Seminar 2 A Model of the Behavior of Stock Prices. Miloslav S. Vosvrda UTIA AV CR

Transcription

1 Seminar A Model of the Behavior of Stock Prices Miloslav S. Vosvrda UTIA AV CR

2 The Black-Scholes Analysis Ito s lemma The lognormal property of stock prices The distribution of the rate of return Estimating volatility from historical data Option valuation

3 Ito s lemma We assume that interest rates are constant and the same for all maturities. The price of a stock option is a function of the underlying stock s price and time. Suppose that the value of a variable x follows an Ito process dx = a( x, t) dt + b( x, t) dz where dz is a Wiener process and a and b are functions of x and t. The variable x has a drift rate of b a and a variance rate of.

4 Ito s lemma says that a function G, of x and t follows the process Thus G also follows an Ito process. It has a drift rate of G G 1 G G dg = ( a + + b ) dt + b dz x t x x G G 1 G a + + b x t x and a variance rate of G ( ) x b

5 Let us apply Ito s lemma on a price S. It follows that process followed by function, G, of S and t is G G 1 G ( µ σ ) S t S dg = S + + S dt + G + σ Sdz S

6 Application to Forward Contracts Assume risk-free rate of interest is constant and equal to r for all maturities Define F as the forward price,i.e. so that F = S e r T t ( ) F S F F = e, = 0, = r S e S t r( T t) r( T t)

7 The process for F is given by rt ( t ) rt ( t) df = ( e µ S r S e ) dt + σ F dz F = S e r T t ( ) Substituting,this becomes df = ( µ r) F dt + σ F dz It has an expected growth rate of rather than µ µ r

8 Application to the logarithm of the stock price Define G = ln S Since G 1 G 1 G =, =, = S S S S t it follows that the process followed by G is dg = ( µ σ ) dt + σ dz µ σ Since and are constants, G follows a generalized Wiener process. This means that 0

9 the change in G between the current time, t,and some future time, T, is normally distributed with mean and variance σ ( µ ) ( T t) σ ( T t) Thus µ σ σ lns ln S N(( ) ( T t), ( T t)) T t

10 The lognormal property of stock prices ln S N(ln S + ( µ σ ) ( T t), σ ( T t)) T t This shows that lns T has a lognormal distribution. Our uncertainty about the logarithm of the stock price is var[ln S ] T t T

11 Example Consider a stock with an initial price of 40Kc, an expected return of 16% per annum, and a volatility of 0% per annum. What is 95% - interval of confidence in 6 month ahead? S T What is a mean value and variance of?

12 %95-% CONFIDENCE INTERVAL FOR STOCK PRICE AFTER 6 MONTHS %initial price cena=40; %expected return p.a. mi=0.16; %volatility p.a. sigma=0.; %six month as a share of year t=6/1; mi=log(cena)+(mi-sigma.^./).*t; sigma=sqrt(sigma.^.*0.5); spodni=mi-1.96.*sigma; horni=mi+1.96.*sigma;

13 %lower interval lower=exp(spodni) %upper interval upper=exp(horni) %mean value of price after 6 months mean=exp(mi) %standard deviation standard=cena.*sigma %copyright kubin-netuka

14 lower = upper = mean = standard =

15 The distribution of the rate of return Define the annualized continuously compounded rate of return between t and T as η It follows that S = S e η ( T t) T and thus and η = T 1 t ln S ln (( t S ), ) ( T t ) 1 η = T N µ σ σ T S T S

16 Example Consider a stock with an expected return of 17% per annum and a volatility of 0% per annum. What is 95% confidence interval for the actual return realized over 10 years?

17 %95-% CONFIDENCE INTERVAL FOR RETURN AFTER 10 YEARS %expected return p.a. mi=0.17; %volatility p.a. sigma=0.; for cas=1:10; i(cas)=cas; %mean value of return vynos(cas)=((mi-sigma.^./)); %lower interval spodni(cas)=vynos(cas)-1.96.*sigma./cas; %upper interval horni(cas)=vynos(cas)+1.96.*sigma./cas; end; plot(i,vynos,i,spodni,'+',i,horni,'*') xlabel('years') ylabel('return') title('95% confidence interval for return after 10 years') %copyright kubin-netuka

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19 Estimating volatility from historical data Define Number of observations: n+1 Stock price at end of ith interval: Length of time interval in years: Put S Since u i u i i = ln for i = 1,, n S i 1 S = S e i i 1 u i S for i=0,, n i τ is the continuously compunded return in the ith interval.

20 The unbiased estimator of the variance of the is given by s Since Var[ ]= the estimator 1 n = n 1 ( u u) i i= 1 u σ i ( T t) s s * = τ u i is the estimator for the parameter σ The standard error of this estimate can be approximately s * n

21 Option valuation We consider a European call option. Suppose that a stock price is currently 00Kc and that it is known that at the end of one month the price will be either 0Kc or 180Kc. Consider to buy the stock for 10Kc in 1 month. If the stock price turns out to be 0Kc the value of the option is 10Kc. If the stock price turns out to be 180Kc, the value of the option is zero.

22 Consider a portfolio consisting of a long position in shares of the stock and a short position in one call option. The value of this portfolio is 0* -10 if stock price moves up and 180* if it moves down. When α is chosen equal to 0.5, these two value are the same: α α 0* -10= 180* =45 For this value of α the portfolio is therefore riskless. The current value of the portfolio is 00*0.5-f=50-f where f is the current value of the call option. α α α

23 Riskless portfolio must earn the risk-free rate of interest. Suppose that risk-free rate is 10% per month. It follows that 1.1(50-f)=45 or f=50-45/1.1=9.09 that is, the current value of the option is 9.09Kc.

24 Next lecture Properties of the Black-Scholes Differential Equation