1. Mathematical Finance An Introduction
Beixi Lei

2. An Introduction... How do random walk connected to stock prices?
Why people are interested in using PDE to forecast option prices?
How to use numerical methods for asset pricing?
How people manage credit risk using lognormal assumptions?
…..

3. Call Options vs. Put options
Call options: A call option gives the holder the right to buy the underlying asset by a certain date for a certain price.
Put options: A put option gives the holder the right to sell the underlying asset by a certain date for a certain price.
The price in the contract is known as exercise price E or strike price.
The date in the contract is known as expiration date t or maturity.

4. European Options vs. American Options
Holder has the right to exercise the option only on the expiration date.
American options:
Holder has the right to exercise the option on or before the expieration date.
Most traded options in the US are American styles.
Exceptions: foreign currency options, and some stock index options.

5. Binomial Method
Basic assumption: The stock price follows a random walk.

6. Binomial Method
we want to calculate the option price at time 0. To do this, we assume that the stock price will increase or decrease by a proportion u or d respectively in order to model all the possibilities of stock price at time T. We know the option price at time T, and we can use backward induction to compute the option price at time 0.

7. Wiener Process
Stock prices are usually assumed to follow a Markov process, where it is a type of stochastic process only the present value of a varibale is relavant for predicting the future.
We say a stochastic process {X_t (ω ), t ≥ 0} is a standard Wiener process if it satisfies the following conditions:
(a) X_0 (ω ) = 0 for all ω ;
(b) the map t→ Xt (ω ) is a continuous function for t ≥ 0 ;
(c) for every t and h ≥ 0, the change [X_t+h (ω ) − X_t (ω )] ∼ N (0, h ); and
(d) X_u (ω ) − X_v (ω ) and X_t (ω ) − X_s (ω ) are independent for all 0 ≤ v ≤ u ≤ s ≤ t .
omega

8. Geometric Brownian Motions
We consider dX(t) (=dX_t) as the limit of X(t+dt) - X(t) (i.e. small change in time t)
As dt → 0:
(i) d_X (t ) is a random variable, drawn from a normal distribution;
(ii) the mean of dX (t ) is zero, E [dX (t )] = 0;
(iii) the variance of dX (t ) is dt , Var(dX (t )) = dt
The stock price is modeled as follows: ΔS/S= μΔt + σεΔt
where ε ~ N (0, 1) is a random number drawing from a standard normal distribution.
epsilon

9. Ito’s Process
A generalized Wiener provess in which the parameter a and b are functions of the value of the underlying variable Y and time t. An Ito’s process can be written as:
dY=a(Y,t)dt+b(Y,t)dX(t)
If the volatility of the stock price σ is always 0, then this model implies that: ΔS=μSΔt
As Δt→ 0, dS( t)/S( t)= μdt + σdX( t), it can also be expressed as:
dS( t) = μS( t) dt + σS( t) dX( t)
i.e. standard deviation of change in Δt should be porportional to the stock price.

10. Programming
μ=2,σ=1,n=10
μ=1,σ=2,n=100
μ=1,σ=1, n=1000

11. Black-Scholes Partial Differential Equation
Basic assumptions:
i) The asset price follows the geometric Brownian motion.
That is, dS (t ) = μS (t )dt + σS (t )dX (t );
ii) The risk-free interest rate r and the asset volatility σ are known functions.
iii) There are no transaction costs.
iv) The asset pays no dividends during the life of the option.
v) There are no arbitrage possibilities.
vi) Trading of the asset can take place continuously.
vii) Short selling is permitted.
viii) We can buy or sell any number (not necessarily an integer) of the asset.

12. Black-Scholes Partial Differential Equations
Using It^o's lemma and noting that S (t ) follows dS( t) = μS( t) dt + σS( t) dX( t) :
where V is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, and σ is the volatility of the stock. The equation becomes:
The boundary final conditions for European call option will be:
K=Exercise price, C=price of the call option

13. Monte Carlo Method
A procedure for a sampling random outcomes for the processes.
we need to simultate many other stock prices at T to get enough samples for the option price c( S, T ) at T, then we take the average of these samples to get a good estimate of the expected value of c( S, T ). By discounting back to t, and given the fact that X^(T−t) ~N(0 , T − t), then:
S_T = S_t e^(μ−σ^2/2)(T−t)+σε^(√T−t) ,
where ε is a random variable drawn from a standard normal distribution N~(0 , 1).
The current value of the option can be calculated by discounting the expected payoff value, where V = e^−r(T−t)E( V_T ).
Epsilon

14. Finite Difference Method
The main idea is to approximate the differential operators in the differential equation by difference operators.
Suppose we solve the B-S equation for an European option V(s,t), recall that it satisfies
and we replace the boundary S = S_max, and the option price by V ( S_max , t).

15. Finite Difference Method
We partition [0, S_max ] into N equal sub- intervals, each of length δS; and [0,T] into M equal sub-intervals, each of length δt. Let S_j = jδS, t_i = iδt for 0 <=j <=N and 0 <=i <=M . By repeating the process M times, we can obtain the option prices V (S_j , t0 ) = V (S_j , 0) at the current time for all Sj .
delta

16. Finite Difference Method
Backward approximation:
df/ds=(f_i,j - f_i,j-1)/ΔS
Forward approximation:
df/dt=(f_i+1,j - f_i,j)/Δt
The value at time iΔt is related to valur to the value at time (i+1)Δt.

17. References
J.C. Hull, Options, Futures and Other Derivatives, (8th edition), Pearson, (2011)
R. McDonald, Derivatives Markets, (3rd edition), Pearson, (2009)
K. Redhead, Financial Derivatives, Prentice Hall, (1997)
R. Chan, Financial Mathematics, Department of Mathematics, Chinese University of Hong Kong (2011)

18. Thanks for watching. Any questions?