Download Monte-Carlo simulation with Black-Scholes

Monte-Carlo simulation with Black-Scholes
ANALYTICAL FINANCE I
2011-10-15
Teacher:
Jan Röman
Group:
Wang Xi
Shi Sha
Zhang Chong
Contents
1 Introduction………………………………………………………..3
2 Case information…………………………………………………..4
3 Theory
3.1 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6
3.2 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 7
Delta neutral hedging
Gamma Delta Hedging
3.3 Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Case Analysis
5 Conclusion
Introduction
In our report, we are going to use Monte Carlo Simulation to analyse the
Black-scholes model. The first part is the theory part, which we introduce
the definition of Greeks, Hedging, Monte Carlo Simulation and
Black-scholes model. The second part is the case analyzing. In this part,
we analyze the possibilities of hedging a portfolio for many options and
stocks with in a assumed time.
Case information
Make a Monte-Carlo simulation with Black-Scholes with parameters
given (initial underlying price, strike price, volatility, time to maturity
and risk-free interest rate). Simulate the situation where you buy 10000
underlying stock and at start, hedge the position with an option. Each
such option has 100 stocks as underlying. So, make the best hedge.
During the price movements of the underlying, change the hedge each
time you need to buy or sell an option (on 100 underlying stocks). Finally,
calculate the outcome of this trading strategy. Make 100 000 simulation
and present a histogram of the result and calculate the mean value and the
variance. What happen if you also introduce a trading cost of 0.2%?
The Greeks
The Greeks measure the sensitivity to change of the option price under a
slight change of a single parameter while holding the other parameters
fixed. Formally, they are partial derivatives of the option price with
respect to the independent variables. The Greeks give the investor a better
idea of how a stock has been performing. They are very helpful in
deciding on options strategies choosing. These statistics forecast the
trends of stock according to the past and these trends can change
completely based on new stock data.
The Greeks for Black-Scholes are Delta, Gamma, Vega, Theta, Rho.
Financial institutions will typically set limits for the Greeks that their
trader cannot exceed. In trading actives, Delta is the most important
Greek and traders will zero it in the end of a day.
What
Δ
γ
Calls
Puts
Delta is a measure of the relationship between an option price and the
underlying stock price. For a call option, a Delta of .50 means a half-point
rise in premium for every dollar that the stock goes up. For a put option
contract, the premium rises as stock prices fall. As options near expiration,
in the money contracts approach a Delta of 1.
Gamma is the rate of changing in Delta with respect to a change in the
value of the underlying stock price, and Gamma indicates an absolute
change in delta. For example, a Gamma change of 0.150 indicates the
delta will increase by 0.150 if the underlying price increases or decreases
by 1.0. Results may not be exact due to rounding.
Hedging
A hedge is an investment position intended to offset potential losses that
may be incurred by a companion investment.
A hedge can be constructed from many types of financial instruments,
including stocks, ETFs, insurance, forward contracts,swaps, options, many types
of over-the-counter and derivative products, and futures contracts. Public futures
markets were established in the 19th century[1] to allow transparent, standardized,
and efficient hedging of agricultural commodity prices; they have since expanded
to include futures contracts for hedging the values of energy, precious
metals, foreign currency, and interest rate fluctuations.
Delta hedging
i.e. establishing the required hedge - may be accomplished by buying or
selling an amount of the underlie that corresponds to the delta of the
portfolio. By adjusting the amount bought or sold on new positions, the
portfolio delta can be made to sum to zero, and the portfolio is then delta
neutral.
Here we useΔ=N(d1) to calculate the number of shares we should hedge：
Ns=Nc*Δ
Ns-amount of stock
Nc-amount of call option
Black–Scholes model
The Black–Scholes model is a mathematical model of a financial market
containing certain derivative investment instruments. It explains how
volatility can be either estimated from historical data or implied from
option prices using the model. It is widely used by options market
participants.
The Black-Sholes model also assumes stocks move in a manner referred
to as a random walk at any given moment, they are as likely to move up
as they are to move down. These assumptions are combined with the
principle that options pricing should provide no immediate gain to either
seller or buyer.
The Black-Scholes formula;
Sn+1=Sn
Monte Carlo simulation
Monte Carlo methods are a class of computational algorithms that rely
on repeated random sampling to compute their results. These methods are
most suited to calculation by a computer and tend to be used when it is
infeasible to compute an exact result with a deterministic algorithm.
Monte Carlo methods are especially useful for simulating systems with
many coupled degrees of freedom.
In order to simulate different kinds of scenarios in our report with random
variables attached to the functions, we will have to use Monte Carlo
simulation.
In this case our ε is random variable .We make ε 100,000 times and put
them in the function find 100,000 stock prices.
Case analyze
Sn+1=Sn
S0=100
K=110
Δt=T/N
T=1 year
r=0.1
σ=0.2
ε=random variable
N=100
Make 1000 path of stock price
Nc=the number of option
Ns=the number of stock
1. to simulate the stock price
we use Sn+1=Sn
2. to calculate the number of stocks we should hedge each time
Ns=Nc*Δ
Δ=N(d1)
Ns=10 000*Δ
3. to evaluate the value of the portfolio including trading cost 2%
C=P*Nc+C(trading cost)
V=S(T)*Ns(T)+max{S(T)-K,0}*100 000-C
Stock price：
With delta-hedge:
Ns=Nc*Δ
Ns=100 000*Δ
Conclusion
After this seminar, we learn what’s the Greeks, such as Delta and Gamma.
With hedging the Greeks components, we can solve our case by using
the Monte Carlo Simulation method for Black-Scholes model. That
helps us to analyzing cases with large amount stocks and options.