1. Option prices and the Black-Scholes-Merton formula
Gabor Molnar-Saska
3 October 2006

2. Morgan Stanley
Morgan Stanley is a leading global financial services firm, offering a wide variety of products and services. A partial list of these products and services includes:
Investment banking services such as advising, securities underwriting
Institutional sales and trading, including both equity and fixed income investments
Research services
Individual investor services such as credit, private wealth management, and financial and estate planning
Traditional investments such as mutual funds, unit investment trusts and separately managed accounts
Alternative investments such as hedge funds, managed futures, and real estate

3. Morgan Stanley
Morgan Stanley is an industry leader in underwriting Initial public offerings of stock worldwide.
Morgan Stanley reported net revenues of $ billion in 2005.
Morgan Stanley ranks as the 30th largest U.S. corporation in 2005.
In 2004, Morgan Stanley held the #1 industry rank for the following categories: Global Equity and Equity-Related Underwriting Market Share, Global IPO Market Share, and Global Equity Trading Market Share.
Morgan Stanley had 53,760 total employees worldwide as of August 31, 2005.

4. The binomial model 2 1 0.5 S0 is the initial stock price (at time t=0)
S1 is the stock price at time t=1
Assume P(S1=2)=0.5 and P(S1=0.5)=0.5
r: continuously compounded interest rate, i.e. 1$ at time zero will grow to exp(rt)
(assume now r=0)
Call option: strike K=1
payout (S1-K)+
What is the price of this option?

5. The binomial model 2 X=1 1 0.5 X=0
Buy a portfolio consisting of 2/3 of a unit of stock and a borrowing of 1/3 of
a unit of bond.
The cost at time zero: 2/3*1$-1/3*1$=0.33$
After an up-jump: 2/3*2$-1/3*1$=1$
After a down-jump: 2/3*0.5$-1/3*1$=0$
The correct price: 0.33$

6. The binomial model s1up Xup s0 s1down Xdown
Consider a general portfolio (a,b)
The cost at time 0: as0+bB0
After an up-jump: as1up+bB0exp(rt)=Xup
After a down-jump: as1down+bB0exp(rt)=Xdown

7. The binomial model
The value of the portfolio:

8. The binomial model 0<q<1 and the value of the portfolio is
Expectation under a new measure!

9. The binomial tree model
Si is the value of stock at time i (binomial tree model)
Bi is the value of the bond at time i.

10. The binomial tree model
Find a new measure under which Zi = Bi-1Si is a martingale (Q),
where Bi-1 is the discount process.
Binomial representation theorem: Suppose Q is such that the binomial price process Z is a Q-martingale. If N is any other Q-martingale, then there exists a previsible process such that
where
is the change in Z from tick time i-1 to i, and
is the value of
at the appropriate node at tick-time i.
Let

11. The binomial tree model
Consider the following construction strategy: at tick-time i, buy the portfolio Hi, consisting of
ai+1 units of the stock
bi+1=Ni-ai+1Bi-1Si units of the cash bond
At time zero the portfolio worth a1S0+b1B0=N0=EQ(BT-1X)
One tick later: a1S1+b1B1=B1(N0+a1(B1-1S1-B0-1S0))=B1N1
Self-financing strategy,
At the end we have BTBT-1X=X, whatever actually happened to S.

12. The binomial tree model
Conclusion:
The price of the claim X is
where Q is the risk-neutral measure

13. Continuous time models
Let be deterministic
where r is the riskless interest rate, is the stock volatility and is the stock drift. Both instruments are freely and instantaneously tradable either long or short at the price quoted.
Let X be a payout at time T.

14. Continuous time models
Discrete approximation:
if up jump
if down jump

15. Continuous time models
Three steps to replication
Find a measure Q under which the discounted stock price exp(-rt)St is a martingale
Form the process Nt=exp(-rT)EQ(X|Ft)
Find a previsible process such that
Price of the claim: exp(-rT)EQ(X)
Price of the call option: (X=(ST-K)+) is
exp(-rT)EQ((ST-K)+)

16. Continuous time models
What is the dynamics under the risk neutral measure?
Ito’s formula: If X is a stochastic process, satisfying and f is a deterministic twice continuously differentiable function, then Yt=f(Xt) is also a stochastic process and is given by

17. Continuous time models
Let Xt=log(St). Then we have
Using the Ito’s lemma we get
Under the risk neutral measure

18. Continuous time models
Thus, we know that ST is log-normally distributed under the risk-neutral measure Q.
The price of the call option: (if X=(ST-K)+) is

19. Connection with partial differential equations
Consider an agent who at each time t has a portfolio valued at X(t). This portfolio invests in a money market account paying a constant rate of interest r and in a stock modeled by the geometric Brownian motion:
Suppose the investor holds shares of stock and the remainder of the portfolio is invested in the money market account.
Then

20. Connection with partial differential equations
Using the Ito formula we have
and

21. Connection with partial differential equations
Let c(t,x) denote the value of the call option at time t if the stock price at that time is S(t)=x.
According to the Ito formula we have

22. Connection with partial differential equations
A hedging portfolio starts with some initial capital X(0) and invests in the stock and money market account so that the portfolio value X(t) at each time agrees with c(t,S(t)). This happens if and only if
for all t. One way to ensure this is to make sure that
and X(0)=c(0,S(0)).

23. Connection with pde We get Equate the dW(t) terms:
Equate the dt terms:

24. Connection with pde
In conclusion we should seek a continuous function c(t,x) that is a solution to the Black-Scholes-Merton partial differential equation
for all and that satisfies the terminal condition

25. Challenges Volatility is stochastic Interest rate r is stochastic
Claim is path dependent (exotic options)
The dynamics of the stock process is not geometric Brownian
Correlation between the dynamics of different market processes