Instructions
INPUTS - The user is required to enter eight inputs being:Option Type: A Call or a Put
Underlying Price: Price of underlying e.g. KLCI
Exercise Price of Option: Strike price of the Option e.g. 720
Dividend Yield: In percentage. e.g. 1.72%
Interest Rate: In percentage. e.g. 3.12%
Volatility: In percentage. e.g. 25%
Valuation Date: e.g. 9-Oct-00
Exercise Date: e.g. 7-Jan-01
| Option Parameters (INPUTS) | ||||
|---|---|---|---|---|
| Option Type | CALL | Underlying Price (e.g. KLCI) | $720.0000 | |
| Dividend Yield | 1.7200% | Exercise Price of Index Option | $720.0000 | |
| Interest Rate | 3.120% | Valuation Date (dd-mmm-yyyy) | 9-Oct-00 | |
| Volatility | 25.0000% | Exercise Date (dd-mmm-yyyy) | 7-Jan-01 | |
OUTPUTS - The output being generated immediately after the user enters the eight inputs are as follows:
| Valuation Model (OUTPUTS) | |||||||
|---|---|---|---|---|---|---|---|
| Option air Value | Delta | Gamma | 1-Day Theta | 7-Day Theta | Vega | Rho | |
| Black-Scholes | 36.6723 | 0.5335 | 0.0044 | -0.2086 | -1.4602 | 1.4143 | 0.8568 |
| Binomial Model | 36.6910 | 0.5335 | 0.0044 | -0.2079 | -1.4555 | 1.4130 | 0.8713 |
| Trinomial Model | 36.6917 | 0.5335 | 0.0044 | -0.2264 | -1.5847 | 1.4142 | 0.8714 |
Additional Valuation like Intrinsic Value , Time Value and the Implied Volatility of the Index Options will be calculated immediately upon input of the Index Options Market Value.
| Additional Valuation | ||||
|---|---|---|---|---|
| Input below: | Intrinsic Value | Time Value | Implied Volatility | |
| Option Market Price | Black-Scholes | 0.0000 | 50.0000 | 34.421% |
| Binomial Model | 0.0000 | 50.0000 | 34.419% | |
| 50.0000 | Trinomial Model | 0.0000 | 50.0000 | 34.410% |
Black-Scholes Model
The Black-Scholes model developed in 1972 was the original option-pricing model for the valuation of European style options. European style options are options that have as a characteristic that they cannot be exercised before the expiration date. Its principles serve as the foundation in almost all options formulas used today.
Fischer Black and Myron Scholes developed their option pricing model under the assumptions that the underlying prices change continuously and that the returns of the underlying follow a log-normal distribution. Also, they assume that the interest rate and the volatility of the underlying remain constant over the life of the option.
The Black-Scholes model as originally developed pertained only to options on underlying with no dividend payment. The calculator used here has been adjusted for the Black-Scholes model to account for dividends.
The Black -Scholes formula for the European option is:
where
| C t | = | theoretical call value at time t |
| S t | = | index level at time t |
| X | = | exercise price of the option |
| T-t | = | time between the valuation date and the expiration date of the option |
| r | = | risk free interest rate |
| N(d) | = | standard normal cumulative distribution in point d. |
| s | = | volatility of the underlying index |
Binomial Model
The binomial model, developed by Cox and Rubinstein, breaks down the time to expiration into potentially a very large number of time intervals, or steps. A tree of the underlying prices is initially produced working forward from the present to expiration.
At each step it is assumed that the underlying price will move up or down by an amount calculated using volatility and time to expiration. This produces a binomial distribution, or recombining tree, of underlying prices. The tree represents all the possible paths that the underlying price could take during the life of the option.
At the end of the tree -- i.e. at expiration of the option -- all the terminal option prices for each of the final possible stock prices are known, as they simply equal their Intrinsic Values .
The option prices at each step of the tree are calculated working back from expiration to the present. The option prices at each step are used to derive the option prices at the next step of the tree using risk neutral valuation based on the probabilities of the underlying prices moving up or down, the risk free rate and the time interval of each step. At the top of the tree you are left with one option price, which is known as the theoretical or fair value of the option.
For European options, the binomial model converges on the Black-Scholes formula as the number of steps in the binomial calculation increases. In fact the Black-Scholes model for European options is really a special case of the binomial model where the number of binomial steps is infinite. In other words, the binomial model provides discrete approximations to the continuous process underlying the Black-Scholes model.
To derive the formula for Binomial pricing model, we first start by dividing the life of an option into a large number of small time intervals of length dt. Assuming that the initial value of the Index is S, the value S can increase to Su or decrease to Sd when comes to the next time interval. Hence index can move from its initial value of S to one of two new values, Su and Sd. The movement from S to Su is therefore an "up" movement and the movement from S to Sd is a "down" movement.
The probability of an up movement will be denoted by p while the probability of a down movement is (1-p).
A diagram on the two-period binomial tree is illustrated below:
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| d | = |
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| p | = |
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Trinomial Model
The Trinomial Model is very similar to the Binomial Model except that at each time interval it is assumed that the underlying index (S) will move up (Su) or down (Sd) by an amount or remain the same (S). The initial Index level, interest rates and the volatility define the nature of the trinomial lattice. If we denote the probability of an up movement pu, the probability of a down movement by pd, so the probability for the across movement will be (1-pu-pd).
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| d | = |
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| a | = |
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| pu | = |
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| pd | = |
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Once the array of the underlying index has been set up by working forwards through the trinomial tree, the option price array is calculated by working backwards from the option expiry.
At option expiry, the options are initialized to their intrinsic value . In discounting back from the expiry to the present, the option price at each interval is calculated as the minimum of the exercise (strike) price and the discounted value of holding the option over the time period. Once the option price array has been populated, the theoretical (fair) option value is the value of the option at t=0 or at present.
Intrinsic Value and Time Value
The intrinsic value of a call is the amount by which the index is above the call’s strike price. The intrinsic value of a put is the amount by which the index is below the put’s exercise price.
Time value is that portion of an option'’ total price in excess of intrinsic value. As the intrinsic value increases, the time value decreases
Let’s look at an illustration below:
Consider a call and a put on the same underlying has the same exercise price of 700. Current underlying price is at 720, the call costs RM 25 and the put costs RM 5. The Intrinsic value of the call is 20 (=720-700) and of the put is 0 (since the index is above the put’s exercise price). The Time value of the call is 5 (=25-20) while that of the put is 5 (=5-0).
Implied Volatility
Implied Volatility is the volatility percentage that explains the current market price of an option. As the forces of supply and demand determine the market price of an index option, the volatility percentage must be adjusted to explain the market price of an option. The Implied Volatility that produces the option’s market price as the theoretical value is the implied volatility.
Option Type
Option type means a Call or a Put.
A Call options give the buyers the right to buy some underlying instrument at a specified price (known as the exercise price or strike price) until a specific date (known as the expiration date). A Put options give the buyers the right to sell.
Options buyers are not obligated; they have rights. Options sellers, however, are obligated to fulfill the terms of the contract if the options buyers exercise their rights.
Underlying
The underlying is the asset or instrument on which an options payoff is based. For example, if purchasing an option on current index level (Index option), the underlying is the index. The underlying price is the current index level of the underlying assets or instrument. The underlying price must be entered in dollar amounts. For example, 720.00
Option Price
The theoretical or fair value of the option, which is used to determine if arbitrage opportunities exist in the market place by checking if market value of the option is trading over, under or equal to its fair value.
Exercise Price
This is the price at which a transaction in the underlying is created. For example, a call (equity) option with a strike price of RM 50 allows the buyer to purchase the share at RM 50 when the market price of the share is at RM 60.
Since index options are settled by cash payment instead of purchase or sale of the underlying instrument, the exercise or strike price of an index option is a reference option, the level of an index that determines whether an option is in-the-money, at-the-money, or out-of-the-money.
In-the-money, at-the-money, or out-of-the-money
An index call is in-the-money if the index level is above the exercise price of the call; an index call is at-the-money if the index level is equal to the exercise price of the call; an index call is out-of-the-money if the index level is below the exercise price of the call.
For puts, the opposite relationship is true. An index put is in-the-money if the index level is below the exercise price of the put; an index put is at-the-money if the index level is equal to the exercise price of the put; an index put is out-of-the-money if the index level is below the exercise price of the put.
Valuation Date and Exercise Date
The Valuation date is the day the user values the options. The exercise date is the day in which the user exercises the options. As the Pricing Calculator are use to value European-style options, the Exercise date here is treated as the Expiration Date, the date on which an option will cease to exist.
In this Pricing Calculator, the difference between the Valuation date and the Exercise date is the days to expiration. The days to expiration represents the amount of time the option has before it becomes worthless.
Dividend Yield
The calculator assumes that the dividend flow is continuous and even throughout an option's life.
Interest Rate
The interest rate is the current risk-free interest rate.
Volatility
With regards to stock index levels, volatility is a measure of changes in price expressed in percentage terms without regard to direction of the underlying. It is a measure of the speed of the market. Markets that move slowly are low-volatility markets; markets that move quickly are high-volatility markets.
Volatility has a significant influence on the price of an option contract. Small variations in the estimate of volatility can result in significantly different options prices.
Tax considerations, transaction costs and margin requirements
Options involve tax considerations and transaction costs that can significantly affect the profit or loss results of buying and writing (selling) options. Certain options transactions also involve margin requirements which can significantly affect the economics of the transaction. None of these factors are taken into account in this pricing Calculator.
For tax considerations, you should seek the advice of a tax professional.
For transaction costs and margin requirements, consult your broker. Transaction costs are an especially important consideration in strategies involving multiple option positions due to the number of opening transactions and the potential for a number of closing transactions.
Greeks
The option risk parameters (or sometimes referred as Sensitivity) called "Greeks" are used to gauge how different movements in the market will affect the price of the option.
Delta
Delta is an estimate of the change in option value given a small change in the underlying price, assuming other factors remain constant. Delta indicates a percentage change. For example, an option with a delta of 0.5 (i.e. 50%) will move half a cent for every full cent movement in the underlying stock.
A deeply out-of-the-money call will have a delta very close to zero; an at-the-money call 0.5; a deeply in-the-money call will have a delta very close to 1.
Call deltas are positive; put deltas are negative, reflecting the fact that the put option price and the underlying stock price are inversely related. The put delta equals the call delta - 1.
The delta is often called the hedge ratio. E.g. if you have a portfolio of n shares of a stock then n divided by the delta gives you the number of calls you would need to be short (i.e. need to write) to create a riskless hedge – i.e. a portfolio which would be worth the same whether the stock price rose by a very small amount or fell by a very small amount. In such a "delta neutral" portfolio any gain in the value of the shares held due to a rise in the share price would be exactly offset by a loss on the value of the calls written, and vice versa.
Note that as the delta changes with the stock price and time to expiration the number of shares would need to be continually adjusted to maintain the hedge. How quickly the delta changes with the stock price is given by gamma (see "Greeks" below).
Gamma
Gamma indicates a change in delta. It measures how fast the delta changes for small changes in the underlying stock price. i.e. the delta of the delta. If the underlying changed by a given amount the option delta would change by a fraction (the gamma) of that amount. For example, a gamma change of '0.150' indicates the delta will increase by 0.150 if the underlying price is increased by 1.000 or decrease by 0.150 if the underlying price is decreased by 1.000.
If you are hedging a portfolio using the delta-hedge technique described under "Delta", then you will want to keep gamma as small as possible as the smaller it is the less often you will have to adjust the hedge to maintain a delta neutral position. If gamma is too large a small change in stock price could wreck your hedge. Adjusting gamma, however, can be tricky and is generally done using options -- unlike delta, it can't be done by buying or selling the underlying asset as the gamma of the underlying asset is, by definition, always zero so more or less of it won't affect the gamma of the total portfolio.
Theta
Theta indicates a change in option value for a 'one unit' reduction in time to expiration. Basically it is a measure of time decay. As an option moves closer to expiration the absolute value of Theta increases.
Some pricing programs assume one unit to be 1 Day while some pricing programs assume one unit to be 7 Days. The Pricing Calculator in this Website provides both 1-Day Theta and 7-Day Theta.
Vega
Vega is the change in option price given a one percentage point change in volatility. For example, a Vega of '0.090' indicates the option value will increase by 0.090 if the volatility percentage is increased by 1.000 or decrease by 0.090 if the volatility percentage is decreased by 1.000.
Rho
Rho indicates the change in option value for a one percent change in the interest rate. For example, a Rho of '0.060' indicates the option value will increase by 0.060 if the interest rate is increased by 1.000