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Outline of Discussion

- Modeling Stock Prices
- Geometric Brownian Motion
- Time Series Models
- Binomial Trees
- Valuing Assets
- Option Pricing
- European Call
- European Puts and American Options
- Estimating Volatility

Brownian Motion

- B(t) or W(t) also referred to as a Wiener process is a stochastic process with the following properties:
- B(0)=0
- {B(t); t≥0} has independent increments

i.e. B(tn)-B(tn-1), B(tn-1)-B(tn-2), . . ., B(t2)-B(t1),B(t1) are independent of one another

- {B(t); t≥0} has stationary increments

i.e. B(t+s)-B(s) behaves in the same manner as B(t)-B(0)

when s≥0

- For all t≥0 , B(t) is normal with mean 0 and variance t

Brownian motion is the continuous version of a random walk processes.

Other Notes About B(t)

- Brownian motion was first realized by botanist Robert Brown when trying to describe the motion exhibited by particles when immersed in a gas or liquid.

The particles were essentially being bombarded by the molecules present within the matter causing the displacement or movements.

- A Brownian motion process with drift µ and variance σ2 can be written as:

X(t)=σB(t) + µt

So this process X(t) is normal with mean µt and variance σ2t.

- Due to independent increments argument B(t)-B(s) is independent of

Fs= σ(B(r); r≤s) (filtration of the process)

- B(t) is a Continuous Time Martingale with respect Ft

E(B(t)| Fs)=E(B(t)-B(s)+B(s)|Fs)=B(s)

- B(t)2-t is also a Martingale with respect to Ft
- B(t) is continuous but not differentiable

A Deductive Approach

- Osborne’s Contribution in 1959 was to finally make the connection between Brownian motion and stock prices via an analogy.

It was clear that some kind of connection existed based upon movement of daily closing prices in relation to Brownian motion.

A Deductive Approach

- Osborne’s Contribution in 1959 was to finally make the connection between Brownian motion and stock prices via an analogy.

It was clear that some kind of connection existed based upon movement of daily closing prices in relation to Brownian motion.

- Like the particle being bombarded, stock prices deviates from the steady state as a result of being jolted by trades. Essentially a macroscopic version of Brownian motion.

A Deductive Approach

- Osborne’s Contribution in 1959 was to finally make the connection between Brownian motion and stock prices via an analogy.

It was clear that some kind of connection existed based upon movement of daily closing prices in relation to Brownian motion.

- Like the particle being bombarded, stock prices deviates from the steady state as a result of being jolted by trades. Essentially a macroscopic version of Brownian motion.
- First, we determine the steady state of the prices by examining the closing prices on a single day.

A Deductive ApproachIt was clear that some kind of connection existed based upon movement of daily closing prices in relation to Brownian motion.

- Like the particle being bombarded, stock prices deviates from the steady state as a result of being jolted by trades. Essentially a macroscopic version of Brownian motion.
- First, we determine the steady state of the prices by examining their value on a single day.

The distribution of prices appears to follow a lognormal distribution.

So the log of the prices are well approximated by a normal.

A Deductive Approach

Focusing now on an individual stock:

- Weber-Fechner law states that equal ratios of physical stimulus corresponds to equal intervals of subjective sensation. This argument implies that one should not study the absolute level of the price, but rather focus on the change in the price.

The focus then changes to log of the price ratios:

Also referred to as the return on the stock over the period from ti to ti+1.

A Deductive Approach

- For a buyer, the estimated values of the return is positive, and for the seller, the opposite, the estimated value must be negative. So for the market as a whole the expected value must balance out to be zero.
- Bachelier observed in the early 1900s that the long-run standard deviation of the return varies according to the square root of time elapsed multiplied by short-run volatility.
- Essentially, log(Sti+1/Sti) follows a Brownian motion with drift of 0 and a variance of σ2.

This means that for t1<t2< . . . <ti<ti+1< . . . :

Mathematical Approach

- We want to form a connection between Brownian motion and the price of a security.
- Brownian motion can be negative while the security price is strictly positive, so it is clear that the security will not be a multiple of Brownian motion.
- Profit or loss is a measure of the proportional increase, so the quantity ΔSt/St is of interest.
- Different stocks have different volatilities, and thus they have different risks. Depending upon the risk, one expects to be compensated a mean rate of return μ>r (risk-free rate of return, rate of return demanded of securities which have relatively no risk).

Mathematical Approach

- We obtain the following stochastic differential equation:

or

Recall that Brownian motion is not differentiable with respect to t, the derivative is only understood with respect to the stochastic integral. The integral itself defined as the limit in L2 of Riemann sums.

Mathematical Approach

- For the stochastic differential equation:

or

In order to solve this differential equation to obtain St, we need to apply a Ito’s formula. This formula is the fundamental theorem of calculus expanded to stochastic integration.

Which results from Taylor’s theorem.

Mathematical Approach

- In this formula:

The first sum resembles a stochastic integral as previously defined and the second sum is approximately the quadratic variation. As a result, so long as the first two derivatives of the function f are continuous, then Ito’s formula says:

Mathematical Approach

- Similarly for a semimartingale, Xt, which has the form which is Bt+t:

From the previous study, we already have a feeling that in our case,

Log(St/S0) is the Xt and that the f(x)=ex. Say Xt=σBt + (µ-(σ2/2))t, a Brownian motion with variance σ2 and drift (µ-(σ2/2)). We claim that by applying Ito’s formula that we can verify that this forms the solution to our SDE or that:

Mathematical Approach

- Ito Calculation:

Mathematical Approach

So the solution below states that the ratio St/S0 follows geometric Brownian motion:

In comparison to Osborne’s result,

we now have

Why (µ-(σ2/2))?

- Why not just µ as the drift parameter?
- Due to the jolts to the price, the rate of return is discounted by the amount of variability.
- If a positive return is followed by a negative of the same degree, the overall return will be depressed by the degree x2, since (1+x)(1-x)=1-x2.
- The Brownian motion determines the E(x2)= σ2.
- Divided over two periods of time, we have an average degree of depression σ2/2.

The Model

- So the model for the stock price is:

This model is fairly simple, and clearly defines that the returns for non-overlapping time periods are independently normal:

Problems

- Due to its simplicity and easy application to option evaluation, this model was highly regarded.

Until . . .

In 1987, there was a stock market crash, and the inadequacy of applying a model with a constant variance, or constant volatility, became apparent.

Stochastic Volatility

- Johnson & Shano, Wigins, Hull & White (1987) all introduced a SDE for the volatility measure. The following is the equations as selected by Hull and White in 1987:

- Stein & Stein (1991):

The second equation expresses the volatility like an AR(1) process. That means the volatility at time ti has a value that depends on the volatility at time ti-1.

ARCH models

- The October 1987 stock crash suggested that the volatility of stocks could encounter sudden change, yet it was also noted, as the previous model attempted to introduce, that volatility has positive serial correlation.
- i.e. The degree of volatility has a tendency to cluster.
- In the early 80s, Engle developed a model called the ARCH (Autoregressive Conditional Heteroscedasticity)

The ARCH is able to capture that serial autocorrelation observed for the volatilities.

Other Issues Discovered

- Other factors that are true about the stocks that we want to include within our model:
- Both the trading days as well as the non-trading days contribute to volatility. (It is observed that Monday tends to be the most volatile day of the week.)
- When the return is negative, or when the price of a stock drops, the volatility tends to rise. (If the equity of the firm drops, the firm becomes more leveraged.)
- Volatility tends to be high during times of financial crisis and recessions. (It is very difficult to distinguish this effect from the leverage effect above, since both occur simultaneously.
- High interest rates are often associated with high volatility.

Generalized ARCH or GARCH

- The Generalized ARCH was first introduced by Bollerslev in 1986.

This model is obviously more parsimonious, and the intercept term can made time dependent to incorporate any seasonal or non-trading days effects. Unfortunately, it still fails to capture the leverage effects of stocks prices.

Exponential ARCH model

- Nelson introduced in 1988 his exponential ARCH model:

We can let ωt=ω+ln(1+Ntδ), Nt=#of non-trading days between ti-1 and ti, and δ –contribution of a non-trading day.

The zt measure the shock or effect of the trades.

Here the parameter γ>0 indicates that if the |zt| rises then so will the volatility.

For the parameter θ<0, a negative ztwill increase the volatility to a greater degree than a positive zt.

This model performs well for extended periods of high volatility, but it is not as adequate when the periods of large fluctuation are short.

Exponential GARCH model

- Nelson introduced in 1991 his exponential GARCH model:

- Nelson, under certain conditions, showed the weak convergence of the AR(1)-EGARCH model to:

Binomial Trees

- This very basic approach was introduced in 1979 by Cox, Ross, and Rubenstein, and it connects very nicely to the asset valuation techniques.
- Binomial model of stock price changes is a simplified discrete process, which is much more flexible than the continuous geometric Brownian motion model.
- There are two types of trees:
- Standard trees – directly connected to the geometric Brownian motion, still subject to restriction
- Flexible trees – free from many restrictions, can adjust to control the level of volatility

General Binomial Tree

- We set a period of time in which we are interested in modeling the stock, an initial starting time to a termination date. This time span is then divided up into smaller intervals.
- At the end of each period, we can state what the possible values of the stock price.
- From each state, the stock only has two possible outcomes, to move up or to move down.
- The degree to which a stock moves up or down should be a measure determined by the volatility of the stock.
- There is an up-transitional probability and a down-transitional probability
- u=(Su/S0)=up ratio d=(Sd/S0)=down ratio

General Binomial Trees

- Recombination tree – the result of a down-up move is the same as a up-down move

Su

S0

Sud

Sd

- Each place where two lines cross is referred to as a node of the tree.

Standard Binomial Trees

- The up ratio and down ratio are fixed as are the transitional probabilities.
- The length of each period of time is also fixed.
- The tree is guaranteed to be a recombination tree.
- The tree will be centered:

d=1/u

Centering condition:

ud = e2rΔt

This way an up-down movement will take a forward value.

Determining Parameters

- Expected rate of return influences the expected value of the stock at later periods in time. We should have:

Where p is the up transitional probability.

Determining Parameters

- The volatility should be the means of determination.

In a standard tree, this σ is the same at every node, since:

Cox, Ross, Rubinstein Method

- From the volatility equation under a standard tree:

- Under the centering assumption:

d=1/u

- Due to the expected value of the stock at one period ahead:

Equal Probability Method

- Assume that p=.5, thus under the expected value:

- The volatility further causes:

- So together,

Option Evaluation

- The models previously introduced were not created with the purpose of predicting the value of the stock at some point in the future. Rather, we employ this type of modeling to find probabilistic distributions of the future stock prices.

Option Evaluation

- The models previously introduced were not created with the purpose of predicting the value of the stock at some point in the future. Rather, we employ this type of modeling to find probabilistic distributions of the future stock prices.
- Our intension is to use this distribution to value assets being sold today. Specifically, we would like to determine the fair value of financial contracts which involve the trading of stocks.
- Derivative Security-an instrument whose value depends on the underlying asset
- Long position in a security means that the individual owns that security. This person benefits if the price of the security increases.
- Short position in a security means that the individual borrowed the security and sold it in the market. Eventually, they will need to buy it back an return it to the owner. Any dividends, payments generated by the security must be paid to the original owner. This person benefits if the price of the security decreases.

Different Types of Financial Contracts

- Call Option – affords the buyer the right to purchase an underlying asset for a fixed price in the future.
- The fixed price of the underlying asset is referred to as the strike price.
- Put Option – affords the buyer the right to sell the underlying asset for a fixed price in the future.
- European Option - can only be exercised on one day, the expiration date.
- American Option – can be exercised at any point prior to expiration.

Other options are just variations and complications of the above forms.

Assumptions

- In order to determine the value of the previously listed options, we must make the following assumptions:
- Arbitrage – there are no opportunities to make risk-free profit
- Liquidity - there are enough buyers to satisfy sellers, and enough sellers to satisfy buyers
- No bid-ask spread – the price at which a person can sell a security is the same price at which someone can buy that security
- Constant interest rate
- No market impact
- Be able to borrow money at the risk-free rate of interest
- No payouts from underlying asset, no dividends
- For the following equations, we are working with the European Option

Breakdown of Value

- The cost of the option at any point in time is broken into two pieces:
- Intrinsic value – the value if the option were to exercised immediately
- Time value – whatever is not explained above

(The spot price is the current value of the underlying asset.)

- Intrinsic value is:
- Positive if the option is in the money
- Negative if the option is out of the money
- 0 if the option is at money or the strike and spot prices are equal

Put-Call Parity

- Assume that we have two investments:
- Buy the call option and sell the put option. Current Value: C-P
- Go long on the stock and sell a riskless, zero-coupon bond which would mature at T to a value of K. Current Value: S-e-r(T-t)K
- At time T, either the call or the put will be in the money. If the call, then C is worth ST-K, and the 1st investment is also worth ST-K. If the put, then P is worth K-ST, and the 1st investment is worth –(K-ST)= ST-K
- At time T , the 2nd investment is worth ST-K.
- Both investments have the same value at T and cost nothing to maintain, therefore they must also be equal at time t. i.e.

C-P=S-e-r(T-t)K

Expectations of the Value

- The European call option’s value is greater if:
- The price of the stock is higher, especially compared to K
- The volatility of the stock is larger
- The farther the expiration date
- As the call option nears the expiration date, the value of the option should be close to max(ST-K,0).
- If the expiration date is far away, then the value of the option should be close to ST.

Hedging Strategy

- The hedging portfolio is created to defray the risks associated with the selling of the option. It will be a combination of stocks and riskless bonds which are intended replicate the payoff at the time at which the option is exercised.
- The hedging portfolio’s value at any point in time should be the same value as the option. Therefore the hedge must be dynamic, constantly adjusting to rebalance.
- There are costs incurred with hedging portfolio
- Set-up costs or the original investment
- Maintenance costs
- Infusion of funds / additional monies needed to rebalance
- Transaction costs / fees or taxes

Hedging Strategy

- The hedging portfolio with only set-up costs is set to be self-financing. We call Δ the rate of change in the value of the option with respect to the change in value of the underlying asset.

- This same Δ is the number of shares of the underlying stock that need to be held in the hedging portfolio.
- Further, the value of the bond e-r(T-t)Bt in the hedging portfolio should be adjusted along with the delta. Bt is the value at expiration.

Portfolio

- Sold a Call Option, short the call
- Purchased Δ shares of the underlying, long that # of shares
- Loaned out the money, short the bond
- The idea: The investor can hold other assets to decrease risk, and therefore the risk influencing the discount rate will only be the risk which can not be diversified away.

Black Scholes

- In 1973, Black Scholes were the first to use the idea of a hedging strategy used by an investor in order to create a riskless portfolio.

As a result, the following option value is produced when applying geometric Brownian motion model.

- This calculation is done under the risk neutral assumption, meaning that μ=r

Black Scholes Calculation

- Samuelson and Merton 1969 recognize discounting method

Under GARCH & EGARCH Model

- Closed form solutions to the option valuation formula can not be determined, so most often simulation is implemented.
- For a particular GARCH process, Heston and Nandi(2000) have developed a closed form solution for the European Call Options

- The option valuation will not only be a function of the current or spot price, but also past prices.
- The average return is allowed to depend upon risk.

One Step Binomial Tree

- Our hedge portfolio has two possible values at time t1:

ΔSu+B=Cu or ΔSd+B=Cd

- Since B is a common feature, we can set:

ΔSu-Cu =ΔSd-Cd

- Thus we can solve for the Δ of the hedge:

Δ=(Cu-Cd)/(Su-Sd)

- We can also solve for B:

B=(SuCd-SdCu)/(Su-Sd)

- Since this hedge is created with the intention of mirroring the option:

C0= ΔS0+ e-r(Δ t)B

One Step Binomial Tree

C0= ((Cu-Cd)/(Su-Sd))S0+ e-r(Δ t) ((SuCd-SdCu)/(Su-Sd))

- It would appear that the option value does not depend on p, the transitional probability, however, we can manipulate the above formula to read:

C0= e-r(Δ t) [((S0-e-r(Δ t)Sd)/(Su-Sd))Cu + ((Sue-r(Δ t)-S0)/(Su-Sd))Cd]

C0= e-r(Δ t) [pCu + (1-p)Cd]

C0= e-r(Δ t) [C1]

- In order to find the current value, we can find the transitional probabilities and the values of Cu and Cd, then compute the expected value discounted back one time period.
- Multiple Step Binomial Tree can be computed by following this process repeatedly working backwards from time tn to tn-1 then to tn-2, etc. At each node one calculates the present value of the option.

Valuing the Other Options

- By the put-call parity, C-P=S-e-r(T)K, the European put can be calculated.

Therefore, under geometric Brownian motion, the value of put is:

- Under the Multiple Step Binomial Tree follow the same process as for the call, but compute the current value of the put at each node by using the transitional probabilities.
- Under the assumption of no dividends the American Call Option should have a value the same as the European Call

Dividend Effects

- So far we have assumed that the underlying asset does not produce dividends.
- There are two kinds of dividend payments that can be made:
- Lumpy dividends - dividends are awarded according to a schedule
- Continuous dividends
- For lumpy dividends in the geometric Brownian motion model, we can calculate the present value of the payment, D, and reduce initial value of the stock by that amount, St0-e-r(t1-t0)D. This also requires that the volatility measure be adjusted σt= ((St0)/St0-e-r(t1-t0)D) σ.

For continuous, we assume a µ=r-q, and we make the adjustment:

- When we have a multiple step binomial model and the dividends are lumpy, if the dividend is percentage of the spot price, then at the ex-dividend date, alter the tree to have Su(1-q) and Sd(1-q). If the lump payments act as a fixed dollar amount, the nodes will need to be shifted on that date by a fixed amount.

Estimating Volatility

- Use observed stock prices to estimate volatility, based upon:

Where the ti=i/N, so

From the likelihood, we create a MLE for the constant volatility.

Estimating Volatility

- Use observed option prices, we could also attempt to compute the value of the measure of the volatility that created it. Since the formula created from the geometric Brownian motion is not invertible, then iterative procedures are required to produce estimates.
- Bisections method
- Guess at the value of σ0.
- Compute the option value using the formula.
- If at the kth attempt, the formula value exceeds market value, then

σk=σk-1- σ0/(2k) is the new guess, otherwise

σk=σk-1+ σ0/(2k)

- Newton-Raphson method
- Guess at the value of σ0.
- Compute V(σ)=rate in change of option per rate in change in volatility =S(T(1/2))φ(d1)
- σk=σk-1 – [(C(σk-1)-C)/V(σk-1)]

Estimating Volatility

- The GARCH/EGARCH models assist in the estimation of unobserved volatility. Such calculations can be less burdensome than the implied volatility methods.
- These estimates of volatilities will be based solely upon the underlying asset returns rather than other options.
- Can produce out-of-sample estimates.
- Nelson has established the consistency of such estimators.

Goals to Estimating Volatility

- We would like to produce an estimator that is consistent and is asymptotically normal. So from the option formula under geometric Brownian motion, which is essentially a nonlinear stochastic regression, we would like to show that the least squares estimator will carry these two properties.
- Further, we would like to combine the information from the stock prices, as well as the option prices, to obtain the form of MLE for the volatility which can be shown to have the properties desired.

References

- Black, F. and Scholes, M. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81 (May 1973) 637-54.
- Chriss, N. A. Black-Scholes and Beyond Option Pricing Models. Chicago: Irwin, 1997.
- Dumas, B., Fleming, J., and Whaley, R.E. “Implied Volatility Functions: Empirical Tests.” The Journal of Finance, Vol. 53, No.6 (Dec., 1998), 2059-2106.
- Heston, S.L. and Nandi, S. “A Closed –Form GARCH Option Valuation Model.” The Review of Financial Studies, Vol.13, No. 3 (Autumn, 2000), 585-625.
- Hull, J. and White, A. “The Pricing of Options on Assets tih Stochastic Volatilities.” The Journal of Finance, Vol. 42, No. 2 (Jun, 1987), 281-300.
- Karatzas, I and Sheeve, S. Brownian Motion and Stochastic Calculus. New York: Springer Verlag, 1988.
- Merton, R.C. “Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science 4 (Spring 1973) 141-83.
- Osborne, M.F.M. “Brownian Motion in the Stock Market.” Operations Research, Vol.7, No.2. (Mar.-Apr., 1959), 145-173.
- Rossi, P.K. Edt. Modeling Stock Market Volatility Bridging the Gap to Continuous Time. San Diego: Academic Press, 1996.

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