Option Derivative Final

Project Assignment Derivatives & Market SUBMITTED BY Rohan Gholam Roll No – 222 MFM 2014-17

Option Derivative:

on

Financial

An option is a contract that gives the buyer the right, but not the obligation, to buy or sell an underlying asset at a specific price on or before a certain date. An option, just like a stock or bond, is a security. It is also a binding contract with strictly defined terms and properties. Still confused? The idea behind an option is present in many everyday situations. Say, for example, that you discover a house that you'd love to purchase. Unfortunately, you won't have the cash to buy it for another three months. You talk to the owner and negotiate a deal that gives you an option to buy the house in three months for a price of $200,000. The owner agrees, but for this option, you pay a price of $3,000. Now, consider two theoretical situations that might arise: 1. It's discovered that the house is actually the true birthplace of Elvis! As a result, the market value of the house skyrockets to $1 million. Because the owner sold you the option, he is obligated to sell you the house for $200,000. In the end, you stand to make a profit of $797,000 ($1 million - $200,000 - $3,000). 2. While touring the house, you discover not only that the walls are chock-full of asbestos, but also that the ghost of Henry VII haunts the master bedroom; furthermore, a family of super-intelligent rats have built a fortress in the basement. Though you originally thought you had found the house of your dreams, you now consider it worthless. On the upside, because you bought an option, you are under no obligation to go through with the sale. Of course, you still lose the $3,000 price of the option. This example demonstrates two very important points. First, when you buy an option, you have a right but not an obligation to do something. You can always let the expiration date go by, at which point the option becomes worthless. If this happens, you lose 100% of your investment, which is the money you used to pay for the option. Second, an option is merely a contract that deals with an underlying asset. For this reason, options are called derivatives, which means an option derives its value from something else. In our example, the house is the underlying asset. Most of the time, the underlying asset is a stock or an index.

Calls and Puts

The two types of options are calls and puts: 1. A call gives the holder the right to buy an asset at a certain price within a specific period of time. Calls are similar to having a long position on a stock. Buyers of calls hope that the stock will increase substantially before the option expires. 2. A put gives the holder the right to sell an asset at a certain price within a specific period of time. Puts are very similar to having a short position on a stock. Buyers of puts hope that the price of the stock will fall before the option expires. Participants in the Options Market There are four types of participants in options markets depending on the position they take: 1. Buyers of calls 2. Sellers of calls 3. Buyers of puts 4. Sellers of puts People who buy options are called holders and those who sell options are called writers; furthermore, buyers are said to have long positions, and sellers are said to have short positions. Here is the important distinction between buyers and sellers: 

Call holders and put holders (buyers) are not obligated to buy or sell. They have the choice to exercise their rights if they choose.



Call writers and put writers (sellers), however, are obligated to buy or sell. This means that a seller may be required to make good on a promise to buy or sell.

Don't worry if this seems confusing - it is. For this reason we are going to look at options from the point of view of the buyer. Selling options is more complicated and can be even riskier. At this point, it is sufficient to understand that there are two sides of an options contract.

The Call Option Let us now attempt to extrapolate the same example in the stock market context with an intention to understand the ‘Call Option’. Do note, I will deliberately skip the nitty-gritty of an option trade at this stage. The idea is to understand the bare bone structure of the call option contract. Assume a stock is trading at Rs.67/- today. You are given a right today to buy the same one month later, at say Rs. 75/-, but only if the share price on that day is more than Rs. 75, would you buy it?. Obviously you would, as this means to say that after 1 month even if the share is trading at 85, you can still get to buy it at Rs.75! In order to get this right you are required to pay a small amount today, say Rs.5.0/-. If the share price moves above Rs. 75, you can exercise your right and buy the shares at Rs. 75/-. If the share price stays at or below Rs. 75/- you do not exercise your right and you do not need to buy the shares. All you lose is Rs. 5/- in this case. An arrangement of this sort is called Option Contract, a ‘Call Option’ to be precise. After you get into this agreement, there are only three possibilities that can occur. And they are1.

The stock price can go up, say Rs.85/-

2.

The stock price can go down, say Rs.65/-

3.

The stock price can stay at Rs.75/Case 1 – If the stock price goes up, then it would make sense in exercising your right and buy the stock at Rs.75/-. The P&L would look like this – Price at which stock is bought = Rs.75 Premium paid =Rs. 5 Expense incurred = Rs.80 Current Market Price = Rs.85 Profit = 85 – 80 = Rs.5/Case 2 – If the stock price goes down to say Rs.65/- obviously it does not makes sense to buy it at Rs.75/- as effectively you would spending Rs.80/(75+5) for a stock that’s available at Rs.65/- in the open market.

Case 3 – Likewise if the stock stays flat at Rs.75/- it simply means you are spending Rs.80/- to buy a stock which is available at Rs.75/-, hence you would not invoke your right to buy the stock at Rs.75/-. At this stage what you really need to understand is this – For reasons we have discussed so far whenever you expect the price of a stock (or any asset for that matter) to increase, it always makes sense to buy a call option! Now that we are through with the various concepts, let us understand options and their associated terms,

Variabl e

Ajay – Venu Transaction

Underlyin 1 acre land g Expiry 6 months Reference Rs.500,000/Price

Stock Exampl e

Remark

1 month

Do note the concept of lot size is applicable in options. So just like in the land deal where the deal was on 1 acre land, not more or not less, the option contract will be the lot size Like in futures there are 3 expiries available

Rs.75/-

This is also called the strike price

Stock

Do note in the stock markets, the premium changes Premium Rs.100,000/- Rs.5/on a minute by minute basis. We will understand the logic soon None, based Stock All options are cash settled, no defaults have Regulator on good faith Exchange occurred until now. Finally before I end this topic, here is a formal definition of a call options contract – “The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument (the underlying) from the seller of the option at a certain time (the expiration date) for a certain price (the strike price). The seller (or “writer”) is obligated to sell the commodity or financial instrument should the buyer so decide. The buyer pays a fee (called a premium) for this right”.

What Are Put Options: In any market, there cannot be a buyer without there being a seller. Similarly, in the Options market, you cannot have call options without having put options. Puts are options contracts that give you the right to sell the underlying stock or index at a pre-determined price on or before a specified expiry date in the future. In this way, a put option is exactly opposite of a call option. However, they still share some similar traits. For example, just as in the case of a call option, the put option’s strike price and expiry date are predetermined by the stock exchange.

WHEN DO YOU BUY A PUT OPTION: There is a major difference between a call and a put option – when you buy the two options. The simple rule to maximize profits is that you buy at lows and sell at highs. A put option helps you fix the selling price. This indicates you are expecting a possible decline in the price of the underlying assets. So, you would rather protect yourself by paying a small premium than make losses.

This is exactly the opposite for call options – which are bought in anticipation of a rise in stock markets. Thus, put options are used when market conditions are bearish. They thus protect you against the decline of the price of a stock below a specified price.

ILLUSTRATION OF A PUT STOCK OPTION: Put options on stocks also work the same way as call options on stocks. However, in this case, the option buyer is bearish about the price of a stock and hopes to profit from a fall in its price.

Suppose you hold ABC shares, and you expect that its quarterly results are likely to underperform analyst forecasts. This could lead to a fall in the share prices from the current Rs 950 per share. To make the most of a fall in the price, you could buy a put option on ABC at the strike price of Rs 930 at a market-determined premium of say Rs 10 per share. Suppose the contract lot is 600 shares. This means, you have to pay a premium of Rs 6,000 (600 shares x Rs 10 per share) to purchase one put option on ABC. Remember, stock options can be exercised before the expiry date. So you need to monitor the stock movement carefully. It could happen that the stock does fall, but gains back right before expiry. This would mean you lost the opportunity to make profits. Suppose the stock falls to Rs 930, you could think of exercising the put option. However, this does not cover your premium of Rs 10/share. For this reason, you could wait until the share price falls to at least Rs 920. If there is an indication that the share could fall further to Rs 910 or 900 levels, wait until it does so. If not, jump at the opportunity and exercise the option right away. You would thus earn a profit of Rs 10 per share once you have deducted the premium costs. However, if the stock price actually rises and not falls as you had expected, you can ignore the option. You loss would be limited to Rs 10 per share or Rs 6,000.

ILLUSTRATION OF PUT STOCK OPTION

Thus, the maximum loss an investor faces is the premium amount. The maximum profit is the share price minus the premium. This is because, shares, like indexes, cannot have negative values. They can be value at 0 at worst. What is a 'Put Option'

“Formal contract between an option seller and an option buyer which gives the optionee the right but not the obligation to sell a specific contract, financial instrument, property, or security, at a specified price (called exercise Price) on or before the option's expiration date.” Option Buyer in a nutshell By now I’m certain you would have a basic understanding of the call and put option both from the buyer’s and seller’s perspective. Buying an option (call or put) makes sense only when we expect the market to move strongly in a certain direction. If fact, for the option buyer to be profitable the market should move away from the selected strike

price. Selecting the right strike price to trade is a major task; we will learn this at a later stage. For now, here are a few key points that you should remember – 1.

P&L (Long call) upon expiry is calculated as P&L = Max [0, (Spot Price – Strike Price)] – Premium Paid

2.

P&L (Long Put) upon expiry is calculated as P&L = [Max (0, Strike Price – Spot Price)] – Premium Paid

3.

The above formula is applicable only when the trader intends to hold the long option till expiry

4.

The intrinsic value calculation we have looked at in the previous topic is only applicable on the expiry day. We CANNOT use the same formula during the series

5.

The P&L calculation changes when the trader intends to square of the position well before the expiry

6.

The buyer of an option has limited risk, to the extent of premium paid. However he enjoys an unlimited profit potential Option seller in a nutshell The option sellers (call or put) are also called the option writers. The buyers and sellers have exact opposite P&L experience. Selling an option makes sense when you expect the market to remain flat or below the strike price (in case of calls) or above strike price (in case of put option). I want you to appreciate the fact that all else equal, markets are slightly favourable to option sellers. This is because, for the option sellers to be profitable the market has to be either flat or move in a certain direction (based on the type of option). However for the option buyer to be profitable, the market has to move in a certain direction. Clearly there are two favourable market conditions for the option seller versus one favourable condition for the option buyer. But of course this in itself should not be a reason to sell options. Here are few key points you need to remember when it comes to selling options –

1.

P&L for a short call option upon expiry is calculated as P&L = Premium Received – Max [0, (Spot Price – Strike Price)]

2.

P&L for a short put option upon expiry is calculated as P&L = Premium Received – Max (0, Strike Price – Spot Price)

3.

Of course the P&L formula is applicable only if the trader intends to hold the position till expiry

4.

When you write options, margins are blocked in your trading account

5.

The seller of the option has unlimited risk but very limited profit potential (to the extent of the premium received) Perhaps this is the reason why Nassim Nicholas Taleb in his book “Fooled by Randomness” says “Option writers eat like a chicken but shit like an elephant”. This means to say that the option writers earn small and steady returns by selling options, but when a disaster happens, they tend to lose a fortune. Well, with this I hope you have developed a strong foundation on how a Call and Put option behaves. Just to give you a heads up, the focus going forward in this module will be on moneyless of an option, premiums, option pricing, option Greeks, and strike selection. Once we understand these topics we will revisit the call and put option all over again. When we do so, I’m certain you will see the calls and puts in a new light and perhaps develop a vision to trade options professionally.

Payoff : There are two basic option type’s i.e. the ‘Call Option’ and the ‘Put Option’. Further there are four different variants originating from these 2 options – 1.

Buying a Call Option

2.

Selling a Call Option

3.

Buying a Put Option

4.

Selling a Put Option Below the pay off diagrams for the four different option variants –

Payoff diagrams in the above fashion helps us understand a few things better. Let me list them for you –

1.

We have stacked the pay off diagram of Call Option (buy) and Call option (sell) one below the other, they both look like a mirror image. The mirror image of the payoff emphasis the fact that the risk-reward characteristics of an option buyer and seller are opposite. The maximum loss of the call option buyer is the maximum profit of the call option seller. Likewise the call option buyer has unlimited profit potential, mirroring this the call option seller has maximum loss potential

2.

We have placed the payoff of Call Option (buy) and Put Option (sell) next to each other. This is to emphasize that both these option variants make money only when the market is expected to go higher. In other words, do not buy a call option or do not sell a put option when you sense there is a chance for the markets to go down. You will not make money doing so, or in other words you will certainly lose money in such circumstances. Of course there is an angle of volatility here which we have not discussed yet; we will discuss the same going forward. The reason why I’m talking about volatility is because volatility has an impact on option premiums

3.

Finally on the right, the payoff diagram of Put Option (sell) and the Put Option (buy) are stacked one below the other. Clearly the payoff diagrams looks like the mirror image of one another. The mirror image of the payoff emphasizes the fact that the maximum loss of the put option buyer is the maximum profit of the put option seller. Likewise the put option buyer has unlimited profit potential, mirroring this the put option seller has maximum loss potential Further, here is a table where the option positions are summarized.

Your View

Market

Bullish Flat or Bullish Flat or Bearish Bearish

Option Type Call (Buy) Put (Sell) Call (Sell) Put (Buy)

Option Option Option Option

Position called Long Call Short Put

also

Other Alternatives

Premiu m

Buy Futures or Buy Pay Spot Buy Futures or Buy Receive Spot

Short Call

Sell Futures

Receive

Long Put

Sell Futures

Pay

It is important for you to remember that when you buy an option, it is also called a ‘Long’

Position. Going by that, buying a call option and buying a put option is called Long Call and Long Put position respectively. Likewise whenever you sell an option it is called a ‘Short’ position. Going by that, selling a call option and selling a put option is also called Short Call and Short Put position respectively. Now here is another important thing to note, you can buy an option under 2 circumstances – 1.

You buy with an intention of creating a fresh option position

2.

You buy with an intention to close an existing short position The position is called ‘Long Option’ only if you are creating a fresh buy position. If you are buying with and intention of closing an existing short position then it is merely called a ‘square off’ position. Similarly you can sell an option under 2 circumstances –

1.

You sell with an intention of creating a fresh short position

2.

You sell with an intention to close an existing long position The position is called ‘Short Option’ only if you are creating a fresh sell (writing an option) position. If you are selling with and intention of closing an existing long position then it is merely called a ‘square off’ position.

Definition of Pricing Model:

the

Option

The Option Pricing Model is a formula that is used to determine a fair price for a call or put option based on factors such as underlying stock volatility, days to expiration, and others. The calculation is generally accepted and used on Wall Street and by option traders and has stood the test of time since its publication in 1973. It was the first formula that became popular and almost universally accepted by the option traders to determine what the theoretical price of an option should be based on a handful of variables. Option traders generally rely on the Black Scholes formula to buy options that are priced under the formula calculated value, and sell options that are priced higher than the Black Schole calculated value. This type of arbitrage trading quickly pushes option prices back towards the Model's calculated value. The Model generally works, but there are a few key instances where the model fails.

The Black Scholes Option Pricing Model: Black Option Pricing Model: The Model or Formula calculates an theoretical value of an option based on 6 variables. These variables are: Whether the option is a call or a put The current underlying stock price The time left until the option's expiration date The strike price of the option The risk-free interest rate The volatility of the stock What you need to know about the Option Pricing For the beginning call and put trader it is NOT necessary to memorize the formula, but it is

important to understand a few implications that the formula or equation has for option pricing and, therefore, on your trading.

Here's what you need to know about the formula: The formula shows the time left until expiration has a direct positive relationship to the value of a call or put option. In other words, the more time that is left before expiration, the higher the expected price will be. Options with 60 days left until expiration will have a higher price than options that only has 30 days left. This is because the more time that is left, the more of a chance the underlying stock price will move. But here is what you really need to understand--every minute that goes by, the cheaper the option price will become. Think of it this way. As time ticks by and as the days tick by, all things being equal, an option with 60 days left will lose about 1/60th of its value tomorrow when it only has 59 days left. That may not seem like a lot, but when we get to expiration week and as Monday changes to Tuesday, options lose 1/5 of their value. As Tuesday slips into Wednesday of expiration week, options lose 1/4 of their value, etc. so you must be careful! While nothing is certain in the stock market, there is ALWAYS one thing that is certain--time ticks by and options lose their value day by day. Please note: Don't take me literally here as the formula for this "time decay" is more complicated than that. It actually indicates that the "time decay" accelerates as you get closer to expiration, but I hope you get the point. The formula suggests the historical volatility of the stock also has a direct correlation to the option's price. By volatility we mean the daily change in a stock's price from one day to the next. The more a stock price fluctuates within a day and from day to day, then the more volatile the stock. The more volatile the stock price, the higher the Model will calculate the value of its options. Think of stocks that are in industries like utilities that pay a high dividend and have been long-term, consistent performers. Their prices go up steadily as the market moves, and they move small percentage points by week. But if you compare those utility stocks' price movements with bio-tech stocks or technology stocks, whose prices swing up and down a few dollars per day, you will know what volatility is. Obviously a stock whose price swings up and down $5 a week has a greater chance of going up $5 then a stocks whose price swings up and down $1 per week. If you are buying options, both put and calls, you LOVE volatility--you WANT volatility. This volatility can be calculated as the

variance of the the prices over the last 60 days, or 90 days, or 180 days. This becomes one of the weaknesses of the model since past results don't always predict future performance. Stocks are often volatile immediately after an earnings release, or after a major press release. Watch out for dividends! If a stock typically pays a $1 dividend, then the day it goes ex-dividend the stock price should drop $1. If you have calls on a stock that you KNOW will drop $1 then you are starting off in the hole $1. Nothing is worse than identifying a stock you are confident will go up, looking at the call prices and thinking "boy those are cheap", buying a few contracts, and then finding the stock go ex-dividend and then you realize why the options were so cheap. Beware of Earnings Releases and Rumors--You can calculate an option price all you want, but nothing can drive a stock price (and its call option prices as well) up more than a positive rumor or a strong earnings release. The Option Pricing Model simply cannot overcome the supply and demand curve of option traders hungry for owing a call option on the day of a strong earnings release or a positive press release.

Black–Scholes in practice: The Black–Scholes model disagrees with reality in a number of ways, some significant. It is widely employed as a useful approximation, but proper application requires understanding its limitations – blindly following the model exposes the user to unexpected risk.Among the most significant limitations are: the underestimation of extreme moves, yielding tail risk, which can be hedged with out-of-the-money options; the assumption of instant, costless trading, yielding liquidity risk, which is difficult to hedge; the assumption of a stationary process, yielding volatility risk, which can be hedged with volatility hedging; the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged with Gamma hedging.

The volatility smile One of the attractive features of the Black–Scholes model is that the parameters in the model other than the volatility (the time to maturity, the strike, the risk-free interest rate, and the current underlying price) are

unequivocally observable. All other things being equal, an option's theoretical value is a monotonic increasing function of implied volatility.

By computing the implied volatility for traded options with different strikes and maturities, the Black–Scholes model can be tested. If the Black– Scholes model held, then the implied volatility for a particular stock would be the same for all strikes and maturities. Despite the existence of the volatility smile (and the violation of all the other assumptions of the Black–Scholes model), the Black–Scholes PDE and Black–Scholes formula are still used extensively in practice. A typical approach is to regard the volatility surface as a fact about the market, and use an implied volatility from it in a Black–Scholes valuation model. This has been described as using "the wrong number in the wrong formula to get the right price. This approach also gives usable values for the hedge ratios (the Greeks). Even when more advanced models are used, traders prefer to think in terms of Black–Scholes implied volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on.

Criticism and comments: EspenGaarderHaug and Nassim Nicholas Taleb argue that the Black– Scholes model merely recasts existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream neoclassical economic theory. They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation.Edward Thorp also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors. Emanuel Derman and NassimTaleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes. In response, Paul Wilmott has defended the model. Living systems need to extract resources to compensate for continuous diffusion. This can be modelled mathematically as lognormal processes. The Black–Scholes equation is a deterministic representation of lognormal processes. The Black–Scholes model can be extended to describe general biological and social systems. British mathematician Ian Stewart published a criticism in which he suggested that "the equation itself wasn't the real problem" and he stated a possible role as "one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation" due to its abuse in the financial industry.

The Option Greeks- Delta Overview There is a remarkable similarity between a Bollywood movie and an options trade. Similar to a Bollywood movie, for an options trade to be successful in the market there are several forces which need to work in the option trader’s favour. These forces are collectively called ‘The Option Greeks’. These forces influence an option contract in real time, affecting the premium to either increase or decrease on a minute by minute basis. To make matters complicated, these forces not only influence the premiums directly but also influence each another. Options Premiums, options Greeks, and the natural demand supply situation of the markets influence each other. Though all these factors work as independent agents, yet they are all intervened with one another. The final outcome of this mixture can be assessed in the option’s premium. For an options trader, assessing the variation in premium is most important. He needs to develop a sense for how these factors play out before setting up an option trade.

So without much ado, let me introduce the Greeks to you – 1.

Delta – Measures the rate of change of options premium based on the directional movement of the underlying

2.

Gamma – Rate of change of delta itself

3.

Vega – Rate of change of premium based on change in volatility

4.

Theta – Measures the impact on premium based on time left for expiry We will discuss these Greeks over the next few topics. The focus of this topic is to understand the Delta. Delta of an Option Notice the following two snapshots here – they belong to Nifty’s 8650 CE option. The first snapshot was taken at 09:18 AM when Nifty spot was at 8692.

A little while later…

Now notice the change in premium – at 09:18 AM when Nifty was at 8692 the call option was trading at 183, however at 10:00 AM Nifty moved to 8715 and the same call option was trading at 198. In fact here is another snapshot at 10:55 AM – Nifty declined to 8688 and so did the option premium (declined to 181).

From the above observations one thing stands out very clear – as and when the value of the spot changes, so does the option premium. More precisely as we already know – the call option premium increases with the increase in the spot value and vice versa. Keeping this in perspective, imagine this – you have predicted that Nifty will reach 8755 by 3:00 PM today. From the snapshots above we know that the premium will certainly change – but by how much? What is the likely value of the 8650 CE premium if Nifty reaches 8755? Well, this is exactly where the ‘Delta of an Option’ comes handy. The Delta measures how an options value changes with respect to the change in the underlying. In simpler terms, the Delta of an option helps us answer questions of this sort – “By how many points will the option premium change for every 1 point change in the underlying?” Please see below 2 images which shows the change in Spot by 1 point how much option premium changes.

With change in Spot by 1 point see below change in premium-

Premium changed to 159.99. Delta of 0.57385 added to premium when nifty moved up by 1 point. Therefore the Option Greek’s ‘Delta’ captures the effect of the directional movement of the market on the Option’s premium.

The delta is a number which varies 1.

Between 0 and 1 for a call option, some traders prefer to use the 0 to 100 scale. So the delta value of 0.55 on 0 to 1 scale is equivalent to 55 on the 0 to 100 scale.

2.

Between -1 and 0 (-100 to 0) for a put option. So the delta value of -0.4 on the -1 to 0 scale is equivalent to -40 on the -100 to 0 scale

3.

We will soon understand why the put option’s delta has a negative value associated with it Delta for a Call Option We know the delta is a number that ranges between 0 and 1. Assume a call option has a delta of 0.3 or 30 – what does this mean?

Well, as we know the delta measures the rate of change of premium for every unit change in the underlying. So a delta of 0.3 indicates that for every 1 point change in the underlying, the premium is likely change by 0.3 units, or for every 100 point change in the underlying the premium is likely to change by 30 points. The following example should help you understand this better – Nifty @ 10:55 AM is at 8688 Option Strike = 8850 Call Option Premium = 133 Delta of the option = + 0.55 Nifty @ 3:15 PM is expected to reach 8710 What is the likely option premium value at 3:15 PM? Well, this is fairly easy to calculate. We know the Delta of the option is 0.55, which means for every 1 point change in the underlying the premium is expected to change by 0.55 points. We are expecting the underlying to change by 22 points (8710 – 8688), hence the premium is supposed to increase by = 22*0.55 = 12.1 Therefore the new option premium is expected to trade around 145.1 (133+12.1) This is the sum of old premium + expected change in premium As you can see from the above example, the delta helps us evaluate the premium value based on the directional move in the underlying. This is extremely useful information to have while trading options. At this stage let me post a very important question – Why is the delta value for a call option bound by 0 and 1? Why can’t the call option’s delta go beyond 0 and 1? To help understand this, let us look at 2 scenarios wherein I will purposely keep the delta value above 1 and below 0. Scenario 1: Delta greater than 1 for a call option Nifty @ 10:55 AM at 8668 Option Strike = 8650 Call Option Premium = 133 Delta of the option = 1.5 (purposely keeping it above 1) Nifty @ 3:15 PM is expected to reach 8710 What is the likely premium value at 3:15 PM? Change in Nifty = 42 points Therefore the change in premium (considering the delta is 1.5)

= 1.5*42 = 63 Do you notice that? The answer suggests that for a 42 point change in the underlying, the value of premium is increasing by 63 points! In other words, the option is gaining more value than the underlying itself. Remember the option is a derivative contract, it derives its value from its respective underlying, hence it can never move faster than the underlying. If the delta is 1 (which is the maximum delta value) it signifies that the option is moving in line with the underlying which is acceptable, but a value higher than 1 does not make sense. For this reason the delta of an option is fixed to a maximum value of 1 or 100. Let us extend the same logic to figure out why the delta of a call option is lower bound to 0.

Scenario 2: Delta lesser than 0 for a call option Nifty @ 10:55 AM at 8688 Option Strike = 8700 Call Option Premium = 9 Delta of the option = – 0.2 (have purposely changed the value to below 0, hence negative delta) Nifty @ 3:15 PM is expected to reach 8200 What is the likely premium value at 3:15 PM? Change in Nifty = 88 points (8688 -8600) Therefore the change in premium (considering the delta is -0.2) = -0.2*88 = -17.6 For a moment we will assume this is true, therefore new premium will be = -17.6 + 9 = – 8.6 As you can see in this case, when the delta of a call option goes below 0, there is a possibility for the premium to go below 0, which is impossible. At this point do recollect the premium irrespective of a call or put can never be negative. Hence for this reason, the delta of a call option is lower bound to zero. Who decides the value of the Delta?

The value of the delta is one of the many outputs from the Black & Scholes option pricing formula. the B&S formula takes in a bunch of inputs and gives out a few key outputs. The output includes the option’s delta value and other Greeks. After discussing all the Greeks, we will also go through the B&S formula to strengthen our understanding on options. However for now, you need to be aware that the delta and other Greeks are market driven values and are computed by the B&S formula. However here is a table which will help you identify the approximate delta value for a given option – Option Type Deep ITM Slightly ITM ATM Slightly OTM Deep OTM

Approx Delta value (CE) Between Between Between Between Between

+ + + + +

0.8 to + 1 0.6 to + 1 0.45 to + 0.55 0.45 to + 0.3 0.3 to + 0

Approx Delta value (PE) Between Between Between Between Between

– – – – –

0.8 to – 1 0.6 to – 1 0.45 to – 0.55 0.45 to -0.3 0.3 to – 0

Of course you can always find out the exact delta of an option by using a B&S option pricing calculator. Delta for a Put Option Parameters Underlying Strike Spot value Premium Delta Expected Nifty Value (Case)

Values Nifty 8700 8668 128 -0.55 8630

Do recollect the Delta of a Put Option ranges from -1 to 0. The negative sign is just to illustrate the fact that when the underlying gains in value, the value of premium goes down. Keeping this in mind, consider the following details – Note – 8668 is a slightly ITM option, hence the delta is around -0.55 (as indicated from the table above). The objective is to evaluate the new premium value considering the delta value to be -0.55. Do pay attention to the calculations made below. Case: Nifty is expected to move to 8630

Expected change = 8668 – 8630 = 38 Delta = – 0.55 = -0.55*38 = -20.9 Current Premium = 128 New Premium = 128 + 20.9 = 148.9 Here I’m adding the value of delta since I know that the value of a Put option gains when the underlying value decreases. I hope with the above Illustrations you are now clear on how to use the Put Option’s delta value to evaluate the new premium value. Also, I will take the liberty to skip explaining why the Put Option’s delta is bound between -1 and 0. Delta versus spot price We looked at the significance of Delta and also understood how one can use delta to evaluate the expected change in premium. Before we proceed any further, here is a quick recap from the previous topic – 1.

Call options has a +ve delta. A Call option with a delta of 0.4 indicates that for every 1 point gain/loss in the underlying the call option premium gains/losses 0.4 points

2.

Put options has a –ve delta. A Put option with a delta of -0.4 Indicates that for every 1 point loss/gain in the underlying the put option premium gains/losses 0.4 points

3.

OTM options have a delta value between 0 and 0.5, ATM option has a delta of 0.5, and ITM option has a delta between 0.5 and 1. Let me take cues from the 3rd point here and make some deductions. Assume Nifty Spot is at 8712, strike under consideration is 8800, and option type is CE (Call option, European).

1.

What is the approximate Delta value for the 8800 CE when the spot is 8712?

a.

Delta should be between 0 and 0.5 as 8800 CE is OTM. Let us assume Delta is 0.4

2.

Assume Nifty spot moves from 8712 to 8800, what do you think is the Delta value?

a.

Delta should be around 0.5 as the 8800 CE is now an ATM option

3.

Further assume Nifty spot moves from 8800 to 8900, what do you think is the Delta value?

a.

Delta should be closer to 1 as the 8800 CE is now an ITM option. Let us say 0.8.

4.

Finally assume Nifty Spot cracks heavily and drops back to 8700 from 8900, what happens to delta?

a.

With the fall in spot, the option has again become an OTM from ITM, hence the value of delta also falls from 0.8 to let us say 0.35.

5. a.

What can you deduce from the above 4 points? Clearly as and when the spot value changes, the moneyness of an option changes, and therefore the delta also changes. Now this is a very important point here – the delta changes with changes in the value of spot. Hence delta is a variable and not really a fixed entity. Therefore if an option has a delta of 0.4, the value is likely to change with the change in the value of the underlying. Have a look at the chart below – it captures the movement of delta versus the spot price. The chart is a generic one and not specific to any particular option or strike as such. As you can see there are two lines –

1.

The blue line captures the behavior of the Call option’s delta (varies from 0 to 1)

2.

The red line captures the behavior of the Put option’s delta (varies from -1 to 0) Let us understand this better – GAMA Gamma is one of the more obscure Greeks. Delta, Vega and Theta generally get most of the attention, but Gamma has important implications for risk in options strategies that can easily be demonstrated.

Gamma measures the rate of change of Delta. Delta tells us how much an option price will change given a one-point move of the underlying. But since Delta is not fixed and will increase or decrease at different rates, it needs its own measure, which is Gamma. Delta, recall, is a measure of directional risk faced by any option strategy. When you incorporate a Gamma risk analysis into your trading, however, you learn that two Deltas of equal size may not be equal in outcome. The Delta with the higher Gamma will have a higher risk (and potential reward, of course) because given an unfavorable move of the underlying; theDelta with the higher Gamma will exhibit a larger adverse change. Figure 9 reveals that the highest Gammas are always found on at-themoney options, with the January 110 call showing a Gamma of 5.58, the highest in the entire matrix. The same can be seen for the 110 puts. The risk/reward resulting from changes in Delta are highest at this point. Delta as we know represents the change in premium for the given change in the underlying price. For example if the Nifty spot value is 8600, then we know the 8800 CE option is OTM, hence its delta could be a value between 0 and 0.5. Let us fix this to 0.2 for the sake of this discussion.Assume Nifty spot jumps 300 points in a single day, this means the 8800 CE is no longer an OTM option, rather it becomes slightly ITM option and therefore by virtue of this jump in spot value, the delta of 8800 CE will no longer be 0.2, it would be somewhere between 0.5 and 1.0, let us assume 0.8. With this change in underlying, one thing is very clear – the delta itself changes. Meaning delta is a variable, whose value changes based on the changes in the underlying and the premium! If you notice, Delta is very similar to velocity whose value changes with change in time and the distance travelled. The Gamma of an option measures this change in delta for the given change in the underlying. In other words Gamma of an option helps us answer this question – “For a given change in the underlying, what will be the corresponding change in the delta of the option?” 1st order Derivative Change in distance travelled (position) with respect to change in time is captured by velocity, and velocity is called the 1st order derivative of position. Change in premium with respect to change in underlying is

captured by delta, and hence delta is called the 1st order derivative of the premium 2nd order Derivative Change in velocity with respect to change in time is captured by acceleration, and acceleration is called the 2nd order derivative of position. Change in delta is with respect to change in the underlying value is captured by Gamma, hence Gamma is called the 2nd order derivative of the premium Calculating the values of Delta and Gamma (and in fact all other Option Greeks) involves number crunching and heavy use of calculus (differential equations and stochastic calculus). Derivatives are called derivatives because the derivative contract derives its value based on the value of its respective underlying. ‘The Gamma’ (2nd order derivative of premium) also referred to as the curvature of the option gives the rate at which the option’s delta changes as the underlying changes. The gamma is usually expressed in deltas gained or lost per one point change in the underlying – with the delta increasing by the amount of the gamma when the underlying rises and falling by the amount of the gamma when the underlying falls. For example consider this – Nifty Spot=8726, Strike = 8800, Option type = CE, Moneyness of Option = Slightly OTM Premium = Rs.122/- , Delta = 0.4848, Gamma = 0.00121, Change in Spot = 170 points New Spot price = 8726 + 170 = 8896

New Moneyness = ATM

When Nifty moves from 8726 to 8896, the 8800 CE premium changed from Rs.122 to Rs.221, and along with this the Delta changed from 0.45 to 0.70. Notice with the change of 170 points, the option transitions from slightly OTM to ITM option. Which means the option’s delta has to change from 0.45 to somewhere close to 0.70. This is exactly what’s happening here. In reality the Gamma also changes with the change in the underlying. This change in Gamma due to changes in underlying is captured by 3rd derivative of underlying called “Speed” or “Gamma of Gamma” or “DgammaDspot”. For all practical purposes, it is not necessary to get into the discussion of Speed, unless you are mathematically inclined or you work for an Investment Bank where the trading book risk can run into several $ Millions. Unlike the delta, the Gamma is always a positive number for both Call and Put Option. Therefore when a trader is long options (both Calls and Puts) the trader is considered ‘Long Gamma’ and when he is short options (both calls and puts) he is considered ‘Short Gamma’. For example consider – The Gamma of an ATM Put option is 0.004, if the underlying moves 10 points, what do you think the new delta is? With 1 point change in spot how much Delta will change is totally depends on the Gamma. Gamma added to the Delta. Please see below chart of change in spot by 1 point how delta changes in respect to Gamma-

Delta is at 0.57385 and Gamma is 0.00118. Premium is at 161.25.

If you notice Delta changed to 0.57503. Gamma of 0.00118 added to Delta value which was at the time of Nifty Spot was at 8800.

Gamma movement

Gamma changes with respect to change in the underlying. This change in Gamma is captured by the 3rd order derivative called ‘Speed’. Have a look at the chart below,

The chart above has 3 different CE strike prices – 80, 100, and 120 and their respective Gamma movement. For example the blue line represents the Gamma of the 80 CE strike price. I would suggest you look at each graph individually to avoid confusion. In fact for sake of simplicity I will only talk about the 80 CE strike option, represented by the blue line. Let us assume the spot price is at 80, thus making the 80 strike ATM. Keeping this in perspective we can observe the following from the above chart – Since the strike under consideration is 80 CE, the option attains ATM status when the spot price equals 80. Strike values below 80 (65, 70, 75 etc) are ITM and values above 80 (85, 90, 95 etx) are OTM options. Notice the gamma value is low for OTM Options (80 and above). This explains why the premium for OTM options don’t change much in terms of absolute point terms, however in % terms the change is higher. For example – the premium of an OTM option can change from Rs.2 to Rs.2.5, while absolute change in is just 50 paisa, the % change is 25%. The gamma peaks when the option hits ATM status. This implies that the rate of change of delta is highest when the option is ATM. In other words, ATM options are most sensitive to the changes in the underlying Also, since ATM options have highest Gamma – avoid shorting ATM options

The gamma value is also low for ITM options (80 and below). Hence for a certain change in the underlying, the rate of change of delta for an ITM option is much lesser compared to ATM option. However do remember the ITM option inherently has a high delta. So while ITM delta reacts slowly to the change in underlying (due to low gamma) the change in premium is high (due to high base value of delta).

Vega Volatility Types The last few topics have laid a foundation of sorts to help us understand Volatility better. We now know what it means, how to calculate the same, and use the volatility information for building trading strategies. It is now time to steer back to the main topic – Option Greek and in particular the 4th Option Greek “Vega”. The different types of volatility that exist – Historical Volatility, Forecasted Volatility, and Implied Volatility. So let’s get going. Historical Volatility is similar to us judging the box office success of ‘The Hateful Eight’ based on Tarantino’s past directorial ventures. In the stock market world, we take the past closing prices of the stock/index and calculate the historical volatility. Do recall, we discussed the technique of calculating the historical volatility in topic. Historical volatility is very easy to calculate and helps us with most of the day to day requirements – for instance historical volatility can ‘somewhat’ be used in the options calculator to get a ‘quick and dirty’ option price. Forecasted Volatility is similar to the movie analyst attempting to forecast the fate of ‘The Hateful Eight’. In the stock market world, analysts forecast the volatility. Forecasting the volatility refers to the act of predicting the volatility over the desired time frame. However, why would you need to predict the volatility? Well, there are many option strategies, the profitability of which solely depends on your expectation of volatility. If you have a view of volatility – for example you expect volatility to increase by 12.34% over the next 7 trading sessions, then you can set up option strategies which can profit this view, provided the view is right. Also, at this stage you should realize – to make money in the stock markets it is NOT necessary to have a view on the direction on the markets. The view can be on volatility as well. Most of the professional options traders trade based on volatility and not really the market direction. I have to mention this – many traders find forecasting volatility is far more efficient than forecasting market direction.

Now clearly having a mathematical/statistical model to predict volatility is much better than arbitrarily declaring “I think the volatility is going to shoot up”. There are a few good statistical models such as ‘Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) Process’. I know it sounds spooky, but that’s what it’s called. There are several GARCH processes to forecast volatility, if you are venturing into this arena, I can straightaway tell you that GARCH (1,1) or GARCH (1,2) are better suited processes for forecasting volatility. Implied Volatility (IV) is like the people’s perception on social media. It does not matter what the historical data suggests or what the movie analyst is forecasting about ‘The Hateful Eight’. People seem to be excited about the movie, and that is an indicator of how the movie is likely to fare. Likewise the implied volatility represents the market participant’s expectation on volatility. So on one hand we have the historical and forecasted volatility, both of which are sort of ‘manufactured’ while on the other hand we have implied volatility which is in a sense ‘consensual’. Implied volatility can be thought of as consensus volatility arrived amongst all the market participants with respect to the expected amount of underlying price fluctuation over the remaining life of an option. Implied volatility is reflected in the price of the premium. For this reason amongst the three different types of volatility, the IV is usually more valued. You may have heard or noticed India VIX on NSE website, India VIX is the official ‘Implied Volatility’ index that one can track. India VIX is computed based on a mathematical formula, here is a whitepaper which explains how India VIX is calculated – If you find the computation a bit overwhelming, then here is a quick wrap on what you need to know about India VIX (I have reproduced some of these points from the NSE’s whitepaper) – 1.

NSE computes India VIX based on the order book of Nifty Options

2.

The best bid-ask rates for near month and next-month Nifty options contracts are used for computation of India VIX

3.

India VIX indicates the investor’s perception of the market’s volatility in the near term (next 30 calendar days)

4.

Higher the India VIX values, higher the expected volatility and viceversa

5.

When the markets are highly volatile, market tends to move steeply and during such time the volatility index tends to rise

6.

Volatility index declines when the markets become less volatile. Volatility indices such as India VIX are sometimes also referred to as the ‘Fear Index’, because as the volatility index rises, one should become careful, as the markets can move steeply into any direction. Investors use volatility indices to gauge the market volatility and make their investment decisions

7.

Volatility Index is different from a market index like NIFTY. NIFTY measures the direction of the market and is computed using the price movement of the underlying stocks whereas India VIX measures the expected volatility and is computed using the order book of the underlying NIFTY options. While Nifty is a number, India VIX is denoted as an annualized percentage Further, NSE publishes the implied volatility for various strike prices for all the options that get traded. You can track these implied volatilities by checking the option chain. For example here is the option chain of Cipla, with all the IV’s marked out.

The Implied Volatilities can be calculated using a standard options calculator. We will discuss more about calculating IV, and using IV for setting up trades in the subsequent topic. For now we will now move over to understand Vega. Realized Volatility is pretty much similar to the eventual outcome of the movie, which we would get to know only after the movie is released. Likewise the realized volatility is looking back in time and figuring out the actual volatility that occurred during the expiry series. Realized volatility matters especially if you want to compare today’s implied volatility with respect to the historical implied volatility. We will explore this angle in detail when we take up “Option Trading Strategies”.

Vega Have you noticed this – whenever there are heavy winds and thunderstorms, the electrical voltage in your house starts fluctuating violently, and with the increase in voltage fluctuations, there is a chance of a voltage surge and therefore the electronic equipments at house may get damaged. Similarly, when volatility increases, the stock/index price starts swinging heavily. To put this in perspective, imagine a stock is trading at Rs.100, with increase in volatility; the stock can start moving anywhere between 90 and 110. So when the stock hits 90, all PUT option writers start sweating as the Put options now stand a good chance of expiring in the money. Similarly, when the stock hits 110, all CALL option writers would start panicking as all the Call options now stand a good chance of expiring in the money. Therefore irrespective of Calls or Puts when volatility increases, the option premiums have a higher chance to expire in the money. Now, think about this – imagine you want to write 500 CE options when the spot is trading at 475 and 10 days to expire. Clearly there is no intrinsic value but there is some time value. Hence assume the option is trading at Rs.20. Would you mind writing the option? You may write the options and pocket the premium of Rs.20/- I suppose. However, what if the volatility over the 10 day period is likely to increase – maybe election results or corporate results are scheduled at the same time. Will you still go ahead and write the option for Rs.20? Maybe not, as you know with the increase in volatility, the option can easily expire ‘in the money’ hence you may lose all the premium money you have collected. If all option writers start fearing the volatility, then what would compel them to write options? Clearly, a higher premium amount would. Therefore instead of Rs.20, if

the premium was 30 or 40, you may just think about writing the option I suppose. In fact this is exactly what goes on when volatility increases (or is expected to increase) – option writers start fearing that they could be caught writing options that can potentially transition to ‘in the money’. But nonetheless, fear too can be overcome for a price, hence option writers expect higher premiums for writing options, and therefore the premiums of call and put options go up when volatility is expected to increase. The graphs below emphasizes the same point –

X axis represents Volatility (in %) and Y axis represents the premium value in Rupees. Clearly, as we can see, when the volatility increases, the premiums also increase. This holds true for both call and put options. The graphs here go a bit further, it shows you the behavior of option premium with respect to change in volatility and the number of days to expiry. Have a look at the first chart (CE), the blue line represents the change in premium with respect to change in volatility when there is 30 days left for expiry, likewise the green and red line represents the change in premium with respect to change in volatility when there is 15 days left and 5 days left for expiry respectively. Keeping this in perspective, here are a few observations (observations are common for both Call and Put options) – 1.

Referring to the Blue line – when there are 30 days left for expiry (start of the series) and the volatility increases from 15% to 30%, the premium increases from 97 to 190, representing about 95.5% change in premium

2.

Referring to the Green line – when there are 15 days left for expiry (mid series) and the volatility increases from 15% to 30%, the premium increases from 67 to 100, representing about 50% change in premium

3.

Referring to the Red line – when there are 5 days left for expiry (towards the end of series) and the volatility increases from 15% to 30%,

the premium increases from 38 to 56, representing about 47% change in premium Keeping the above observations in perspective, we can make few deductions – 1.

The graphs above considers a 100% increase of volatility from 15% to 30% and its effect on the premiums. The idea is to capture and understand the behavior of increase in volatility with respect to premium and time. Please be aware that observations hold true even if the volatility moves by smaller amounts like maybe 20% or 30%, its just that the respective move in the premium will be proportional

2.

The effect of Increase in volatility is maximum when there are more days to expiry – this means if you are at the start of series, and the volatility is high then you know premiums are plum. Maybe a good idea to write these options and collect the premiums – invariably when volatility cools off, the premiums also cool off and you could pocket the differential in premium

3.

When there are few days to expiry and the volatility shoots up the premiums also goes up, but not as much as it would when there are more days left for expiry. So if you are a wondering why your long options are not working favorably in a highly volatile environment, make sure you look at the time to expiry So at this point one thing is clear – with increase in volatility, the premiums increase, but the question is ‘by how much?’. This is exactly what the Vega tells us. The Vega of an option measures the rate of change of option’s value (premium) with every percentage change in volatility. Since options gain value with increase in volatility, the vega is a positive number, for both calls and puts. For example – if the option has a vega of 0.15, then for each % change in volatility, the option will gain or lose 0.15 in its theoretical value. Taking things forward It is now perhaps time to revisit the path this module on Option Trading has taken and will take going forward (over the next topic). We started with the basic understanding of the options structure and then proceeded to understand the Call and Put options from both the buyer and

sellers perspective. We then moved forward to understand the moneyness of options and few basic technicalities with respect to options. We further understood option Greeks such as the Delta, Gamma, Theta, and Vega along with a mini series of Normal Distribution and Volatility. At this stage, our understanding on Greeks is one dimensional. For example we know that as and when the market moves the option premiums move owing to delta. But in reality, there are several factors that works simultaneously – on one hand we can have the markets moving heavily, at the same time volatility could be going crazy, liquidity of the options getting sucked in and out, and all of this while the clock keeps ticking. In fact this is exactly what happens on an everyday basis in markets. This can be a bit overwhelming for newbie traders. It can be so overwhelming that they quickly rebrand the markets as ‘Casino’. So the next time you hear someone say such a thing about the markets, make sure you point them to Varsity. Anyway, the point that I wanted to make is that all these Greeks manifest itself on the premiums and therefore the premiums vary on a second by second basis. So it becomes extremely important for the trader to fully understand these ‘inter Greek’ interactions of sorts. This is exactly what we will do in the next topic. We will also have a basic understanding of the Black & Scholes options pricing formula and how to use the same.

This is a very interesting chart, and to begin with I would suggest you look at only the blue line and ignore the red line completely. The blue line represents the delta of a call option. The graph above captures few interesting characteristics of the delta; let me list them for you (meanwhile keep this point in the back of your mind – as and when the spot price changes, the moneyness of the option also changes) – 1.

Look at the X axis – starting from left the moneyness increases as the spot price traverses from OTM to ATM to ITM

2.

Look at the delta line (blue line) – as and when the spot price increases so does the delta

3.

Notice at OTM the delta is flattish near 0 – this also means irrespective of how much the spot price falls ( going from OTM to deep OTM) the option’s delta will remain at 0

a.

Remember the call option’s delta is lower bound by 0

4.

When the spot moves from OTM to ATM the delta also starts to pick up (remember the option’s moneyness also increases)

a.

Notice how the delta of option lies within 0 to 0.5 range for options that are less than ATM

5. 6.

At ATM, the delta hits a value of 0.5 When the spot moves along from the ATM towards ITM the delta starts to move beyond the 0.5 mark

7. a.

Notice the delta starts to fatten out when it hits a value of 1 This also implies that as and when the delta moves beyond ITM to say deep ITM the delta value does not change. It stays at its maximum value of 1.

Theta Time is money Remember the adage “Time is money”, it seems like this adage about time is highly relevant when it comes to options trading. Assume you have enrolled for a competitive exam, you are inherently a bright candidate and have the capability to clear the exam, however if you do not give it sufficient time and brush up the concepts, you are likely to flunk the exam – so given this what is the likelihood that you will pass this exam? Well, it depends on how much time you spend to prepare for the exam right? Let’s keep this in perspective and figure out the likelihood of passing the exam against the time spent preparing for the exam. Number of days for preparation 30 days 20 days 15 days 10 days 5 days 1 day

Likelihood of passing Very high High Moderate Low Very low Ultra low

Quite obviously higher the number of days for preparation, the higher is the likelihood of passing the exam. Keeping the same logic in mind, think about the following situation.

Is there anything that we can infer from the above? Clearly, the more time for expiry the likelihood for the option to expire In the Money (ITM) is higher. Now keep this point in the back of your mind as we now shift our focus on the ‘Option Seller’. We know an option seller sells/writes an option and receives the premium for it. When he sells an option he is very well aware that he carries an unlimited risk and limited reward potential. The reward is limited to the extent of the premium he receives. He gets to keep his reward (premium) fully only if the option expires worthless. Now, think about this – if he is selling an option early in the month he very clearly knows the following –

1.

He knows he carries unlimited risk and limited reward potential

2.

He also knows that by virtue of time, there is a chance for the option he is selling to transition into ITM option, which means he will not get to retain his reward (premium received) In fact at any given point, thanks to ‘time’, there is always a chance for the option to expiry in the money (although this chance gets lower and lower as time progresses towards the expiry date). Given this, an option seller would not want to sell options at all right? After all why would you want to sell options when you very well know that simply because of time there is scope for the option you are selling to expire in the money. Clearly time in the option sellers context acts as a risk. Now, what if the option buyer in order to entice the option seller to sell options offers to compensate for the ‘time risk’ that he (option seller) assumes? In such a case it probably makes sense to evaluate the time risk versus the compensation and take a call right? In fact this is what happens in real world options trading. Whenever you pay a premium for options, you are indeed paying towards –

1.

Time Risk

2.

Intrinsic value of options. So given that we know how to calculate the intrinsic value of an option, let us attempt to decompose the premium and extract the time value and intrinsic value. Have a look at the following snapshot –

Details to note are as follows – 

Spot Value = 8531



Strike = 8600 CE



Status = OTM



Premium = 99.4



Today’s date = 6th July 2015



Expiry = 30th July 2015

Intrinsic value of a call option – Spot Price – Strike Price i.e. 8531 – 8600 = 0 (since it’s a negative value) we know – Premium = Time value + Intrinsic value 99.4 = Time Value positive 0 this implies Time value = 99.4! Do you see that? The market is willing to pay a premium of Rs.99.4/- for an option that has zero intrinsic value but ample time value! Recall time is money. Here is snapshot of the same contract that I took the next day i.e. 7th July Movement of time Time as we know moves in one direction. Keep the expiry date as the target time and think about the movement of time. Quite obviously as

time progresses, the number of days for expiry gets lesser and lesser. Given this let me ask you this question – With roughly 18 trading days to expiry, traders are willing to pay as much as Rs.100/- towards time value, will they do the same if time to expiry was just 5 days? Obviously they would not right? With lesser time to expiry, traders will pay a much lesser value towards time. In fact here is a snap shot that I took from the earlier months – 

Date = 29th April



Expiry Date = 30th April



Time to expiry = 1 day



Strike = 190



Spot = 179.6



Premium = 30 Paisa



Intrinsic Value = 179.6 – 190 = 0 since it’s a negative value



Hence time value should be 30 paisa which equals the premium

With 1 day to expiry, traders are willing to pay a time value of just 30 paisa. However, if the time to expiry was 20 days or more the time value would probably be Rs.5 or Rs.8/-. The point that I’m trying to make here is this – with every passing day, as we get closer to the expiry day, the time to expiry becomes lesser and lesser. This means the option buyers will pay lesser and lesser towards time value. So if the option buyer pays Rs.10 as the time value today, tomorrow he would probably pay Rs.9.5/- as the time value. This leads us to a very important conclusion – “All other things being equal, an option is a depreciating asset. The option’s premium erodes daily and this is attributable to the passage of time”. Now the next logical question is – by how much would the premium decrease on a daily basis owing to the passage of time? Well, Theta the 3 rd Option Greek helps us answer this question.

Theta All options – both Calls and Puts lose value as the expiration approaches. The Theta or time decay factor is the rate at which an option loses value

as time passes. Theta is expressed in points lost per day when all other conditions remain the same. Time runs in one direction, hence theta is always a positive number, however to remind traders it’s a loss in options value it is sometimes written as a negative number. A Theta of -0.5 indicates that the option premium will lose -0.5 points for every day that passes by. For example, if an option is trading at Rs.2.75/- with theta of -0.05 then it will trade at Rs.2.70/- the following day (provided other things are kept constant). A long option (option buyer) will always have a negative theta meaning all else equal, the option buyer will lose money on a day by day basis. A short option (option seller) will have a positive theta. Theta is a friendly Greek to the option seller. Remember the objective of the option seller is to retain the premium. Given that options loses value on a daily basis, the option seller can benefit by retaining the premium to the extent it loses value owing to time. For example if an option writer has sold options at Rs.54, with theta of 0.75, all else equal, the same option is likely to trade at – =0.75 * 3 = 2.25 = 54 – 2.25 = 51.75 Hence the seller can choose to close the option position on T+ 3 day by buying it back at Rs.51.75/- and profiting Rs.2.25 See below chart how change in 1 day keeping other things constant premium changes.

Nifty trading at 8800 date is 4th October, 2016 and Theta is -4.1636, Premium is 161.25

Now next day all things remains the same and premium changes to 157.05 Theta reduced from premium. See the below chart for reference-

Have a look at the graph below – How premium erodes as expiry comes near-

This is the graph of how premium erodes as time to expiry approaches. This is also called the ‘Time Decay’ graph. We can observe the following from the graph –

1.

At the start of the series – when there are many days for expiry the option does not lose much value. For example when there were 120 days to expiry the option was trading at 350, however when there was 100 days to expiry, the option was trading at 300. Hence the effect of theta is low

2.

As we approach the expiry of the series – the effect of theta is high. Notice when there was 20 days to expiry the option was trading around 150, but when we approach towards expiry the drop in premium seems to accelerate (option value drops below 50).

Implication of implied volatility and influence of lV on options Greek What is volatility? “Volatility is the up down movement of the stock market”. If you have a similar opinion on volatility, then it is about time we fixed that . What Is Implied Volatility? It is not uncommon for investors to be reluctant about using options because there are several variables that influence an option's premium. Don't let yourself become one of these people. As interest in options continues to grow and the market becomes increasingly volatile, this will dramatically affect the pricing of options and, in turn, affect the possibilities and pitfalls that can occur when trading them. Implied volatility is an essential ingredient to the option pricing equation. To better understand implied volatility and how it drives the price of options, let's go over the basics of options pricing. How Implied Volatility Affects Options The success of an options trade can be significantly enhanced by being on the right side of implied volatility changes. For example, if you own options when implied volatility increases, the price of these options climbs higher. A change in implied volatility for the worse can create losses, however, even when you are right about the stock's direction!

Each listed option has a unique sensitivity to implied volatility changes. For example, short-dated options will be less sensitive to implied volatility, while long-dated options will be more sensitive. This is based on the fact that long-dated options have more time value priced into them, while short-dated options have less. How to Use Implied Volatility to Your Advantage One effective way to analyze implied volatility is to examine a chart. Many charting platforms provide ways to chart an underlying option's average implied volatility, in which multiple implied volatility values are tallied up and averaged together. For example, the volatility index (VIX) is calculated in a similar fashion. Implied volatility values of near-dated, near-themoney S&P 500 Index options are averaged to determine the VIX's value. The same can be accomplished on any stock that offers options.

Source: http://www.prophet.net/ Figure 1: Implied volatility using INTC options

Figure 1 shows that implied volatility fluctuates the same way prices do. Implied volatility is expressed in percentage terms and is relative to the underlying stock and how volatile it is. For example, General Electric stock will have lower volatility values than Apple Computer because Apple's stock is much more volatile than General Electric's. Apple's volatility range will be much higher than GE's. What might be considered a low percentage value for AAPL might be considered relatively high for GE. Because each stock has a unique implied volatility range, these values should not be compared to another stock's volatility range. Implied volatility should be analyzed on a relative basis. In other words, after you have determined the implied volatility range for the option you are trading, you will not want to compare it against another. What is

considered a relatively high value for one company might be considered low for another.

Source: www.prophet.net

Figure 2 : An implied volatility range using relative values Figure 2 is an example of how to determine a relative implied volatility range. Look at the peaks to determine when implied volatility is relatively high, and examine the troughs to conclude when implied volatility is relatively low. By doing this, you determine when the underlying options are relatively cheap or expensive. If you can see where the relative highs are (highlighted in red), you might forecast a future drop in implied volatility, or at least a reversion to the mean. Conversely, if you determine where implied volatility is relatively low, you might forecast a possible rise in implied volatility or a reversion to its mean. Implied volatility, like everything else, moves in cycles. High volatility periods are followed by low volatility periods, and vice versa. Using relative implied volatility ranges, combined with forecasting techniques, helps investors select the best possible trade. When determining a suitable strategy, these concepts are critical in finding a high probability of success, helping you maximize returns and minimize risk. Using Implied Volatility to Determine Strategy You've probably heard that you should buy undervalued options and sell overvalued options. While this process is not as easy as it sounds, it is a great methodology to follow when selecting an appropriate option strategy. Your ability to properly evaluate and forecast implied volatility will make the process of buying cheap options and selling expensive options that much easier. When forecasting implied volatility, there are four things to consider:

1. Make sure you can determine whether implied volatility is high or low and whether it is rising or falling. Remember, as implied volatility increases, option premiums become more expensive. As implied volatility decreases, options become less expensive. As implied volatility reaches extreme highs or lows, it is likely to revert back to its mean. 2. If you come across options that yield expensive premiums due to high implied volatility, understand that there is a reason for this. Check the news to see what caused such high company expectations and high demand for the options. It is not uncommon to see implied volatility plateau ahead of earnings announcements, merger and acquisition rumors, product approvals and other news events. Because this is when a lot of price movement takes place, the demand to participate in such events will drive option prices price higher. Keep in mind that after the market-anticipated event occurs, implied volatility will collapse and revert back to its mean. 3. When you see options trading with high implied volatility levels, consider selling strategies. As option premiums become relatively expensive, they are less attractive to purchase and more desirable to sell. Such strategies include covered calls, naked puts, short straddles and credit spreads. By contrast, there will be times when you discover relatively cheap options, such as when implied volatility is trading at or near relative to historical lows. Many option investors use this opportunity to purchase long-dated options and look to hold them through a forecasted volatility increase. 4. When you discover options that are trading with low implied volatility levels, consider buying strategies. With relatively cheap time premiums, options are more attractive to purchase and less desirable to sell. Such strategies include buying calls, puts, long straddles and debit spreads. The Bottom Line In the process of selecting strategies, expiration months or strike price, you should gauge the impact that implied volatility has on these trading decisions to make better choices. You should also make use of a few simple volatility forecasting concepts. This knowledge can help you avoid buying overpriced options and avoid selling underpriced ones.

Volatility and Implied Volatility - Explanation Volatility, as applied to options, is a statistical measurement of the rate of price changes in the underlying asset: the greater the changes in a given time period, the higher the volatility. The volatility of an asset will influence the prices of options based on that asset, with higher volatility leading to higher option premiums. Option premiums depend, in part, on volatility because an option based on a volatile asset is more likely to go into the money before expiration. On the other hand, a low volatile asset will tend to remain within tight limits in its price variation, which means that an option based on that asset will only have a significant probability of going into the money if the underlying price is already close to the strike price. Thus, volatility is a measure of the uncertainty in the expected future price of an asset. An option premium consists of time value, and it may also consist of intrinsic value if it is in the money. Volatility only affects the time value of the option premium. How much volatility will affect option prices will depend on how much time there is left until expiration: the shorter the time, the less influence volatility will have on the option premium, since there is less time for the price of the underlying to change significantly before expiration. Higher volatility increases the delta for out-of-the-money options while decreasing delta for in-the-money options; lower volatility has the opposite effect. This relationship holds because volatility has an effect on the probability that the option will finish in the money by expiration: higher volatility will increase the probability that an out-of-the-money option will go into the money by expiration, whereas an in-the-money option could easily go out-of-the-money by expiration. In either case, higher volatility increases the time value of the option so that intrinsic value, if any, is a smaller component of the option premium. Implied Volatility is Not Volatility — It Measures the Demand Over Supply for a Particular Option Because volatility obviously has an influence on option prices, theBlackScholes model of option pricing includes volatility as a component plus the following factors: 

strike price in relation to the underlying asset price;



the amount of time remaining until expiration;



interest rates, where higher interest rates increase the call premium but lower the put premium;



dividends, where a higher dividend paid by the underlying asset lowers a call premium but increases the put premium.

The Black-Scholes formula calculates only a theoretical price for a call premium; the theoretical price for a put premium can be calculated through the put-call parity relationship. However, the actual value — the market price — of an option premium will be determined by the instantaneous supply and demand for the option. When the market is active, the following factors are known: 

the actual option premium



strike price



time until expiration



interest rates



any dividend

Therefore, volatility can be calculated with the Black-Scholes equation or from another option-pricing model by plugging in the known factors into the equation and solving for the volatility that would be required to yield the market price of the call premium. This is what is known as implied volatility. Implied volatility does not have to be calculated by the trader, since most option trading platforms provide it for each option listed.

Implied volatility makes no predictions about future price swings of the underlying stock, since the relationship is tenuous at best. Implied volatility can change very quickly, even without any change in the volatility of the underlying asset. Although implied volatility is measured the same as volatility, as a standard deviation percentage, it does not actually reflect the volatility either of the underlying asset or even of the option itself. It is simply the demand over supply for that particular option, and nothing more. Generally, in a rising market, calls will generally have a higher implied volatility while puts will have a lower implied volatility; in a declining market, puts will have a higher implied volatility over calls. This reflects the increased demand for calls in a rising market and a rising demand for puts in a declining market. A rise in the implied volatility of a call will decrease the delta for an in-themoney option, because it has a greater chance of going out-of-the-money, whereas for an out-of-the-money option, a higher implied volatility will increase the delta, since it will have a greater probability of finishing in the money. Implied volatility is not present volatility nor future volatility. It is simply the volatility calculated from the market price of the option premium. There is an indirect connection between historical volatility and implied

volatility, in that historical volatility will have a large effect on the market price of the option premium, but the connection is only indirect; implied volatility is directly affected by the market price of the option premium, which, in turn, is influenced by historical volatility. Implied volatility is the volatility that is implied by the current market price of the option premium. That implied volatility does not represent the actual volatility of the underlying asset can be seen more clearly by considering the following scenario: a trader wants to either buy or sell a large number of options on a particular underlying asset. A trader may want to sell because he needs the money; perhaps, it is a pension fund that needs to make payments on its pension obligations. Now, a large order will have a direct influence on the pricing of the option, but it would have no effect on the price of the underlying. It is clear to see that the price change in the option premium is not effected by any changes in the volatility of the underlying asset, because the buy or sell orders are for the option itself, not for the underlying asset. As a further illustration, the implied volatility for puts and calls and for option contracts with different strike prices or expiration dates that are all based on the same underlying asset will have different implied volatilities, because the different options will each have a different supply-demand equilibrium. This is what causes the volatility skew and volatility smile. Thus, implied volatility is not a direct measure of the volatility of the underlying asset. Implied volatility varies with the change in the supply-demand equilibrium, which is why it measures the supply and demand for a particular option rather than the volatility of the underlying asset. For instance, if a stock is expected to increase in price, then the demand for calls will be greater than the demand for puts, so the calls will have a higher implied volatility, even though both the calls and the puts are based on the same underlying asset. Likewise, puts on indexes, such as the S&P 500, may have a higher implied volatility, since there is a greater demand by fund managers who wish to protect their position in the underlying stocks. At the same time, the same fund managers generally sell calls on the indexes to finance the purchase of puts on the same index; such a spread is referred to as a collar. This lowers the implied volatility on the calls while increasing the implied volatility for the puts. Because implied volatility is a measure of the instantaneous demandsupply equilibrium, it can indicate that an option is either over- or underpriced relative to the other factors that determine the option premium, but only if implied volatility is not higher because of major news or because of an impending event, such as FDA approval for a drug or the results of an important court case. Likewise, implied volatility may be low because the option is unlikely to go into the money by expiration. If

implied volatility is high because of an impending event, then it will decline after the event, since the uncertainty of the event is removed; this rapid deflation of implied volatility is sometimes referred to as a volatility crush. However, implied volatility that is merely due to the normal statistical fluctuation of supply and demand for a particular option may be used to increase profits or decrease losses, especially for an option spread. If an option has high implied volatility, then it may contract later on, reducing the time value of the option premium in relation to the other price determinants; likewise, low implied volatility may have resulted from a temporary decline in demand or a temporary increase in supply that may revert to the average later. So high implied volatility will tend to decline, while low implied volatility will tend to increase over the lifetime of the option. Thus, implied volatility may be an important consideration when setting up option spreads, where maximum profits and losses are determined by how much was paid for long options and how much was received for short options. When selecting long options for a spread, some consideration should be given to selecting strike prices that have lower implied volatilities, while strike prices for short options should have higher implied volatilities. This lowers the debit when paying for long spreads while increasing the credit received for selling short spreads. Although the implied volatility varies widely among different assets, including different stocks, different indexes, different futures contracts, and so on, the volatility of an index will usually be less than the volatility of individual assets, since an index is a measure of the price changes of all of the individual components of the index, where assets with greater volatility will be offset by other assets with lower volatility. Diagram showing the difference in the expected price distribution about the mean for a volatile and a non-volatile stock.

Implied volatility, like volatility, is calculated as an annual standard deviation, expressed as a percentage, that can be used to compare implied volatility of different options that are not only based on the same asset, but also on different assets, including stocks, indexes, or futures. Moreover, the other factors of the option-pricing model, such as interest and dividends, are also usually expressed as an annual percentage. Most trading platforms calculate the implied volatility for the different options. The standard deviation is a statistical measure of the variability — and, therefore, of volatility — of an underlying asset and can be useful in predicting the probability that the asset will be within a particular price range. In a normal distribution, which characterizes the price variation of most assets, 68.3% of price changes of the underlying asset over a 1year period will be within 1 standard deviation of the mean, 95.4% will be within 2 standard deviations, and 99.7% will be within 3 standard deviations. Volatility determines how wide the standard deviation is. If there is little variability, then the normal distribution will be much narrower, whereas for a highly variable asset, the normal distribution would be much flatter, where 1 standard deviation would encompass a wider variability in pricing over a unit of time. So if a stock has a mean price of $100, and has a volatility percentage of 15%, then during the course of the year, the price of the stock will stay within ±$15 for 68.3% of the time. Vega and Other Measures of Volatility Vega measures a change in the theoretical option price caused by a 1point change in implied volatility. For instance, an option with a vega of . 01 will increase by $10 per contract (which consists of 100 shares) for each point increase in volatility and will lose $10 per contract for each 1% decline in volatility. For the short position, vega would have the exact opposite effect, where a 1-point increase in volatility would decrease the value of the short option by $10. Most options have both intrinsic value and time value. Intrinsic value is a measure of how much the option is in the money; the time value is equal to the option premium minus the intrinsic value. Thus, time value depends on the probability that the option will go into the money or stay into the money by expiration. Volatility only affects the time value of an option. Therefore, vega, as a measure of volatility, is greatest when the time value of the option is greatest and least when it is least. Because time value is greatest when the option is at the money, that is also when volatility will have the greatest effect on the option price. And just as time

value diminishes as an option moves further out of the money or into the money, so goes vega. There are general measures of volatility that represent volatility of entire markets. The Chicago Board Option Exchange (CBOE) Volatility Index (VIX) measures the implied volatility on the options based on the S&P 500 index. This is not the same as implied volatility of the underlying stocks that compose the S&P 500 index, but as a measure of the implied volatility of the options on those stocks. VIX is also known as a fear index, because it presumably measures the amount of fear in the market; in actuality, it probably causes fear rather than reflecting fear, because higher fluctuations in the supply and demand for the options creates more uncertainty. Other measures of general volatility include the NASDAQ 100 Volatility Index (VXN), which measures the volatility of the NASDAQ 100, which includes many high-tech companies. The Russell 2000 Volatility Index (RVX) measures the volatility of the index composed of the 2000 stocks in the Russell 2000 Index. Volatility Skew Volatility skew is a result of different implied volatilities for different strike prices and for whether the option is a call or put. Volatility skew further illustrates that implied volatility depends only on the option premium, not on the volatility of the underlying asset, since that does not change with either different strike prices or option type. How the volatility skew changes with different strike prices depends on the type of skew, which is influenced by the supply and demand for the different options. A forward skew is exhibited by higher implied volatilities for higher strike prices. A reverse skew is one with lower implied volatilities for higher strike prices. A smiling skew is exhibited by an implied volatility distribution that increases for strike prices that are either lower or higher than the price of the underlying. A flat skew means that there is no skew: implied volatility is the same for all strike prices. The options of most underlying assets exhibit a reverse skew, reflecting the fact that slightly out-of-the-money options have a greater demand than those that are in the money. Furthermore, out-of-the-money options have a higher time value, so volatility will have a greater effect for options that only have time value. Thus, a call and a put at the same strike price will have different implied volatilities, since the strike price will likely differ from the price of the underlying, demonstrating yet again that implied volatility is not the result of the volatility of the underlying asset. Options with the same strike prices but with different expiration months also exhibit a skew, with the near months generally showing a higher

implied volatility than the far months, reflecting a greater demand for near-term options over those with later expirations.

Put Call Parity Option: Put/call parity is an options pricing concept first identified by economist Hans Stoll in his 1969 paper "The Relation between Put and Call Prices." It defines the relationship that must exist between European put and call options with the same expiration and strike price (it does not apply to American style options because they can be exercised any time up to expiration). The principal states that the value of a call option, at one strike price, implies a fair value for the corresponding put and vice versa. The relationship arises from the fact that combinations of options can create positions that are identical to holding the underlying itself (a stock, for example). The option and stock positions must have the same return or an arbitrage opportunity would arise. Arbitrageurs would be able to make profitable trades, free of risk, until put/call parity returned. Put-call parity is one of the simplest and best known no-arbitrage relations. It requires neither assumptions about the probability distribution of the future price of the underlying asset, nor continuous trading, nor a host of other complications often associated with option pricing models. Investigations into apparent violations of put-call parity for the most part find that the violations do not represent tradable arbitrage opportunities once one accounts for market features such as dividend payments, the early exercise value of American options, short-sales restrictions, simultaneity problems in trading calls, puts, stocks and bonds at once, transaction costs, lending rates that do not equal borrowing rates, margin requirements, and taxes, to name a few. Prices are not fully efficient in the model and option prices deviate from put-call parity in the direction of the informed investors’ private information. Over time, of course, deviations are expected to be arbitraged away, but this is not instantaneous given that there is private information. In practice one would also expect any tradable violations of put-call parity to be quickly arbitraged away. However, options on individual stocks are American and can be exercised before expiration, therefore put-call parity is an inequality rather than a strict equality; in addition, market imperfections and transactions costs only widen the range within which call and put prices are required to fall so as not to violate arbitrage restrictions.

Let’s investigate whether the relative position of call and put prices within this range matters, using the difference in implied volatility, or “volatility spread,” between call and put options on the same underlying equity, and with the same strike price and the same expiration date, to measure deviations from put-call parity. Our main results are easily summarized. First, we find that deviations from put-call parity contain information about subsequent stock prices. The evidence of predictability that we report is significant, both economically and statistically. For example, between January 1996 and December 2005, a portfolio that is long stocks with relatively expensive calls (stocks with high volatility spreads) and short stocks with relatively expensive puts (stocks with low volatility spreads) earns a value-weighted, four-factor adjusted abnormal return of 50 basis points per week (t-statistic 8.01) in the week that follows portfolio formation. Consistent with the view that deviations from put-call parity are not driven solely by short sales constraints, the long side of this portfolio earns abnormal returns that are as large as the returns on the short side: 29 basis points with a t-statistic of 6.3 for the long side versus -21 basis points (t-statistic -4.47) for the short side. In addition, we present direct evidence that our results are not driven by short sales constraints by using a shorter sample for which we have data on rebate rates, a proxy for the difficulty of short selling from the stock lending market. Also, we find even stronger results when we form portfolios based on both changes and levels of volatility spreads: a portfolio that buys stocks with high and increasing volatility spreads and sells stocks with low and decreasing volatility spreads earns a value-weighted and four-factor adjusted return of 107 basis points per week (t-statistic 7.69) including the first overnight period, and 45 basis points (t-statistic 3.53) excluding it. Again, the long side of this portfolio earns abnormal returns that are as large as the returns on the short side. The four-weekly return on the hedge portfolio, excluding the overnight period, is 85 basis points (t-statistic 2.48). Thus we present strong evidence that option prices contain information not yet incorporated in stock prices that it takes several days until this information is fully incorporated, and that the predictability is not due to short sales constraints. This constitutes our first main result.

Put-Call Parity and Arbitrage Opportunity An important principle in options pricing is called a put-call parity. It says that the value of a call option, at one strike price, implies a certain fair value for the corresponding put, and vice versa. The argument, for this

pricing relationship, relies on the arbitrage opportunity that results if there is divergence between the value of calls and puts with the same strike price and expiration date. Arbitrageurs would step in to make profitable, risk-free trades until the departure from put-call parity is eliminated. Knowing how these trades work can give you a better feel for how put options, call options and the underlying stocks are all interrelated.

Put options, call options and the underlying stock are related in that the combination of any two yields the same profit/loss profile as the remaining component. For example, to replicate the gain/loss features of a long stock position, an investor could simultaneously hold a long call and a short put (the call and put would have the same strike price and same expiration). Similarly, a short stock position could be replicated with a short call plus a long put and so on. If put/call parity did not exist, investors would be able to take advantage of arbitrage opportunities. Options traders use put/call parity as a simple test for their European style options pricing models. If a pricing model results in put and call prices that do not satisfy put/call parity, it implies that an arbitrage opportunity exists and, in general, should be rejected as an unsound strategy. There are several formulas to express put/call parity for European options. The following formula provides an example of a formula that can be used for non-dividend paying securities: c= S + p – Xe – r(T-t) p = c – S + Xe – r(T-t) Where c = call value S = current stock price p = put price X = exercise price e = Euler\'s constant (exponential function on a financial calculator equal to approximately 2.71828 r = continuously compounded risk free rate of interest T = Expiration date t = Current value date