7.0 Brownian Motion, Itô’s Lemma and Differentials Aimi Syairah 7.1 Brownian Motion Brownian Motion Arithmetic Brownian Motion Geometric Brownian Motion 7.1.1 Introduction We will now study the theoretical background for BlackScholes pricing. In order to price options, we need: 1. A model for the price movement of the underlying asset 2. A way to calculate the price movement of a claim on the asset as a function of the price movement of the asset • Brownian motion is a model for price movements • Itô’s lemma is a way to relate changes in values of functions of assets to changes in values of assets 7.1.2 Discrete Random Walks A one-dimensional discrete random walk models somebody starting out on the x axis at the point 0 and then moving left or right at the rate of 1 per unit time, with the direction being chosen randomly at every point Let 𝑋(𝑡) be the position at time 𝑡. For a random walk: 1. 𝑋(0) = 0 2. For 𝑡 > 0, if 𝑋 𝑡 − 1 = 𝑘, then 𝑘 + 1 with probability 1/2 𝑋 𝑡 = 𝑘 − 1 with probability 1/2 7.1.2 Discrete Random Walks To calculate the probability that 𝑋 3 = 1, we note that for 𝑋 to move 1 at time 3, it would have to move up twice and down once from time 0 There are 23 = 8 possibilities for 3 movements and of these 8, 3 of them would end up at 1, since we would need 2 ups and 1 down Hence Pr 𝑋 3 = 1 = 3 8 7.1.2 Discrete Random Walks Suppose that instead of moving 1 per unit of time, we moved ℎ per ℎ units of time and took the limit as ℎ → 0. We then have a continuous random walk The binomial random variable will converge to a normal random variable and hence become Brownian motion 7.1.3 Brownian Motion Brownian motion is denoted by 𝑍(𝑡) It is a random process, a collection of random variables indexed by time 𝑡, defined by the following properties: 1. 𝑍 0 = 0 2. 𝑍 𝑡 + 𝑠 |𝑍 𝑡 has a normal distribution with 𝜇 = 𝑍 𝑡 and 𝜎 2 = 𝑠 3. Increments are independent: 𝑍 𝑡 + 𝑠1 − 𝑍(𝑡) independent of 𝑍 𝑡 − 𝑍(𝑡 − 𝑠2 ) 4. 𝑍(𝑡) is continuous in 𝑡 is Example 7.1.3 The price of a stock follows a Brownian motion. The price of the stock at time 3 is 52. Determine the probability that the price of the stock is at least 55 at time 12 Solution 7.1.3 Pr 𝑍 12 > 55 𝑍 3 = 52 = Pr(𝑌 > 55) 𝑍 12 |𝑍(3) has a normal distribution with mean 𝑍 3 = 52 and variance = 9 Hence, 0.1587 Pr 𝑌 > 55 = 1 − 𝑁 55−52 9 =1−𝑁 1 = 7.1.4 Arithmetic Brownian Motion Arithmetic Brownian motion consists of a Brownian motion scaled by multiplication and shifted by addition If 𝑋(𝑡) is an arithmetic Brownian motion (where 𝑍 𝑡 is a Brownian motion), then 𝑋 𝑡 = 𝑋 0 + 𝛼𝑡 + 𝜎𝑍(𝑡) 𝑋 𝑡 + 𝑠 − 𝑋(𝑡) has a normal distribution with mean 𝛼𝑠 and variance 𝜎 2 𝑠 The parameter 𝛼 is called the drift of the process 7.1.4 Arithmetic Brownian Motion For the distribution of an arithmetic Brownian motion 𝑋 𝑢 : Let 𝑡 be the latest time for which we have the value 𝑋(𝑡) 𝑋 𝑡 + 𝑠 |𝑋 𝑡 , 𝑠 > 0 is normally distributed with mean 𝑋 𝑡 + 𝛼𝑠 and variance 𝜎 2 𝑠 Example 7.1.4 The price of a stock follows arithmetic Brownian motion of the form 𝑋 𝑡 = 𝑋 0 + 𝑡 + 0.2𝑍(𝑡). The current price of the stock is 40. Determine the probability that the price of the stock at time 4 is less than 43 Solution 7.1.4 Given 𝑋 𝑡 = 𝑋 0 + 𝑡 + 0.2𝑍(𝑡) or 𝛼 = 1, 𝜎 = 0.2 We want to evaluate the distribution at time 𝑠 = 4 𝑋 4 − 𝑋(0) is a normal random variable with mean 𝛼𝑠 = 1 4 = 4 and variance 𝜎 2 𝑠 = 0.22 (4) We want the probability that it is less than 43 − 40 = 3 Then, 3−4 Pr 𝑋 4 − 𝑋 0 < 3 = 𝑁 = 𝑁(−2.5) 0.2 4 7.1.5 Geometric Brownian Motion Arithmetic Brownian motion is not a good model for stock price movement because it can go negative and does not scale with stock price Just like we transform a normal distribution to a lognormal distribution, we transform arithmetic Brownian motion to geometric Brownian motion 7.1.5 Geometric Brownian Motion 𝑋(𝑡) follows geometric Brownian motion if ln 𝑋(𝑡) follows arithmetic Brownian motion If ln 𝑋 𝑡 is a normal random variable, then 𝑋(𝑡) is a lognormal random variable If ln 𝑋𝑡 𝑋0 is normal with mean 𝜇𝑡 and variance 𝜎 2 𝑡, then 𝑋𝑡 𝑋0 is lognormal, and its mean and variance are 𝐸[𝑋(𝑡) 𝑋 0 ] = 𝑉𝑎𝑟[𝑋(𝑡) 𝑋(0)] = 2𝑡 𝜇𝑡+0.5𝜎 𝑒 2𝑡 2𝑡 2𝜇𝑡+𝜎 𝜎 𝑒 (𝑒 − 1) 7.1.5 Geometric Brownian Motion Let 𝑆 𝑡 be the time- 𝑡 stock’s price. Suppose the volatility of the stock is 𝜎, hence 𝑉𝑎𝑟[ln 𝑆 𝑡 𝑆 0 = 𝜎 2 𝑡 Suppose the stock pays no dividends, then the only return from the stock is its growth in price Suppose 𝐸 𝑆 𝑡 = 𝑆(0)𝑒 𝛼𝑡 , i.e., the continuously compounded expected rate of return on the stock is 𝛼 7.1.5 Geometric Brownian Motion Now assume that the stock pays dividends at a continuously compounded rate of 𝛿 There are two sources of earnings on the stock: the dividends, which return 𝛿 and the growth in price The sum of these returns is 𝛼, so the rate at which the stock price increases is 𝛼 − 𝛿 The expected stock price increase is 𝐸𝑆 𝑡 =𝑆 0 𝑒 𝛼−𝛿 𝑡 𝐸[𝑆(𝑡) 𝑆(0)] = 𝑒 and 𝛼−𝛿 𝑡 7.1.5 Geometric Brownian Motion But, for a lognormal distribution, the expected value is (𝜇+0.5𝜎 2 )𝑡 𝐸[𝑋 𝑡 𝑋(0)] = 𝑒 It follows that α − 𝛿 = 𝜇 + 0.5𝜎 2 In other words, to go from a geometric Brownian motion to the associated arithmetic Brownian motion, we subtract 0.5𝜎 2 We must go from the geometric Brownian motion to the associated arithmetic Brownian motion if we wish to use the normal tables to look up probabilities or percentiles 7.1.5 Geometric Brownian Motion If we want to calculate probabilities or percentiles for a stock whose return is 𝛼 and pays dividends 𝛿 with volatility 𝜎, set 𝑚 = 𝜇𝑡 = (𝛼 − 𝛿 − 0.5𝜎 2 )𝑡 and 𝑣 = 𝜎 𝑡 and use those parameters to look up the normal distribution in the table Example 7.1.5 The time 𝑡 price of a stock is 𝑆(𝑡). The stock price is modeled as following geometric Brownian motion. Your are given: The stock’s continuously compounded expected rate of return is 0.15 The stock’s continuously compounded dividend yield is 0.04 The stock’s volatility is 0.3 𝑆(0) = 45 𝑆(0.6) = 47 Calculate Pr(𝑆 1 < 45) given the above facts Solution 7.1.5 The price of the stock at time 0 is irrelevant 𝜎 = 0.3 𝑆(1) ln 𝑆 0.6 is normally distributed with parameters 𝑚 = 𝛼 − 𝛿 − 0.5𝜎 2 𝑡 = (0.15 − 0.04 − 0.5 0.32 1 − 0.6 = 0.026 and 𝑣 2 = 𝜎 2 𝑡 = 0.32 1 − 0.6 = 0.036 ln 𝑆(1) 𝑆(0) − 𝑚 Pr( 𝑆 1 < 45) = 𝑁 𝑣 =𝑁 ln 45 47 − 0.026 0.036 = 𝑁 −0.3662 7.2 Differentials and Itô’s Lemma Stochastic Differential Equations Itô’s Lemma Martingales 7.2.1 Differentials In ordinary calculus: 𝑦 = 𝑒 𝐽𝑡 𝑑𝑦 = 𝐽𝑒 𝐽𝑡 𝑑𝑡 Multiplying both sides by 𝑑𝑡 gives us a differential expression: 𝑑𝑦 = 𝐽𝑒 𝐽𝑡 𝑑𝑡 = 𝐽𝑦𝑑𝑡 The last expression says that the change of 𝑦 is proportional to 𝐽𝑦 times the change of 𝑡 7.2.1 Differentials Suppose there were some uncertainty in this rate of change. We could add 𝜎𝑑𝑍(𝑡) to the right hand side and get 𝑑𝑦 = 𝐽𝑑𝑡 + 𝜎𝑑𝑍(𝑡) 𝑦 𝑑𝑍(𝑡) is a differential of a Brownian motion. It is the limit, as ℎ goes to 0, of a random variable equal to ℎ with probability 0.5 and – ℎ with probability 0.5 𝑡 will represent time 𝑍(𝑡) is independent of 𝑡 7.2.1 Differentials The value at time 𝑡 of an arithmetic Brownian motion 𝑋(𝑡) with drift 𝜇 and coefficient of 𝑍(𝑡) equal to 𝜎 can be expressed as: 𝑋 𝑡 = 𝑋 0 + 𝜇𝑡 + 𝜎𝑍(𝑡) The differential of this arithmetic Brownian motion 𝑋(𝑡) is 𝑑𝑋 𝑡 = 𝜇𝑑𝑡 + 𝜎𝑑𝑍(𝑡) This says that a small change in 𝑋 equals 𝜇 times a small change in time, or 𝑑𝑡, plus 𝜎 times a small change in a Brownian motion, or 𝑑𝑍(𝑡) 7.2.1 Differentials The Brownian motion itself, 𝑍(𝑡), is a r.v. normally distributed with mean 0 and variance 𝑡 A small change in a Brownian motion is a r.v. normally distributed with mean 0 and variance 𝑑𝑡 The time value of a GBM 𝑋(𝑡) can be expressed in terms of its logarithm ln 𝑋(𝑡) = ln 𝑋(0) + 𝛼 − 𝛿 − 0.5𝜎 2 𝑡 + 𝜎𝑍(𝑡) (1) The differential of this expression is 𝑑(ln 𝑋(𝑡)) = 𝛼 − 𝛿 − 0.5𝜎 2 𝑑𝑡 + 𝜎𝑑𝑍(𝑡) (2) 7.2.1 Differentials The same GBM can be expressed directly as 𝛼−𝛿−0.5𝜎 2 𝑡+𝜎𝑍(𝑡) 𝑋 𝑡 = 𝑋(0)𝑒 (3) The differential of this GBM 𝑋(𝑡) is 𝑑𝑋 𝑡 = 𝛼 − 𝛿 𝑋 𝑡 𝑑𝑡 + 𝜎𝑋 𝑡 𝑑𝑍(𝑡) If 𝑋(𝑡) satisfies any of equation (1) through (3), then 𝑋(𝑡) follows a GBM with coefficients 𝛼 − 𝛿 𝑋 𝑡 for 𝑑𝑡 and 𝜎𝑋 𝑡 for 𝑑𝑍(𝑡) 7.2.1 Differentials Any process of the form 𝑑𝑆 𝑡 = (𝛼 − 𝛿) 𝑆 𝑡 , 𝑡 𝑑𝑡 + 𝜎 𝑆 𝑡 , 𝑡 𝑑𝑍(𝑡) where (𝛼 − 𝛿) 𝑆, 𝑡 and 𝜎 𝑆, 𝑡 are functions of 𝑆 and 𝑡 is called Itô process The coefficient of 𝑑𝑡 is called the drift process Example 7.2.1A You are given an Itô process of the form 𝑑𝑆 𝑡 = 0.25𝑆 𝑡 𝑑𝑡 + 0.10𝑆 𝑡 𝑑𝑍(𝑡) Calculate the probability that 𝑆(𝑡) is at least 5% higher than 𝑆(0): i. At time 𝑡 = 0.1 ii. At time 𝑡 = 1 Solution 7.2.1A The Itô process is a GBM with 𝛼 − 𝛿 = 0.25 and σ = 0.10 We calculate 𝜇 = 𝛼 − 𝛿 − 0.5𝜎 2 to obtain the 𝜇 of the corresponding arithmetic Brownian motion d ln S (t ) (0.25 0.5(0.10 2 ))dt 0.10dZ (t ) 0.245dt 0.10dZ (t ) Then Pr S (t ) / S (0) 1.05 Pr ln S (t ) ln S (0) ln 1.05 Solution 7.2.1A 1. For 𝑡 = 0.1, 𝑚 = 𝜇 0.1 = 0.0245, 𝑣 = 0.10 0.1 S (1) ln 1.05 0.0245 Pr ln ln 1.05 1 N 0.22121 0.10 0.1 S ( 0) 2. For 𝑡 = 1, 𝑚 = 0.245, 𝑣 = 0.10 ln 1.05 0.245 1 N 0.97512 0.10 Example 7.2.1B You are given that 𝑆(𝑡) follows an Ito process of the form dS (t ) 0.15dt 0.20dZ (t ) S (t ) Given that 𝑆(9) = 40, calculate the probability that 40 < 𝑆(13) < 50 Solution 7.2.1B 𝑆(𝑡) follows GBM, implying 𝑆(𝑡) is lognormally distributed The associated normal distribution has parameters 𝜇𝑡 = 4 0.15 − 0.5(0.22 ) = 0.52 and 𝜎 𝑡 = 0.2 4 = 0.4 Solution 7.2.1B The probability that 40 40 < 𝑆(13) 𝑆(9) < 50 40 or that 0 < ln[𝑆(13) 𝑆(9)] < ln 1.25 is ln 1.25 0.52 0.52 N N 0.13220 0 .4 0.4 7.2.2 Language of Brownian Motion 𝑑𝑆 𝑆 = 𝛼 − 𝛿 𝑑𝑡 + 𝜎𝑑𝑍(𝑡) is a form of an Itô process where: 𝛼 − 𝛿 and 𝜎 are constants The stock follows the Black-Scholes framework The continuous rate of increase in the stock price is the continuously compounded rate of return on the stock (𝛼) minus the continuous dividend rate (𝛿) The volatility is 𝜎 To actually apply the Black-Scholes formula, we will need the dividend rate and the risk-free rate 7.2.3 Itô’s Lemma An Itô process is a random process 𝑋(𝑡) whose differential can be expressed as dX (t ) t , X (t ) dt t , X (t ) dZ (t ) It generalizes arithmetic and geometric Brownian motions For arithmetic Brownian motion, 𝜉 𝑡, 𝑋 𝑡 𝜎 𝑡, 𝑋 𝑡 = 𝜎 with 𝛼 and 𝜎 constants For geometric Brownian motion, 𝜉 𝑡, 𝑋 𝑡 𝜎 𝑡, 𝑋 𝑡 = 𝜎𝑋(𝑡) = 𝛼 and = 𝜉𝑋(𝑡) and 7.2.3 Itô’s Lemma Itô’s lemma is a formula for evaluating 𝑑𝐶(𝑆, 𝑡) if 𝐶(𝑆, 𝑡) is a function of 𝑆 and 𝑡. From ordinary calculus’ chain rule: dC Cs dS Ct dt where 𝐶𝑠 = 𝑆(𝑡) and 𝐶𝑡 = 𝜕𝐶 𝜕𝑡 𝜕𝐶 𝜕𝑆 is the partial derivative of 𝐶(𝑡) wrt is the partial derivative of 𝐶(𝑡) wrt to 𝑡 7.2.3 Itô’s Lemma Itô’s lemma: 𝑑𝐶 = 𝐶𝑠 𝑑𝑆 + 0.5𝐶𝑠𝑠 𝑑𝑆 Where 𝐶𝑠𝑠 = 𝐶(𝑡) wrt 𝑆(𝑡) 𝜕2 𝐶 𝜕𝑆 2 2 + 𝐶𝑡 𝑑𝑡 , the second partial derivative of In ordinary calculus, second order terms such as 𝑑𝑡 2 are zero In stochastic calculus 𝑑𝑍 𝑡 × 𝑑𝑍 𝑡 = 𝑑𝑡 ≠ 0 To multiply differentials, we use the following table: Multiplication table 𝑑𝑡 𝑑𝑍(𝑡) 𝑑𝑡 0 0 𝑑𝑍(𝑡) 0 𝑑𝑡 Example 7.2.3A You are given that 𝑋 𝑡 = 𝛼𝑡 + 𝜎𝑍(𝑡). Calculate 𝑑𝑋 𝑡 using Itô’s lemma Solution: 𝑋(𝑡) is a function of 𝑍(𝑡) so, 𝑋 will play the role of 𝐶 and 𝑍 will play the role of 𝑆 in Itô’s lemma 𝜕𝑋 𝜕𝑍 =𝜎 𝜕2 𝑋 𝜕𝑍 2 𝜕𝑋 1 𝜕2𝑋 𝑑𝑋 = 𝑑𝑍 + 𝑑𝑍 2 𝜕𝑍 2 𝜕𝑍 𝜕𝑋 𝜕𝑡 2 =0 𝜕𝑋 𝜕𝑡 =𝛼 𝜕𝑋 + 𝑑𝑡 = 𝜎𝑑𝑍 + 0 + 𝛼𝑑𝑡 𝜕𝑡 does not involve 𝑍(𝑡) as 𝑍(𝑡) in Itô’s lemma is treated as an independent variable from t Example 7.2.3B You are given that 𝑋 𝑡 = 𝑋(0)𝑒 𝑑𝑋(𝑡) using Itô’s lemma Solution: 2 +𝜎𝑍 𝑡 𝜕𝑋 𝛼−0.5𝜎 = 𝜎𝑋 0 𝑒 𝜕𝑍 𝜕2 𝑋 2 = 𝜎 𝑋(𝑡) 𝜕𝑍 2 𝜕𝑋 = 𝛼 − 0.5𝜎 2 𝑋(𝑡) 𝜕𝑡 𝛼−0.5𝜎 2 𝑡+𝜎𝑍(𝑡) . Calculate = 𝜎𝑋(𝑡) 𝑑𝑋 𝑡 = 𝜎𝑋 𝑡 𝑑𝑍 𝑡 + 0.5𝜎 2 𝑋 𝑡 𝑑𝑍(𝑡) 2 + 𝛼 − 0.5𝜎 2 𝑋 𝑡 𝑑𝑡 = 𝜎𝑋 𝑡 𝑑𝑍 𝑡 + 0.5𝜎 2 𝑋 𝑡 𝑑𝑡 + 𝛼 − 0.5𝜎 2 𝑋 𝑡 𝑑𝑡 = 𝛼𝑋 𝑡 𝑑𝑡 + 𝜎𝑋 𝑡 𝑑𝑍(𝑡) Example 7.2.3C 𝑑𝑋(𝑡) 𝑋(𝑡) You are given using Itô’s lemma. Solution: = 𝜉𝑑𝑡 + 𝜎𝑑𝑍(𝑡). Calculate 𝑑 ln 𝑋(𝑡) Let 𝑌 𝑡 = ln 𝑋(𝑡) 𝑑𝑌(𝑡) 1 = 𝑑𝑋(𝑡) 𝑋(𝑡) 𝑑 2 𝑌(𝑡) 1 =− 2 𝑑𝑋(𝑡) 𝑋(𝑡)2 𝑑𝑌(𝑡) =0 𝑑𝑡 Example 7.2.3C Hence: 1 1 𝑑 𝑌(𝑡) = 𝑑𝑋 𝑡 − 0.5 𝑋 𝑡 𝑋(𝑡)2 = 𝜉𝑑𝑡 + 𝜎𝑑𝑍 𝑡 − 0.5𝜎 2 𝑑𝑡 = 𝜉 − 0.5𝜎 2 𝑑𝑡 + 𝜎𝑑𝑍(𝑡) 𝑑𝑋(𝑡)2 Exercise 7.2.3 Given an Itô process 𝑑𝑋(𝑡) and 𝑑𝑌(𝑡). 𝑑𝑌(𝑡) is defined by 𝑑𝑌(𝑡) = 0.1𝑑𝑡 + 0.5𝑑𝑍(𝑡) 𝑌(𝑡) And 𝑋 = 𝑌𝑒 0.02𝑡 𝑑𝑋(𝑡) can be expressed as 𝑑𝑋 𝑡 = 𝛼𝑋 𝑡 𝑑𝑡 + 𝜎𝑋 𝑡 𝑑𝑍(𝑡) Use Itô’s lemma to find 𝛼 7.2.4 Martingales ( An Itô process is a martingale if and only if the coefficient of 𝑑𝑡, the drift, is identically zero We can use Itô’s lemma to calculate the coefficient of 𝑑𝑡 Example 7.2.4 The process 𝑋 𝑡 = 𝑍(𝑡)3 + 𝑐𝑡𝑍(𝑡) is a martingale. Determine c. (Easier way to solve) Solution: dX (t ) 3Z (t ) 2 ct dZ (t ) d 2 X (t ) 6 Z (t ) 2 dZ (t ) dX (t ) cZ (t ) dt dX (t ) 3Z (t ) 2 ct dZ (t ) 0.56 Z (t ) dt cZ (t )dt 3Z (t ) 2 ct dZ (t ) (3 c) Z (t )dt To make (3 c) Z (t ) 0, we need c 3 7.2.5 Sharpe Ratio If an Itô process is expressed as dS (t ) t , S (t ) t , S (t ) dt t , S (t ) dZ (t ) S (t ) We use 𝛼 𝑡, 𝑆(𝑡) and 𝜎 𝑡, 𝑆(𝑡) to obtain the Sharpe ratio of 𝑆. The dividend rate is not subtracted from 𝛼 (Follow geometric) If an unusual asset followed an arithmetic Brownian motion, we have to express the process as the equation above, by dividing the coefficients of 𝑑𝑡 and 𝑑𝑍(𝑡) by 𝑆(𝑡) because 𝑑𝑆(𝑡) is divided by 𝑆(𝑡) Example 7.2.5A The price of a nondividend paying stock 𝑆 satisfies the following stochastic differential equation: d ln S (t ) 0.1dt 0.3dZ (t ) The continuously compounded risk-free rate is 0.04. Calculate the Sharpe ratio for 𝑆 (𝐹𝑜𝑙𝑙𝑜𝑤 𝐴𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐) Solution: dS (t ) 0.1 0.5(0.32 ) dt 0.3dZ (t ) 0.145dt 0.3dZ (t ) S (t ) r 0.145 0.04 Sharpe ratio 0.35 0 .3 7.2.5 Sharpe Ratio The Sharpe ratio may vary with time (𝑡), the risk-free rate 𝑟(𝑡) which itself may vary with time, or with the Brownian motion 𝑍(𝑡) that is part of the 𝑆(𝑡) At any time 𝑡, for two Itô processes depending on the same 𝑍(𝑡), the Sharpe ratios are equal Example 7.2.5B (This Ito process follow geometric) You are given two nondividend paying assets 𝑋(𝑡) and 𝑌(𝑡) satisfying the following stochastic differential equations: dX (t ) 0.15 X (t )dt 0.30 X (t )dZ (t ) dY (t ) AY (t )dt 0.20Y (t )dZ (t ) The continuously compounded risk-free rate is 0.03. Determine A. Solution: 0.15 0.03 0 .4 0 .3 A 0.03 For Y (t ) : 0.4, A 0.11 0 .2 From X (t ) : Next Agenda Bond Pricing and Binomial Interest Rates