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MATH 4514 Financial Economics in Actuarial Science: Lecture Note 2 - Binomial Tree Pricing Model | The Hong Kong University of Science and Technology

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MATH 4514 Financial Economics in Actuarial Science Lecture Note 2 Binomial Tree Pricing Model | The Hong Kong University of Science and Technology ntroduction In Chapter 1, we have examined various properties of option prices and the application of options in risk management. It remains to obtain the pricing formula of options. As mentioned earlier, future price movement of asset price will significantly affect the options price. As an example, we consider 1-year European call option with strike price $51 and consider the following two scenarios: $48 𝑺𝟏 = $𝟓𝟓 𝑆1 = $41 Now (Time 0) Maturity (Year 𝟏) 𝑆1 = $48 The option will be exercised when 𝑺𝟏 = $𝟓𝟓 (yield a profit of $4). So the option price will be positive. $48 𝑆1 = $50 𝑆1 = $46 Now (Time 0) Maturity (Year 𝟏) 𝑆1 = $48 The option is never exercised at maturity date and the options price is simply 𝟎.3 MATH4514 Financial Economics in Actuarial Science Lecture Note 2: Binomial Tree Pricing Model In order to derive the pricing formula of options, it is essential to develop a mathematical model to simulate the future price movement of the asset price. In this chapter, we shall introduce a binomial tree pricing model which is the simplest model to model the asset price movement. The model can be described as follows: time 0 𝑇 ⋮ ⋮ ⋮ (Period 0) (Period 1) (Period 2) (Period 𝑛) Δ𝑡 The time horizon ሾ0, 𝑇ሿ is divided into 𝑛 periods. Given 𝑆𝑡, the asset price either move up to 𝑢𝑆𝑡 and 𝑑𝑆𝑡 in the next period4 MATH4514 Financial Economics in Actuarial Science Lecture Note 2: Binomial Tree Pricing Model Based on this binomial tree model, our goal is to develop a pricing theory under this model:  We will study how to price an option (or other derivatives) using no arbitrage pricing principle and replication technique.  We will also develop another equivalent pricing approach, which is known as risk-neutral valuation principle. The principle states that the option price at time 0 (or at any time 𝑡 < 𝑇) can be expressed as the present value of expected option payoff (or value) at future time 𝑇: 𝑉0 = 𝑉0(𝑆0) = 𝑒−𝑟𝑇𝔼𝑄ሾ𝑉𝑇(𝑆𝑇)|𝑆0ሿ.  We will also study how to calculate the prices for American options and path-dependent options. In addition, we will also study how to construct a binomial tree model using historical path of asset prices (i.e. model calibration).  Cox-Ross-rubinstein (CIR) model  Forward tree model (also called standard binomial tree mode

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