*The Business Professor*, updated April 4, 2019, last accessed August 10, 2020, https://thebusinessprofessor.com/lesson/put-option-definition/.

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**Put Option Definition**

A Put Option is a time bound contract that awards a shareholder the right to sell underlying assets at an agreed price. It is the converse of a ‘Call Option’ that awards a shareholder a time bound contract to buy underlying assets at an agreed price before its expiry.

**A Little More on What is a Put Option**

Underlying assets like commodities, stocks, indices and currencies can all be traded as ‘Put Options’. The price at which the holder can sell the underlying assets is called the ‘Strike Price’.

The value of Put Options is inversely proportional to the depreciation value of the underlying stock in relation to the strike price. On the other hand, as the relative value of the underlying stock increases, the Put Option decreases in value as it nears its date of expiry.

**What is Time Decay**

A Put Option decreases in value as it approaches its expiry date. The more the time lapses, the higher the probability of the value of the underlying asset increasing in relation to the strike value. The difference between the strike price and the underlying stock price is called the intrinsic value of the Put Option.

- Loss of time value leaves the Put Option with the intrinsic value, which is called being In The Money (ITM).
- Options Out of The Money (OTM) and At The Money (ATM), do not have an intrinsic value rending the Put Options useless, as the strike price is undesirable and the sale would be lossy.

The predetermined premium at which the Put Option is agreed upon is reflected in it’s Time Value. If the value of a Put Option is $40 while the underlying asset is being traded at $34, the intrinsic value of the Put Option is $6. If this Put Option ends up being traded at a value higher than $6, say $8, the extra $2 is considered its Time Value. These are variable values that fluctuate with the value of the underlying stock options.

**An Example of a Long Put Option**

If an investor owns a Put Option with a strike price of $10, to be sold in 30 days, with a premium of $1 per share, the investor can sell the shares at the higher price of $10 per share only until the expiration date.

If, within the duration of these 30 days, the value of the underlying assets depreciates to $5, the investor will be able to buy the shares at a reduced cost, considerably increasing his profit margins. The more the value of the underlying assets falls compared to the strike price of $10, the higher the profits.

**An Example of a Short Put Option**

A Short Put Option, a.k.a a ‘Written Option’ requires taking delivery for the purchase of underlying stocks.

If the outlook on a stock is bullish, and it is trading at $50 in the market with a stop loss value of $47 for the next 30 days, an investor can enter a Short Put Option to buy the shares of this stock at a specified premium with the strike rate of $47.

If the value of these shares stays above $47, the investor collects the total premium price equal to the maximum profits possible, even as the Put Option expires after 30 days.

Conversely, if the value of these shares falls below the $47 level, the investor would incur losses as he’s still obligated to purchase the stipulated number of shares at the strike rate of $47.

**Exercising Put Options**

Writers and buyers aren’t obligated to hold on to Put Options until their expiry. The right to sell the Put Option can be exercised anytime within the stipulated period either to lock in maximum profits in the ITM stage, or cut losses during OTM and ATM stages.

Option Writers can similarly buy back shares of Options that are performing poorly to avoid heavy losses.

**References for Put Option**

- https://www.investopedia.com/terms/p/putoption.asp
- https://en.wikipedia.org/wiki/Put_option
- http://www.businessdictionary.com/definition/put-option.html

**Academic Research on Put Option**

The American **put option **valued analytically, **Geske, R., & Johnson, H. E. (1984). ***The Journal of Finance***, ***39***(5), 1511-1524. **This article takes an analytical look at the valuation of Put Options in the United States.

Motivations for bank mergers and acquisitions: Enhancing the deposit insurance **put option **versus earnings diversification, **Benston, G. J., Hunter, W. C., & Wall, L. D. (1995). ***Journal of money, credit and banking***, ***27***(3), 777-788. **This paper presents an empirical analysis of data from 302 mergers to test the hypotheses of earning diversifications and Put Option insurance deposits.

On the discounted penalty at ruin in a jump-diffusion and the perpetual **put option**, **Gerber, H. U., & Landry, B. (1998). ***Insurance: Mathematics and economics***, ***22***(3), 263-276. **This paper examines the most optimal boundary to exercise a perpetual Put Option.

Optimal exercise boundary for an American **put option**, **Kuske, R. A., & Keller, J. B. (1998). ***Applied Mathematical Finance***, ***5***(2), 107-116. **This paper examines the optimal boundary to exercise a Put Option in the American stock valuation system.

The American **put option **and its critical stock price, **Bunch, D. S., & Johnson, H. (2000). ***The Journal of Finance***, ***55***(5), 2333-2356. **This paper examines the critical price of stocks in American Put Options valuation system.

Pricing the American **put option**: A detailed convergence analysis for binomial models, **Leisen, D. P. (1998). ***Journal of Economic Dynamics and Control***, ***22***(8-9), 1419-1444. **This paper presents a detailed look at conversion analysis of binomial stock models in the American Put Option pricing system.

A new evaluation of mean value for fuzzy numbers and its application to American **put option **under uncertainty, **Yoshida, Y., Yasuda, M., Nakagami, J. I., & Kurano, M. (2006). ***Fuzzy Sets and Systems***, ***157***(19), 2614-2626. **This articles presents a new evaluation system for arriving at the mean value of fuzzy numbers in the context of American Put Option valuation system.

Value of a **put option **to the risk-averse newsvendor, **Chen, F., & Parlar, M. (2007). ***IIE Transactions***, ***39***(5), 481-500. **This paper discusses the perceived value of Put Options to newsvendors who avoid risky investments.

**Put option **premiums and coherent risk measures, **Jarrow, R. (2002). ***Mathematical Finance***, ***12***(2), 135-142. **This paper takes a look at the risk factors inherent with the premiums on Put Options.

A note on a moving boundary problem arising in the American **put option**, **Knessl, C. (2001). ***Studies in Applied Mathematics***, ***107***(2), 157-183. **This paper takes a look at the problem of moving boundaries in the American Put Option valuation system.

The British **put option**, **Peskir, G., & Samee, F. (2011). ***Applied Mathematical Finance***, ***18***(6), 537-563. **This paper does a deep dive on the ‘Put Option’ scenario in the British trading indices.