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MATH 4514 Financial Economics in Actuarial Science: Lecture Note 3 -Continuous time options pricing model: Black-Scholes formulation | The Hong Kong University of Science and Technology

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MATH 4514 Financial Economics in Actuarial Science: Lecture Note 3 -Continuous time options pricing model: Black-Scholes formulation | The Hong Kong University of Science and Technology Introduction In previous chapter, we consider a multi-period binomial tree model to simulate the price movement of the asset price: The asset price at any period 𝑆𝑡 either climbs up to 𝑢𝑆𝑡or drops down to 𝑑𝑆𝑡 in the next period with certain probabilities. 𝑆 (period 0) 𝑢𝑆 𝑑𝑆 (period 1) 𝑢2𝑆 𝑢𝑑𝑆 𝑑2𝑆 (period 2) 𝑢3𝑆 𝑢2𝑑𝑆 𝑢𝑑2𝑆 𝑑2𝑆 (period 3)3 MATH4514 Financial Economics in Actuarial Science Lecture Note 3: Continuous time options pricing model One limitation of the binomial tree model is that the stock price either goes up or goes down by a fixed percentage over each period. In reality, the return rate of stock can be arbitrary and the price should evolve continuously. In this chapter, we shall present a continuous time model proposed by Black and Scholes. We would lie to study the following issues: 1. How to model the continuous evolution of stock price? (Continuous “random walk” and geometric Brownian motion) 2. Under the continuous time framework, the calculation of derivative price requires some calculus technique and it is clear that the price depends on the underlying asset’s price which is a random variable. How do we perform “calculus” on random variables? (Ito’s calculus) 3. How do we price the derivatives using no arbitrage pricing principle and risk-neutral valuation approach under continuous time framework?4 MATH4514 Financial Economics in Actuarial Science Lecture Note 3: Continuous time options pricing model Price dynamic of asset price: Geometric Brownian Motion In order to model the price dynamic of stock price, we first study the historical price movement of Google stock (Jan 19, 2005 to Feb 25, 2008). Given the set of historical stock price at each closing day (denoted by 𝑆𝑖), we proceed to compute the daily (1 day), weekly (5 days) and monthly (22 days) rate of return of the stock. These quantities can be calculated using the following formula: 𝑅𝑖 = 𝑆𝑖 − 𝑆𝑖−1 𝑆𝑖−1 where 𝑆𝑖 is the stock price at the end of 𝑖𝑡ℎ day (for daily return)/ 𝑖𝑡ℎ week (for weekly return)/ 𝑖𝑡ℎ month (for monthly return). The sample mean and sample standard deviation of 𝑅𝑖 can be calculated as 𝑅̅ = 1 𝑁 ∑ 𝑅𝑖 𝑁 𝑖=1 , 𝑆𝐷 = √ 1 𝑁 − 1 ∑(𝑅𝑖 − 𝑅̅)2 𝑁 𝑖=1 , where 𝑁 is the number of samples.5 MATH4514 Financial Economics in Actuarial Science Lecture Note 3: Continuous time options pricing model We finally compute the scaled return (denoted by 𝑅𝑖′) as 𝑅𝑖′ = 𝑅𝑖 − 𝑅̅ 𝑆𝐷 . We observe from the figure below that the distribution of 𝑅𝑖′ looks like the normal distribution. So we conjecture (or assume) that the return rate of the stock 𝑅𝑖 follows normal distribution. That is, 𝑅𝑖 − 𝔼[𝑅𝑖] 𝜎𝑅𝑖 ~𝑁(0, 1).

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