.. slideconf:: :slide_classes: appear ============================================================================== Binomial Trees ============================================================================== Random Walk ============================================================================== A *random walk* is a stochastic process the evolves in the following manner: .. math:: \begin{align} Y_t & = Y_{t-1} + \varepsilon_t. \end{align} .. raw:: - If :math:`\smash{\varepsilon_t \sim N(0,1)}`, then :math:`\smash{Y_t}` is a continuous random variable and is a referred to as a Gaussian random walk. .. raw:: - If :math:`\smash{\varepsilon_t}` is drawn from a discrete distribution, then :math:`\smash{Y_t}` is also a discrete random variable. .. raw:: - We will refer to :math:`\smash{\varepsilon_t}` as the *innovation* or *shock*. Binomial Distribution ============================================================================== Suppose a random variable :math:`\smash{X}` can only take one of two values :math:`\smash{X_u}` and :math:`\smash{X_d}`. .. raw:: - If :math:`\smash{P(X = X_u) = p}` and :math:`\smash{P(X = X_d) = 1-p}`, then :math:`\smash{X}` follows a Bernoulli distribution with parameter :math:`\smash{p}`. .. raw:: - In notation: :math:`\smash{X \sim Bernoulli(p)}`. .. raw:: - The sum of :math:`\smash{n}` independent Bernoulli random variables is a Binomial random variable with parameters :math:`\smash{n}` and :math:`\smash{p}`. .. raw:: - In notation: if :math:`\smash{Y_t = \sum_{i=t-n}^t X_i}` then :math:`\smash{Y_t \sim Binomial(n,p)}`. Bernoulli Random Walk ============================================================================== A very simple model of stock prices assumes that they follow a random walk with Bernoulli innovations: .. math:: \begin{align} P_t & = P_{t-1} + \varepsilon_t, \,\,\,\, \varepsilon_t \sim Bernoulli(p). \end{align} .. raw:: - Interpretation: The stock price can only move to one of two values at each time period. .. raw:: - This model is useful for valuing options on the asset. Binomial Tree Example ============================================================================== Suppose a stock price is currently :math:`\smash{\$20}` and that in three months it will either be :math:`\smash{\$18}` or :math:`\smash{\$22}`. .. raw:: - You would like to buy a call option with strike of :math:`\smash{\$21}`. .. raw:: - If the price moves up to :math:`\smash{\$22}`, the option will be worth :math:`\smash{\$1}` at expiry. .. raw:: - If the price moves down to :math:`\smash{\$18}`, the option will be worthless. Binomial Tree Example ============================================================================== .. image:: BinomialTrees/binomialTree1.png :width: 8in :align: center A Riskless Portfolio ============================================================================== Consider the following portfolio: .. raw:: - Buy :math:`\smash{\Delta}` shares of the stock. .. raw:: - Sell one call option. .. raw:: - If the prices moves up, the value of the portfolio is :math:`\smash{\$22 \Delta - 1}`. .. raw:: - If the prices moves down, the value of the portfolio is :math:`\smash{\$18 \Delta}`. .. raw:: - If :math:`\smash{\Delta}` can be chosen so that :math:`\smash{\$22 \Delta - 1 = \$18 \Delta}`, the portfolio is risk free. A Riskless Portfolio ============================================================================== Clearly, if :math:`\smash{\Delta = 0.25}` the payoffs are equal and the portfolio is riskless. .. raw:: - Thus a portfolio which is long 0.25 shares and short 1 option is riskless. .. raw:: - If the prices moves up, the value of the portfolio is :math:`\smash{\$22 \Delta - 1 = \$4.5}`. .. raw:: - If the prices moves down, the value of the portfolio is :math:`\smash{\$18 \Delta = \$4.5}`. .. raw:: - Although it's not possible to buy 0.25 shares, you can buy 100 shares and short 400 calls. Value of the Riskless Portfolio ============================================================================== Since the portfolio has no risk, it must earn the risk-free rate of return. .. raw:: - Suppose the continuously compounded risk-free interest rate is 12\% per annum. .. raw:: - The value of the portfolio today must be the present value of the riskless :math:`\smash{\$4.5}` payoff: .. math:: \begin{align} 4.5 e^{-0.12 \times 3/12} & = 4.367. \end{align} Value of the Riskless Portfolio ============================================================================== - The cost of the portfolio is :math:`\smash{\$20 \Delta - f = 5 -f}`, where :math:`\smash{f}` is the current price of the call. .. raw:: - The cost and value of the portfolio must be equal, otherwise an arbitrage opportunity would exist: .. math:: \begin{align} 5 - f & = 4.367 \\ \Rightarrow f & = 0.633. \end{align} One-Step Binomial Tree ============================================================================== We can generalize the previous example. .. raw:: - The current stock price is :math:`\smash{S_0}`. .. raw:: - The current option price is :math:`\smash{f}`. .. raw:: - Time periods are :math:`\smash{T}` years. .. raw:: - At the end of the next time period, the stock price will be either :math:`\smash{S_0 u}` or :math:`\smash{S_0 d}`, where :math:`\smash{u > 1}` and :math:`\smash{d < 1}`. .. raw:: - :math:`\smash{u-1}` represents a percentage increase and :math:`\smash{d - 1}` represents a percentage decrease. .. raw:: - The option values at the end of next time period are either :math:`\smash{f_u}` or :math:`\smash{f_d}`. One-Step Binomial Tree ============================================================================== .. image:: BinomialTrees/binomialTree1General.png :width: 8in :align: center One-Step Portfolio ============================================================================== Consider a portfolio that is long :math:`\smash{\Delta}` shares and short 1 call option. .. raw:: - If the stock price moves up, the value is :math:`\smash{S_0 u \Delta - f_u}`. .. raw:: - If the stock price moves down, the value is :math:`\smash{S_0 d \Delta - f_d}`. .. raw:: - Find :math:`\smash{\Delta}` such that the payoffs are equal: .. math:: \begin{align} \Delta & = \frac{f_u - f_d}{S_0 u - S_0 d}. \end{align} One-Step Portfolio ============================================================================== Since the payoffs are equivalent in each state, the portfolio is riskless. .. raw:: - Suppose the continuously compounded risk-free rate is :math:`\smash{r}`. .. raw:: - The present value of the portfolio payoff is :math:`\smash{\left(S_0 u \Delta - f_u\right) e^{-rT}}`. .. raw:: - The current cost of the portfolio is :math:`\smash{S_0 \Delta - f}`. .. raw:: Call Option Price ============================================================================== Equating cost and value of the portfolio: .. math:: \begin{align} S_0 \Delta - f & = \left(S_0 u \Delta - f_u\right) e^{-rT} \\ \Rightarrow f & = S_0 \Delta \left(1-u e^{-rT}\right) + f_u e^{-rT} \\ & = S_0 \frac{f_u-f_d}{S_0 u - S_0 d} \left(1-u e^{-rT}\right) + f_u e^{-rT} \\ & = \frac{f_u\left(1-d e^{-rT}\right) + f_d\left(u e^{-rT} - 1\right)}{u-d} \\ & = e^{-rT} \left(p f_u + (1-p) f_d\right), \end{align} where :math:`\smash{p = \frac{e^{rT} - d}{u-d}}`. Revisiting One-Period Example ============================================================================== Recall the previous example: .. raw:: - :math:`\smash{u = 1.1}`, :math:`\smash{d = 0.9}`, :math:`\smash{f_u = 1}`, :math:`\smash{f_d = 0}`, :math:`\smash{r = 0.12}` and :math:`\smash{T = 3/12}`. .. raw:: - Thus :math:`\smash{p = \frac{e^{0.12\times 3/12} - 0.9}{1.1-0.9} = 0.6523}`. .. raw:: - :math:`\smash{f = e^{-0.12 \times 0.25} \left(0.6523 \times 1 + 0.3477 \times 0\right) = 0.633}`. Two-Step Binomial Tree ============================================================================== Consider a two-period example, where the stock price can move up or down during the first period, and then can move up or down during the second period. .. raw:: - There will be either three or four possible stock prices at the end of time two. .. raw:: - We can solve the problem backwards: - First solve for the prices of the options in each state of the world at the end of period 1 (two separate problems). - Use those prices to solve for the price at the current time (single problem). Two-Step Binomial Tree ============================================================================== .. image:: BinomialTrees/binomialTree2General.png :width: 8in :align: center Two-Step Binomial Tree ============================================================================== Let us now denote a time period as :math:`\smash{\Delta t}`. .. raw:: - The value (price) of the call option at the end of the first period in the high state is: .. math:: \begin{align} f_u & = e^{-r \Delta t} \left(p f_{uu} + (1-p) f_{ud}\right) \end{align} .. raw:: - The value (price) of the call option at the end of the first period in the low state is: .. math:: \begin{align} f_d & = e^{-r \Delta t} \left(p f_{ud} + (1-p) f_{dd}\right) \end{align} Two-Step Binomial Tree ============================================================================== - The value (price) of the call option at the current time is: .. math:: \begin{align} f & = e^{-r \Delta t} \left(p f_{u} + (1-p) f_{d}\right) \\ & = e^{-2r \Delta t}\left(p^2 f_{uu} + 2*p(1-p) f_{ud} + (1-p)^2 f_{dd} \right). \end{align} Two-Step Binomial Tree Example ============================================================================== Continuing with the previous example: :math:`\smash{u = 1.1}`, :math:`\smash{d = 0.9}`, :math:`\smash{r = 0.12}` and :math:`\smash{\Delta t = 3/12}`. .. raw:: - If the stock price moves up twice :math:`\smash{S_0 u^2 = 24.2}` and :math:`\smash{f_{uu} = 3.2}`. .. raw:: - If the stock price moves down twice :math:`\smash{S_0 d^2 = 16.2}` and :math:`\smash{f_{dd} = 0}`. .. raw:: - If the stock price moves up and down :math:`\smash{S_0 ud = 19.8}` and :math:`\smash{f_{ud} = 0}`. Two-Step Binomial Tree Example ============================================================================== Given the numbers above: .. math:: \begin{align} f_u & = e^{-0.12 \times 0.25} \left(0.6523 \times 3.2 + 0.3477 \times 0 \right) = 2.0257 \\ f_d & = e^{-0.12 \times 0.25} \left(0.6523 \times 0 + 0.3477 \times 0 \right) = 0 \\ f & = e^{-0.12 \times 0.25} \left(0.6523 \times 2.0257 + 0.3477 \times 0 \right) = 1.2823. \end{align} Valuing Put Options ============================================================================== The foregoing treatment was not unique to call options. .. raw:: - We likewise could have determine the number of shares necessary to create a riskless portfolio that is long the stock and short a put. .. raw:: - This would result in a symmetric solution for :math:`\smash{\Delta}`, but using the final values of the put in each state. .. raw:: - The same formulas for the one- and two-period trees can be used to value the option. - The only difference is that the option payoffs at each node are different than the call. Two-Step Put Option Example ============================================================================== The current price of a stock is :math:`\smash{S_0 = \$50}`, :math:`\smash{u = 1.2}`, :math:`\smash{d = 0.8}`, :math:`\smash{r = 0.05}` and :math:`\smash{\Delta t = 1}`. Consider a put with strike price :math:`\smash{\$52}`. .. raw:: - If the stock price moves up twice :math:`\smash{S_0 u^2 = 72}` and :math:`\smash{f_{uu} = 0}`. .. raw:: - If the stock price moves down twice :math:`\smash{S_0 d^2 = 32}` and :math:`\smash{f_{dd} = 20}`. .. raw:: - If the stock price moves up and down :math:`\smash{S_0 ud = 48}` and :math:`\smash{f_{ud} = 4}`. .. raw:: - :math:`\smash{p = \frac{e^{r \Delta t} - d}{u-d} = \frac{e^{0.05 \times 1} - 0.8}{1.2-0.8} = 0.6282}`. Two-Step Put Option Example ============================================================================== Given the numbers above: .. math:: \begin{align} f_u & = e^{-0.05 \times 1} \left(0.6282 \times 0 + 0.3718 \times 4 \right) = 1.4147 \\ f_d & = e^{-0.05 \times 1} \left(0.6282 \times 4 + 0.3718 \times 20 \right) = 9.4635 \\ f & = e^{-0.05 \times 1} \left(0.6282 \times 1.4147 + 0.3718 \times 9.4635 \right) = 4.1923. \end{align} American Options ============================================================================== To value an American option with a binomial tree: .. raw:: - At each node, compute the value of the call implied by the subsequent nodes (using the foregoing methods). .. raw:: - Compute the value of immediate exercise. .. raw:: - Assign the value as the maximum of these two quantities. .. raw:: - Proceed in reverse fashion until the value is computed at the first node, taking into account the possibilities of early exercise. American Option Example ============================================================================== Suppose that the put in the previous example is an American option. .. raw:: - If the stock price moves down, the value of immediate exercise is :math:`\smash{\$12}`. .. raw:: - This means :math:`\smash{f_d = \max(9.4626,12) = 12}`. .. raw:: - As a result: .. math:: \begin{align} f & = e^{-0.05 \times 1} \left(0.6282 \times 1.4147 + 0.3718 \times 12 \right) = 5.0894. \end{align} .. raw:: - Note that the value of the American option is greater than that of the European option. Delta ============================================================================== Recall that :math:`\smash{\Delta}` is the number of shares that must be purchased to create a riskless portfolio with a short option. .. raw:: - It is the ratio of possible option values over the possible stock prices. .. raw:: - It is often written and referred to as *delta*. .. raw:: - Following this investment strategy is called *delta hedging*. .. raw:: - Call option deltas are positive and put option deltas are negative. Call Deltas ============================================================================== In the two-period call option example: .. math:: \begin{align} \Delta_0 & = \frac{2.0257 - 0}{22-18} = 0.5064 \\ \Delta_u & = \frac{3.2 - 0}{24.2-19.8} = 0.7273 \\ \Delta_d & = \frac{0 - 0}{19.8-16.2} = 0, \end{align} where :math:`\smash{\Delta_0}` is the delta for the first time period and :math:`\smash{\Delta_u}` and :math:`\smash{\Delta_d}` are the deltas for the second periods if the price moves up and down. .. raw:: - Notice that delta changes with time - i.e you need to rebalance your portfolio to maintain a riskless hedge. .. raw:: - This is called dynamic hedging. Put Deltas ============================================================================== In the two-period put option example: .. math:: \begin{align} \Delta_0 & = \frac{1.4147 - 9.4636}{60-40} = -0.4024 \\ \Delta_u & = \frac{0-4}{72-48} = -0.1667 \\ \Delta_d & = \frac{4-20}{48-32} = -1. \end{align} Stock Return Volatility ============================================================================== Suppose :math:`\smash{\sigma^2}` is the annualized variance of the stock returns. .. raw:: - Assuming the returns are independent, :math:`\smash{\Delta t \sigma^2}` is a good approximation of the variance for period :math:`\smash{\Delta t}`. .. raw:: - This means a good approximation of the standard deviation, or volatility, for period :math:`\smash{\Delta t}` is :math:`\smash{\sqrt{\Delta t} \sigma}`. Binomial Tree Parameters ============================================================================== Given the current stock price, :math:`\smash{S_0}`, the option strike price and the risk-free rate, :math:`\smash{r}`: .. raw:: - The binomial tree is completely determined by three parameters: :math:`\smash{u}`, :math:`\smash{d}` and :math:`\smash{\Delta t}`. .. raw:: - :math:`\smash{u}` and :math:`\smash{d}` are often set so that :math:`\smash{u = e^{\sigma \sqrt{\Delta t}}}` and :math:`\smash{d = e^{-\sigma \sqrt{\Delta t}}}`. .. raw:: - In doing so, it can be shown that the volatility of the returns in the binomial tree approximate :math:`\smash{\sigma}`. Binomial Tree Parameters ============================================================================== Simple binomial tree models with long time periods are unrealistic. .. raw:: - We can simply increase the number of steps in the model. .. raw:: - For example, for a three-month period with daily time steps, there would be a total of roughly 66 period (22 trading days per month). .. raw:: - For the recombining trees we considered above, this amounts to 66 possible terminal values and :math:`\smash{\sum_{i=1}^{65} i = 2145}` individual options nodes to value. .. raw:: - For the non-recombining trees, this amounts to :math:`\smash{2^{65}}` possible terminal values and :math:`\smash{\sum_{i=1}^{65} 2^{i-1}}` individual options nodes to value. Options on Other Assets ============================================================================== We have only valued options on a stock that doesn't pay dividends. .. raw:: - To value other assets, we only need to adjust the interest rate :math:`\smash{r}` used in the calculation of :math:`\smash{p}` to account for any earnings or costs associated with the underlying asset. .. raw:: - Discounting to present value is always done with :math:`\smash{r}`. Options on Other Assets ============================================================================== For stock paying dividends: .. math:: \begin{align} p & = \frac{e^{(r-q)\Delta t}-d}{u-d}. \end{align} .. raw:: For currencies: .. math:: \begin{align} p & = \frac{e^{(r-r_f)\Delta t}-d}{u-d}. \end{align} .. raw:: For futures: .. math:: \begin{align} p & = \frac{1-d}{u-d}. \end{align}