S0 is the initial price of the underlying asset and ST is its price at expiration. If the share price is higher than X the call option will be exercised. Else, cash will be retained. If the share price is lower than X, the put option will be exercised. Else, the underlying asset will be retained. This gives us the put-call parity equation.
Equation for put-call parity:.
Arbitrage:
We can summarize the payoffs of both the portfolios under different conditions as shown in the table below. From the above table, we can see that under both scenarios, the payoffs from both the portfolios are equal. For put-call parity to hold, the following conditions should be met. However, in the real world, they hardly hold true and put-call parity equation may need some modifications accordingly. For the purpose of this blog, we have assumed that these conditions are met.
Hence, put-call parity will hold in a frictionless market with the underlying stock paying no dividends. In options trading, when the put-call parity principle gets violated, traders will try to take advantage of the arbitrage opportunity. An arbitrage trader will go long on the undervalued portfolio and short the overvalued portfolio to make a risk-free profit. Let us now consider an example with some numbers to see how trade can take advantage of arbitrage opportunities. In this case, the value of portfolio A will be,.
Portfolio B is overvalued and hence an arbitrageur can earn by going long on portfolio A and short on portfolio B. The following steps can be followed to earn arbitrage profits. Return from the zero coupon bond after three months will be This stock will be used to cover the short. So far, we have gone through the basic concepts in options trading and looked at an options trading strategy as well.
Options are attractive instruments to trade in because of the higher returns. This way, the holder can restrict his losses and multiply his returns. While it is true that one options contract is for shares, it is thus less risky to pay the premium and not risk the total amount which would have to be used if we had bought the shares instead. Thus your risk exposure is significantly reduced. However, in reality, options trading is very complex and that is because options pricing models are quite mathematical and complex.
So, how can you evaluate if the option is really worth buying? The key requirement in successful options trading strategies involves understanding and implementing options pricing models. In this section, we will get a brief understanding of Greeks in options which will help in creating and understanding the pricing models. Recall the moneyness concept that we had gone through a few sections ago. For OTM call options, the stock price is below the strike price and for OTM put options; stock price is above the strike price. The price of these options consists entirely of time value.
It is based on the time to expiration.
Step By Step Guide On Options Trading in India 2021: Indicators, Strategies
You can enroll for this free online options trading python course on Quantra and understand basic terminologies and concepts that will help you in options trading. We know what is intrinsic and the time value of an option. We even looked at the moneyness of an option. But how do we know that one option is better than the other, and how to measure the changes in option pricing. Greeks are the risk measures associated with various positions in options trading.
The common ones are delta, gamma, theta and vega. With the change in prices or volatility of the underlying stock, you need to know how your options pricing would be affected. Greeks in options help us understand how the various factors such as prices, time to expiry, volatility affect the options pricing. Delta is dependent on underlying price, time to expiry and volatility.
While the formula for calculating delta is on the basis of the Black-Scholes option pricing model, we can write it simply as,. Here, we should add that since an option derives its value from the underlying stock, the delta option value will be between 0 and 1. While the delta for a call option increases as the price increases, it is the inverse for a put option. Think about it, as the stock price approaches the strike price, the value of the option would decrease.
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Thus, the delta put option is always ranging between -0 and 1. Gamma measures the exposure of the options delta to the movement of the underlying stock price. Hence, gamma is called the second-order derivative. Let's see an example of how delta changes with respect to Gamma. In this way, delta and gamma of an option changes with the change in the stock price. We should note that Gamma is the highest for a stock call option when the delta of an option is at the money.
Since a slight change in the underlying stock leads to a dramatic increase in the delta. Similarly, the gamma is low for options which are either out of the money or in the money as the delta of stock changes marginally with changes in the stock option. You can watch this video to understand it in more detail. Theta measures the exposure of the options price to the passage of time.
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It measures the rate at which options price, especially in terms of the time value, changes or decreases as the time to expiry is approached. Vega measures the exposure of the option price to changes in the volatility of the underlying. Generally, options are more expensive for higher volatility. So, if the volatility goes up, the price of the option might go up to and vice-versa. Vega increases or decreases with respect to the time to expiry? What do you think? You can confirm your answer by watching this video.
One of the popular options pricing model is Black Scholes, which helps us to understand the options greeks of an option.
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C is the price of the call option P represents the price of a put option. N x is the standard normal cumulative distribution function. To calculate the Greeks in options we use the Black-Scholes options pricing model. Delta and Gamma are calculated as:. In the example below, we have used the determinants of the BS model to compute the Greeks in options. At an underlying price of If we were to increase the price of the underlying by Rs. As can be observed, the Delta of the call option in the first table was 0. Hence, given the definition of the delta, we can expect the price of the call option to increase approximately by this value when the price of the underlying increases by Rs.
The new price of the call option is If you observe the value of Gamma in both the tables, it is the same for both call and put options contracts since it has the same formula for both options types. If you are going long on the options, then you would prefer having a higher gamma and if you are short, then you would be looking for a low gamma. Thus, if an options trader is having a net-long options position then he will aim to maximize the gamma, whereas in case of a net-short position he will try to minimize the gamma value. The third Greek, Theta has different formulas for both call and put options.
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These are given below:. In the first table on the LHS, there are 30 days remaining for the options contract to expire. We have a negative theta value of He has to be sure about his analysis in order to profit from trade as time decay will affect this position. This impact of time decay is evident in the table on the RHS where the time left to expiry is now 21 days with other factors remaining the same.
As a result, the value of the call option has fallen from If an options trader wants to profit from the time decay property, he can sell options instead of going long which will result in a positive theta. We have just discussed how some of the individual Greeks in options impact option pricing. However, it is very essential to understand the combined behaviour of Greeks in an options position to truly profit from your options position. If you want to work on options greeks in Excel, you can refer to this blog.
Let us now look at a Python package which is used to implement the Black Scholes Model. Mibian is an options pricing Python library implementing the Black-Scholes along with a couple other models for European options on currencies and stocks. In the context of this article, we are going to look at the Black-Scholes part of this library. Mibian is compatible with python 2. This library requires scipy to work properly.
The function which builds the Black-Scholes model in this library is the BS function. The syntax for this function is as follows:. The first input is a list containing the underlying price, strike price, interest rate and days to expiration.