Hull, Options, Futures, and Other Derivatives, 9 th Edition

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1 P1.T4. Valuation & Risk Models Hull, Options, Futures, and Other Derivatives, 9 th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Sounder

2 Hull, Chapter 13: Binomial Trees CALCULATE THE VALUE OF A EUROPEAN CALL OR PUT OPTION USING THE ONE STEP AND TWO STEP BINOMIAL MODEL

3 Hull, Chapter 13: Binomial Trees Calculate the value of an American and a European call or put option using a one-step and two-step binomial model. Describe how volatility is captured in the binomial model. Describe how the value calculated using a binomial model converges as time periods are added. Explain how the binomial model can be altered to price options on: stocks with dividends, stock indices, currencies, and futures. The basic approaches to option valuation are binomial (a simulation) and Black- Scholes (analytical) The two basic approaches to option valuation are Black-Scholes (analytical or closed-form) and Binomial (simulation or open lattice). The binomial is a tree of future price possibilities; it is very flexible and highly intuitive but tedious to generate. The Black-Scholes is efficient, elegant, logical, robust and super-quick, but for most people it remains cryptic. c S N( d ) Ke N( d ) rt Binomial (discrete time, aka lattice) Black-Scholes (continuous time & closed form) Calculate the value of a European call or put option using the onestep and two step binomial model. A binomial tree is a useful and popular technique for pricing an option, which is a diagram representing different possible random paths that might be followed by the stock price over the life of an option. In each time step, the stock has a certain probability of moving up by a certain percentage amount and a certain probability of moving down by a certain percentage amount. 3

4 To price an option in a binomial model, we assume that arbitrage opportunities do not exist. We set up a portfolio of the stock and the option in such a way that there is no uncertainty about the value of the portfolio at the end of the time period. We then argue that, because the portfolio has no risk, the return it earns must equal the risk-free interest rate. This enables us to work out the cost of the setting up the riskless portfolio and the option s price. We use the following notations: = price/value of option = proportional up jump ( >1) = stock price = proportional down jump ( <1) = number of shares of stock = option payoff if stock jumps up t = length of the time step = option payoff if stock jumps down Let us consider a portfolio consisting of a long position in shares of a stock and a short position in one option. The portfolio is riskless if the value of is chosen so that the final value of the portfolio is the same for both alternatives. = This equation shows that is the ratio of the change in the option price to the change in the stock price as we move between the nodes at time T. Risk-neutral Valuation In a risk neutral world all individuals are indifferent to risk, and investors would require no compensation for risk as against the real world, where investors require higher returns for assuming higher risks. Though options are risky investments, the principle of pricing an option under the risk-neutral assumption is possible in the real-world because when pricing an option relatively in terms of the price of the underlying stock, risk preferences are unimportant. In a risk-neutral world, the expected return on a stock (or any other instrument) would be the risk- free rate. ( ) = If the risk-neutral probability of an up movement is denoted by so that 1 is the riskneutral probability of a down movement. We assume > so that 0 < < 1. = = where = The expected future payoff from the option in a risk-neutral world is + (1 ). This probability then plugs into the equation that solves for the option price: = [ + (1 ) ] This equation states that the value of the option today is its expected future payoff in a riskneutral world discounted at the risk-free rate. This is an application of risk-neutral valuation. 4

5 Binomial Tree In any binomial pricing model, we need to calculate the option price at the initial node of the tree. This can be done by (i) building the paths forward and (ii) backward induction. The option prices at the final nodes of the tree are first calculated. For this we need to build the build the paths forward, where forward stock prices are a function of their respective up or down movements. Stock prices can move up from an initial level of to a new level, or down to in a single time step. As time steps increases they may move to new levels like, or. Then the option values at final nodes are calculated as the payoffs from the option, such as: for a call option: Max (0, Stock price at final node strike price) for a put option: Max (0, Strike price stock price at the final node) If at any node, the option is out of the money, then its value is zero. Then, working backward by discounting the value of the two subsequent nodes at the riskfree rate, (that is, weighing the final node option values by their respective risk neutral probabilities and then discounting them back for a single time period at the risk-free rate), we calculate the option price ( ) at the initial node. A one-step binomial model (as shown above) will have a single node from which stock prices will move to either up or down state at the end of the given time period one. This can be extended to a two-step model wherein from each branch node at time period one, the stock prices may move up or down by the end of the time period two. For example, after two movements, the value of the option can be. If is the length of the time step, then the value at the initial node, can be found by backward induction (as above) or by directly using the formula below, which discounts the value for two periods: = 2 [ (1 ) + (1 ) ] where, 2 (1 ) (1 ) 2 are the probabilities that the upper, middle and final nodes will be reached, respectively. 5

6 Two step Binomial Trees Below is the two-step binomial for a European call option on a stock. The assumptions are: Current asset price = $20, Strike = $21, Time step= 0.25 years, Volatility = n.a, Riskless rate = 12%, and Dividend Yield = 0%. We consider a 6-month option, so each time step is three months long, or 0.25 years. Here we assume the stock price (at each of the two time steps) will either go up by 10% or down by 10%. The stock prices at the end of period one are $22.00 and $ The possible final stock prices are: $24.20, $19.80, and $ The risk-neutral probability ( = ) is found first. The option prices at the final nodes are calculated as payoffs from the option. At the topmost node of time period two (T=0.5), the option value ( ) is 3.2 (stock price of 24.2 minus strike price of 21). In the middle and lowermost nodes, the option is out of the money and so its value is zero (for and ). Moving backwards to time period one (T=0.25), we find that at the lowermost node the option price ( ) is zero, because this node leads to forward nodes where the option is out of money ( and ). The value of the topmost node ( ) during this time period is found to be by discounting back the risk neutral probability weighted option values at top and middle nodes ( and ) in time period two. Repeatedly moving one step back to time period zero(t=0) at the initial node, the value of the call ( =1.2822) can once again be found by discounting the risk neutral probability weighted option values obtained from the two nodes ( = and = 0) found at time period one. Hull s Example 13.4: Two-step European call option, with up and down simply given as inputs. Here volatility does not inform up and down. Consequently, this model does not implicitly assume lognormal prices. Here the assumption is simply that the jump is+/- 10%. Asset price $20.00 Params Strike price $21.00 u % given (normal arithmetic) Time/step, Δt 0.25 yrs d % given (normal arithmetic) Volatility, σ n.a. a a = exp[(r-q)*δt] Riskfree rate, r 12.0% p < probability of up jump Dividend, q 0.0% 1-p < probability of down jump Node Time (yrs) S = c = Stock Option $