3.1 Itô s Lemma for Continuous Stochastic Variables

Transcription

1 Lecture 3 Log Normal Distribution 3.1 Itô s Lemma for Continuous Stochastic Variables Mathematical Finance is about pricing (or valuing) financial contracts, and in particular those contracts which depend of the value of another contract. Now we are going to write down a result which tells us how a function that depends on a random variable changes in time. We assume that the function f(s, t) is a smooth function of S and t. Then if S satisfies the SDE ds = µsdt + σsdw Itô s Lemma states that the distribution of the change in f is given by ( f f df = + µs t S + 1 ) 2 σ2 S 2 2 f S 2 dt + σs f dw (3.1) S where df = f(s + ds, t + dt) f(s, t). There are two things to note here. Firstly, the resulting df is a distribution characterised by the term in blue containing the dw term. Because the function f depends on something that is random and unpredictable, the value of the function itself will also be random and unpredictable in the future. Secondly, we have this extra red term appearing from nowhere, so where does it come from? Well, we are not going to spend time going through this but if we think about a Taylor series approximation normally the ds 2 which this term is related to is much much smaller than all the other terms as ds 2 << ds. However, in stochastic calculus we find that ds 2 = O(dt) 22

2 MATH20912 Lecture 3 which comes from the earlier result that E[W 2 ] = t. Expanding the above we have ds 2 = (µsdt + σsdw )(µsdt + σsdw ) = σ 2 S 2 dw 2 + o(dt) σ 2 S 2 dt and we can see where the red term has come from. You are not required to know how to derive this result for the purpose of this course only how to apply Itô s Lemma in practical situations. Later on in the course we will need to apply it in order to derive the value of contracts. Example 3.1. Find the SDE satisfied by f = S 2. Solution

3 3.2 Log Normal Distribution Example 3.2. Show that the stochastic differential equation (SDE) for f = ln S is: ) ln(s(t)) = ln(s 0 ) + (µ σ2 t + σw (t) (3.2) 2 Hint: remember that adding together normal distributions results in another normal distribution. This means that constant coefficient SDE s can be integrated using the result Solution 3.2. t 0 dw = W (t). (3.3) 24

4 From (3.2), we can deduce that the logarithm of the share price ln(s(t)) is normally dis- 25

5 ( tributed with mean ln(s 0 ) + µ σ2 2 ln(s(t)) N ) t and variance σ 2 t. It follows then that ( ln(s 0 ) + ) ) (µ σ2 t, σ 2 t 2 (3.4) Example 3.3. Consider a share with an initial price of 40, an expected return of 16% and a volatility of 20%. Find the probability distribution of ln S in six months. Solution

6 MATH20912 Lecture Geometric Brownian Motion Definition A stochastic process S(t) is said to follow a Geometric Brownian Motion if ds = µsdt + σsdw where W is a Wiener process, and µ and σ are constants. Given that our share price model is a GBM, we have some properties and results that can be useful: 1. We have an exact formula for the share price at time t given by S(t) = S(0)e (µ σ2 /2)t+σW (t) 2. There is a closed form expression for the probability density function at time t given by ( ( 1 ln s f St (s; µ, σ, t) = sσ 2πt exp ln(s0 ) (µ σ 2 /2)t ) ) 2 2σ 2. t The two results stated here are particularly important when it comes to evaluating financial contracts depending on S using numerical methods. The first result is particularly useful when generating random samples for S at some future time t as we only need a single random number to be generated for each path. The second result is used in more complex numerical methods that take advantage of the fact that expectations can be evaluated as some integral involving the probability distribution. Example 3.4. Write down a formula for S(t) in terms of the standard normal distribution. Solution

7 Example 3.5. Try to draw a sketch of the log-normal distribution for S(t). Solution 3.5. Fill in Figure Is It A Good Model? How good is GBM as a model of the share price? Well, it certainly has a lot of the properties that we would like to see in a good model of the share price such as independent increments and it works very well for the most part but there are two main flaws. They are: the observed volatility σ in real markets is not constant in time, there are often large non normal jumps in the stock market, when external events cause a large number of participants to want to either buy or sell at the same time. Trying to come up with models that can overcome these flaws has been the focus of much research over the years, however in the light of recent market failures academics and practitioners are beginning to question whether such a simple model can ever really capture everything about the market. 28

8 MATH20912 Lecture 3 Figure 3.1: Drawing a log-normal distribution. 29