Options valuation simplified. Very useful for CFA and FRM level 1 preparation candidates. For a more detailed understanding, you can watch the webinar video on this topic. The link for the webinar video on this topic is https://www.youtube.com/watch?v=i9ZI7PBrpl4
JIMS Rohini News - Introduction to Derivatives by Mr. N.P. Singh (Associate Professor - JIMS Rohini Sector 5) - PGDM Programme . Derivatives are defined as contracts which derive their value from an underlying asset.
- Stock
- Index
- Commodity
- Currency
- Interest Rate or
- Any other asset
Jagan Institute of Management Studies has evolved as an institution of excellence and commitment in the field of Management and Technical education. The institute from the very outset focused on professional studies at the Post- Graduate level with a view to tap, direct and channelize the enormous talent pool in the country. We offer Post Graduate Diploma in Management (PGDM) (two year - Full Time and three year - Part Time).
JIMS Rohini News - Introduction to Derivatives by Mr. N.P. Singh (Associate Professor - JIMS Rohini Sector 5) - PGDM Programme . Derivatives are defined as contracts which derive their value from an underlying asset.
- Stock
- Index
- Commodity
- Currency
- Interest Rate or
- Any other asset
Jagan Institute of Management Studies has evolved as an institution of excellence and commitment in the field of Management and Technical education. The institute from the very outset focused on professional studies at the Post- Graduate level with a view to tap, direct and channelize the enormous talent pool in the country. We offer Post Graduate Diploma in Management (PGDM) (two year - Full Time and three year - Part Time).
Want to understand how options work but don\'t have time to go through books? Read this presentation I prepared with couple of my classmates for a case study in Advanced Finance at AIM
List of 13 most common, important and anticipated questions that will be shot at you if you are going for a SEO job interview. Equip yourself with its answers and help yourself in earning your dream SEO job. ALL THE BEST!!!
10 Best MS Excel Experts you can follow on Twitter to get the valuable Tips and Tricks and priceless knowledge from them which will help you in developing your Microsoft Excel knowledge and skills.
To Understand competitive analysis
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
#Financial Modeling: Growing needs of financial modeling skills in financial ...13 Llama Interactive
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
The art of Equity Investing
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
Want to understand how options work but don\'t have time to go through books? Read this presentation I prepared with couple of my classmates for a case study in Advanced Finance at AIM
List of 13 most common, important and anticipated questions that will be shot at you if you are going for a SEO job interview. Equip yourself with its answers and help yourself in earning your dream SEO job. ALL THE BEST!!!
10 Best MS Excel Experts you can follow on Twitter to get the valuable Tips and Tricks and priceless knowledge from them which will help you in developing your Microsoft Excel knowledge and skills.
To Understand competitive analysis
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
#Financial Modeling: Growing needs of financial modeling skills in financial ...13 Llama Interactive
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
The art of Equity Investing
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
About EduPristine
Trusted by Fortune 500 Companies and 10,000 Students from 40+ countries across the globe, EduPristine is one of the leading Training providers for Finance Certifications like CFA, PRM, FRM, Financial Modeling, Business Analytics etc. EduPristine holds a profound history in training Risk Professionals across the globe. It has been an International Authorized Training provider for FRM & PRM trainings since past 4 years and has helped 250+ FRM aspirants clear the Exam. It is Registered with GARP & CFA Institute as an Approved Provider of Continuing Professional Education (CPE) credits.
http://www.edupristine.com/ca
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
3. 3
Session Agenda
Options Valuation
• Brief Introduction to Options
• Complications in Valuing Options
• Binomial Method of Valuing Options
• Replicating Call Option
• Replicating Put Option
• Risk Neutral Valuation
• Change in future stock price
• Generalizing Binomial Method
• Black Scholes Model
• Limitations of Black Scholes Model
• Summary
Expect around 4-5 questions in the exam from today’s lecture
4. • Options are contracts that give its buyer the right to buy or sell a particular asset
– In future
– At a pre-decided price (i.e. exercise or strike price)
– Without any obligations
• The seller of the option collects a payment (Option Premium) from the buyer for providing the
option
• Types of options:
– Call or Put Options
• Call Option: gives option holder the right to buy the asset at an agreed price
• Put Options: gives option holder the right to sell the asset at an agreed price
– European or American Options
• European options are those that can only be exercised on expiration.
• American options may be exercised on any trading day on or before expiration
• Positions:
• Long position: An option buyer is said to be in a long position
• Short Position: An option writer (or seller) is said to be in a short position
4
What are Options?
5. 5
Complications in Valuing Options
• Standard approach for valuing any asset:
– Figure out expected cash flows and
– Discount them at risk adjusted cost of capital reflecting the opportunity cost of capital
• Complications that arise in valuing Options
– Impossible to quantify risks associated with the Option cash flows
– Risks associated with the Options change every time there is change in the price of the underlying
6. 6
Valuation Method 1: Binomial Method
• Binomial method entails
– Assuming the price of the underlying can take only two values in any given interval of time
– Determining Option pay-offs at these prices
– Valuing the option in one of the following ways:
• Replicating the same pay-offs in a package consisting of assets that can be valued
• Alternatively, determining probability of each pay-off to arrive at a certainty equivalent expected cash-
flow and discounting it to the present value at the risk-free rate (Risk Neutral Method)
S0
Su
Su
2
Sud
Sd
Sd
2
IV1 = Max[(Su
2-X), 0]
IV2
IV3
p
1 - p
1 - p
p
p
1 - p
7. 7
Question
• Consider a six-month European call and put option on non-dividend paying stock with identical
exercise prices of Rs 85. This option is at the money. The short-term, risk-free interest rate was a bit
less than 4 percent per year, or about 2 percent for six months. The stock either falls to Rs 63.75 or
rises to Rs113.33 after six months. Determine their pay-offs at expiration:
• Solution: The pay-offs are as follows
Stock price=Rs63.75 Stock price=Rs113.33
1 Call option Rs0 Rs28.33
1 Put option Rs21.25 Rs0
8. 8
Question - Replicating Call Option
• Determine the value of the call option in previous question by replicating the call option .
• Solution: Lets look at the pay-offs from a package consisting of 0.5714 stocks and borrowing a
principal of Rs35.71 from the bank. The total amount to be repaid is Rs36.42 (including interest)
• The pay-offs are exactly the same as in the previous example for the call option. It follows that the
value of the call today should be equal to the value of 0.5714 shares less Present Value of Rs 36.42
• Thus, value of Call = (0.5714*85)- PV(36.42)= Rs 12.86
Stock price=Rs63.75 Stock price=Rs113.33
0.5714 Shares Rs36.42 Rs64.75
Repayment of loan + interest -36.42 -36.42
Total Payoff 0 Rs28.33
9. 9
Replicating Call Option
• Two questions remain, how did we determine the the number of stocks i.e. 0.5714 and how did we
determine the amount to be borrowed?
– The number of shares to be held is give by the option delta, given by:
• The amount to be borrowed is equal to the present value of the difference between the pay-offs from
the option and pay offs from the delta shares, i.e. 0.5714 share. In our example:
• The amount to be borrowed equals Present Value or PV of 36.43
Stock Price
Scenario 1
63.75
Scenario 2
113.33
Option value
Value of ∆Stock
0
36.43
28.33
64.76
Payoff from Option 0 -28.33
Portfolio Value 36.43 36.43
5714.0
75.6333.113
033.28
pricessharepossibleofSpread
priceoptionofPossibleSpread
10. 10
Replicating Put Option
• The Pay-offs from a put option can be replicated by selling delta share and setting aside a sum of
money in a risk-free investment
• In our example, the delta for the put option is given by:
• The amount to be placed in risk-free investment is PV(48.57). Calculated as shown below:
4286.0
75.6333.113
25.210
pricessharepossibleofSpread
priceoptionofPossibleSpread
Scenario 1 Scenario 2
Stock Price 63.75 113.33
Option value 21.25 0
Value of ∆Stock 27.32 48.57
Payoff from Option 21.25 0
Portfolio Value 48.57 48.57
11. 11
Replicating Put Option
• The put can be replicated as shown below:
• The value of put therefore is,
• Value of put = -0.4286 shares + PV( Rs48.57) (safe loan)
= - (0.4286 * 85) + 47.62= Rs11.19
Stock price=Rs63.75 Stock price=Rs113.33
Sale of 0.4284 Shares -Rs27.32 -Rs48.57
Repayment of loan + interest 48.57 48.57
Total Payoff 21.25 Rs0
12. 12
Risk Neutral Method
• The assumption is that investors are indifferent to risk.
– Step 1: Determine the probabilities associated with the different pay-offs
– Step 2: Determine expected cash flow under the assumption that investors are indifferent to risk
– Step 3: Discount the expected cash flow at the risk-free rate to arrive at the present value
• In our example, since the risk-free rate for six months is 2%, and investors are indifferent to risk, it
follows that:
– Expected return= [probability of rise * 33.33] + [(1- probability of rise) * (-25)] = 2.0 percent
– Therefore the probability of rise, p, = 0.463 or 46.3%
• Expected future value of the call option after six months is given by
– [Probability of rise * 28.33] + [(1- probability of rise) * 0]
= (0.463* 28.33) + (0.537 * 0)
= Rs 13.16
• The value today therefore is PV(13.16) = Rs 12.86
13. 13
Change in future stock price
• The following formula that relates the up and down changes to the standard deviation of stock returns:
– 1 + upside change = u = eσ√h
– 1 + downside change = d = 1/ u
• Where, e = base of natural logarithms = 2.718
• σ = standard deviation of (continuously compounded) stock returns
• h = time interval as fraction of year
• In our example, standard deviation of stock returns, σ = 40.69%, h = 0.5
– u= e 0.4069√0.5 = 1.3333, => upside change = 33%
– d= 1/u = 1/1.3333 = 0.75, => downside change = 25%
• Thus stock price takes the following two values
– Rs85x1.3333 = Rs113.33
– Rs85x0.75 = 63.75
14. 14
Generalizing the Binomial Method
• One step Binomial Method is simplistic
– Assumes just two values for the asset price is possible in the future
• More realism can be added by shortening the time intervals so that the calculations can allow for
greater number of values for the asset price at expiration.
– In our example if we allowed the stock to take values at the end of three months, we would have three
values at the end of six months:
• To work out the equivalent upside and downside changes when we divide the period into two three-
month intervals (h = 0.25), we use the same formula:
– 1 + upside change (3 months interval) = u = e 0.4069√0.25 = 1.226,=> upside change = 22.6%
– 1 + downside change = d = 1/ u = 1/1.226 = 0.816, => downside change = 18.6%
• We get the following tree
3
Months
6
Months -18.6%
56.6
-18.6%
69.36
85
85
104.21
127.76
+22.6 or -18.6% +22.6
+22.6
15. 15
Generalizing the Binomial Method
• If the time intervals could be made extremely small, we would be able to account for a large number of
changes in the share price
• With the help of computer programs available today the binomial method can be used with very small
time intervals
16. 16
Question
Currently, shares of ABC Corp. trade at USD 100. The monthly risk neutral probability of the price
increasing by USD 10 is 30%, and the probability of the price decreasing by USD 10 is 70%.What are
the mean and standard deviation of the price after 2 months if price changes on consecutive months
are independent? (FRM 2010 Sample Paper)
Solution:
Develop a 2 step tree.
Mean = 9% (120) + 42% (100) + 49% (80) = 92
Variance = 9% (120 – 92)2 + 42% (100 – 92)2 + 49% (80 – 92)2 = 168
Thus, standard deviation = 12.96
17. 17
Black and Scholes Model
• Black and Scholes formula allows for infinitesimally small intervals as well as the need to revise
leverage for European options on Non Dividend paying stocks
• The formula is
– Value of call option = [delta * share price] – [bank loan]
– Where,
• Log is the natural log with base e
– N (d) = cumulative normal probability density function
– EX = exercise price option; PV(EX) is calculated by discounting at the risk- free interest rate rf
– t = number of periods to exercise date
– P =present price of stock
– σ = standard deviation per period of (continuously compounded) rate of return on stock
• Value of Put = [N (-d2) * PV (EX)] - [N (-d1) * P]
)](*)2([]*)1([ EXPVdNPdN
tdd
t
t
EXPVp
d
12
2
)](/log[
1
18. 18
Question: Black and Scholes Model
• Calculation of the value of call option
Price of stock now (P) 85
Exercise price (EX) 85
Standard deviation of continuously compounded annual returns (σ) 0.4069
Year to maturity (t ) 0.5
Risk-free interest rate per annum, rf 4%
log [P/PV (EX)] 0.02
log [P/PV (EX)]/σ√t 0.07
σ√t/2 0.14
d1 = log [P/PV (EX)]/σ√t+ σ√t/2 0.2134
d2 = d1 - σ√t -0.0743
N(d1) - Can be calculated by using NORMSDIST(d1) in excel 0.5845
N(d2) - Can be calculated by using NORMSDIST(d2) in excel 0.4704
PV(EX) = 85*e-4%/2
83.3169
Value of Call 10.49
19. 19
Question: Black and Scholes Model
• For European Options on dividend paying stocks, the present value of expected dividends during the
life of the option needs to be reduced from the present price of the stock:
Without dividend With dividend
Price of stock now 85 85
Present Value of Dividend 0 1.99
Price of stock adjusted for dividend (P) 85 83.01
Exercise price (EX) 85 85
Standard daviation of continuously compounded annual returns (σ) 0.4069 0.4069
Year to maturity (t ) 0.5 0.5
Risk-free interest rate per annum, rf 4% 4%
log [P/PV (EX)] 0.02 -0.004
log [P/PV (EX)]/σ√t 0.07 -0.01
σ√t/2 0.14 0.14
d1 = log [P/PV (EX)]/σ√t+ σ√t/2 0.2134 0.1309
d2 = d1 - σ√t -0.0743 -0.1568
N(d1) 0.5845 0.5521
N(d2) 0.4704 0.4377
PV(EX) = 85*e
-4%/2
83.316 83.316
Value of Call 10.49 9.36
20. 20
Limitations of Black and Scholes Model
• Limitations:
– The model does not allow for early exercise
– Not suitable for valuing American Options that can be exercised any time during their life
– The stepwise binomial method is superior for valuing American Options, particularly American Puts and
American Calls on stocks that pay dividends
– Not suitable for valuing warrants as warrants are long term options and it is quite likely that the underlying
stock will pay dividends during the life of the warrant
– Also, when exercised warrants increase the total number of shares which adds another level of
complication in valuing warrants using Black and Scholes formula
21. 21
Summary
• Complications arise in valuing options because its impossible to quantify risks associated with options
• Options can be valued using the binomial method
– Replicating options
– Risk neutral method
• European options on non dividend paying stocks can be valued using the Black Scholes method
• Option Delta is defined as:
pricessharepossibleofSpread
priceoptionofPossibleSpread
22. 22
• Replicating a call option
• Construct a package containing
– Buy delta stocks and
– Borrow a sum of money which is equal to the difference between the pay-offs from the option and pay offs
from the delta shares
• This package has the same pay-off as a call option
• The value of the package is the value of the call option
• Replicating a put option
• Construct a package containing
– Sell delta stocks and
– Deposit a sum of money which is equal to the difference between the pay-offs from the option and pay offs
from the delta shares
• This package has the same pay-off as of a put option
• The value of the package is the value of the put option
Summary (Cont...)
23. 23
• Risk Neutral Method
• Determine the probability of upside and downside changes in stock price
• Assume investors are risk neutral
• Discount the future expected pay-off at the riskfree rate to derive the option value
Summary (Cont...)
24. 24
• Black Scholes Model
• Assumes log normal distribution of stock prices
• Provides a model for valuing European options on non dividend paying stocks:
– Value of call option = [delta * share price] – [bank loan]
– Where,
• Log is the natural log with base e
– N (d) = cumulative normal probability density function
– EX = exercise price option; PV(EX) is calculated by discounting at the risk- free interest rate rf
– t = number of periods to exercise date
– P =present price of stock
– σ = standard deviation per period of (continuously compounded) rate of return on stock
• Value of Put = [N (-d2) * PV (EX)] - [N (-d1) * P]
)](*)2([]*)1([ EXPVdNPdN
tσd1d2
2
tσ
tσ
X)]log[p/PV(E
d1
Summary (Cont...)
25. Other Webinars
Here are the links for the blogs of the other recent webinars on our website to
help you with CFA/FRM preparation
Linear regression analysis (11/04/2013)
Blog: http://www.edupristine.com/blog/demystifying-linear-regression-analysis-
for-frm-level-1-exam/
Understanding Income statement (12/04/2013)
Blog: http://www.edupristine.com/blog/cfa-tutorial-understanding-income-
statement-from-cfa-perspective/
Hedging strategies using futures (13/04/2013)
Blog: http://www.edupristine.com/blog/frm-tutorial-hedging-strategies-using-
futures-for-frm-level-1-exam/
You can find many more blogs on our website: www.edupristine.com/blog
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26. Upcoming Webinars
Look forward to more webinars from our side on the topics of your choice!! Just
drop a mail to us to suggest a topic! You can check for updates on our site:
http://www.edupristine.com/webinars
CLASSROOM TRAINING IN NEWYORK, BOSTON, CHICAGO IN US
AND LONDON IN UK
FOR MORE DETAILS, VISIT:
http://www.edupristine.com/ca/courses/frm-program/
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