions of Black-Scholes Model The stock follows geometric Brownian motion No dividends are paid out on the underlying stock
during the option life. The option can only be exercised at expiration time
(European style) Efficient markets (No arbitrage) No transaction cost Risk free interest rates do not change over the life of
the option (and are known)
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
The Problem
Black Scholes ModelRequires different volatilities for different strikes
and maturities to price the option
Ideally want a 1 – 1 correspondenceOne volatility assigned to each stock
BS assumes a log normal distributionBy adding jumps we correct this flaw
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
The Problem
Solution : Merton’s Jump Diffusion ModelThis model attempts to solve the problems
associated with a log normal distribution
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Research
Merton’s Jump Diffusion Model
Show how Black- Scholes fails
Show how Jump Diffusion works
Derivation
Research using NDX-100 andIBM
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Failure of lognormal distribution assumption
Research
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Research
Introduction to the Jump Diffusion ModelAllows for larger moves in asset prices
caused by sudden events.
The jump component represents non-systematic risk, a type of risk that affects aparticular company or industry.
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Research
Jump Diffusion Model DerivationMerton includes a discontinuity of underlying
stock returns called a jump.
The following formula describes a relative changeof stock price with the jump factor: q
€
Si+1 − SiSi
= µΔt +σΔω i + Δq
€
Δq =0y −1
withoutwith
jumpsjumps
where
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Research
Jump Diffusion Model DerivationThe following is the modified Black-Scholes
equation in the jump diffusion model.
The solution can be represented by a seriesconsisting of terms
€
12σ
2S2Fss + (r − λk)SFs − Fτ − rF + λE[F(sY,τ) − F(s,τ)] = 0
€
Fn =e−λτ (λτ )n
n!E[W (x,τ,K,σ 2)],n = 0,1,2,...
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Research
Volatility Smile Implied volatility-is the volatility used in Black
–Scholes to match the market price.
The Black-Scholes formula uses different impliedvolatilities for different strikes and maturities.
At-the-money options tend to have lower impliedvolatilities.
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Volatility Smile
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Volatility smile for an option on the NDX-100
Research
Volatility Smile(option for the NDX stock)
17
18
19
20
21
22
23
24
1520 1540 1560 1580 1600 1620
strike
implied volatility
implied volatility (jump)
implied volatility (market)
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Volatility smile for an option on IBM stock
Research
Volatility Smile(Option for the IBM stock expired on Dec 6th 2006)
12
13
14
15
16
17
18
19
80 85 90 95 100 105 strike
implied volatility
implied vol (jump)implied vol (market)
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Option Pricing Errors of Jump Diffusion Modeland Black-Scholes Model (IBM)
Price Errors
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
80 85 90 95 100 105
strike
price difference
Jump model
σ=0.1 σ=0.11 σ=0.12 σ=0.13 σ=0.14 σ=0.15 σ=0.16
BS model σ=0.12
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Summary
Merton included the impact of a sudden largestock fluctuations
Model works better on individual stocksrelative to indices
Non-systematic jump risk assumption isimportant in this model.
The University of Utah VERTICAL INTEGRATION OF RESEARCH AND EDUCATION
Future Direction
Estimate the jump risk on particular stocksand indices
Analyze the volatility smile to determinesystematic or nonsystematic risks
![Jump-diffusion models: a practitioner’s guide · Starting with Merton’s seminal paper [21] and up to the present date, various aspects of jump-diffusion models have been studied](https://img.dokumen.tips/doc/110x75/5b5c1b9f7f8b9a68368c0904/jump-diusion-models-a-practitioners-guide-starting-with-mertons-seminal.jpg)
