Question

Consider the S&P/ASX 200 option contracts that expire on 17th September 2020, with a strike price of 6050. On 24th July 2020, the S&P/ASX 200 index was priced at 6019.8. The annual standard deviation of S&P/ASX 200 stocks is 26%. The risk-free rate is 2.25% with annual compounding. Assume no dividends are paid on any of the underlying securities in the index.

Using a three-step binomial tree model, construct a Theotrical price for European calls and puts please show all working out

Answer 1

binomial options pricing model (BOPM) is model to use valuation of options. Essentially, As price of stock is varying over the time of the underlying financial instrument. This Model provide Three Step valuation methods which is used to determine Call Option and Put option.-

Step 1: Create the binomial price tree

The tree of prices is produced by working forward from valuation date to expiration.

At each step, it is assumed that the underlying instrument will move up or down by a specific factor or per step of the tree (where, by definition, } and ). So, if is the current price, then in the next period the price will either be or .

The up and down factors are calculated using the underlying volatility, , and the time duration of a step, , measured in years (using the day count convention of the underlying instrument). From the condition that the variance of the log of the price is , we have:

Step 2: Find option value at each final node

At each final node of the tree—i.e. at expiration of the option—the option value is simply its intrinsic, or exercise, value:

Max [ (S_n− K), 0 ], for a call option

Max [ (K − S_n), 0 ], for a put option,

where K is the strike price and is the spot price of the underlying asset at the n^th period.

Step 3: Find option value at earlier nodes

Once the above step is complete, the option value is then found for each node, starting at the penultimate time step, and working back to the first node of the tree (the valuation date) where the calculated result is the value of the option.

In overview: the "binomial value" is found at each node, using the risk neutrality assumption; see Risk neutral valuation. If exercise is permitted at the node, then the model takes the greater of binomial and exercise value at the node.

The steps are as follows:

1. Under the risk neutrality assumption, today's fair price of a derivative is equal to the expected value of its future payoff discounted by the risk free rate. Therefore, expected value is calculated using the option values from the later two nodes (Option up and Option down) weighted by their respective probabilities—"probability" p of an up move in the underlying, and "probability" (1−p) of a down move. The expected value is then discounted at r, the risk free rate corresponding to the life of the option.

   The following formula to compute the expectation value is applied at each node:

   , or

   

   where

   is the option's value for the node at time t,

   is chosen such that the related binomial distribution simulates the geometric Brownian motion of the underlying stock with parameters r and σ,

   q is the dividend yield of the underlying corresponding to the life of the option. It follows that in a risk-neutral world futures price should have an expected growth rate of zero and therefore we can consider for futures.

   Note that for p to be in the interval the following condition on has to be satisfied .

2. This result is the "Binomial Value". It represents the fair price of the derivative at a particular point in time (i.e. at each node), given the evolution in the price of the underlying to that point. It is the value of the option if it were to be held—as opposed to exercised at that point.
3. Depending on the style of the option, evaluate the possibility of early exercise at each node: if (1) the option can be exercised, and (2) the exercise value exceeds the Binomial Value, then (3) the value at the node is the exercise value.
   • For a European option, there is no option of early exercise, and the binomial value applies at all nodes.