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DERIVATIVES

OPTIONS

KAUSHIK  DESARKAR

MBA

GOA INSTITUTE  OF  MANAGEMENT 

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

• The Binomial Option Pricing Model

• The Replicating Portfolio Approach

•

The Risk

 Neutral

 Valuation

• The replicating Portfolio approach focuses on developing a portfolio that replicates thecash flows from the option position and then valuing the cash flows to determine thevalue of the option.

• The Risk Neutral Valuation approach focuses on valuing the Option in a risk neutralenvironment where exposure to uncertainty does not lead to any risk‐premium. Thereason for choosing this approach is because options being highly leveraged, one cannotascertain the right risk‐adjusted discount rate.

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

• TO ASCERTAIN THE RISK‐ADJUSTED DISCOUNT RATE FOR A CALL OPTION

• STEP

1. VALUE THE CALL OPTION USING THE RISK NEUTRAL PROBABILITIES

2. NOW CALCULATE THE REAL WORLD PROBABILITIES USING THE VOLATILITY AND THE RADR OF THE UNDERLYING [USE  THE  RADR TO CALCULATE  a in (a‐d)/(u‐d)]

3. BASED ON THE UP‐PRICE AND THE DN‐PRICE  – CALCULATE THE VALUE OF THE CALL AND HENCE THE EXPECTED VALUE IN THE FUTURE STATE.

4. CONNECT THE CALL’S EXPECTED VALUE IN THE FUTURE STATE TO THE VALUE FROM 1.5. NOTE THE RADR OF THE CALL

6. CHANGE THE SPOT/STRIKE AND NOTICE HOW THE RADR CHANGES.

7. EXAMPLE 

• SPOT = 200, STRIKE = 200, T = 1, UNDERLYING RADR = 16.13%, RF = 10% & VOL = 35%•WHAT IS THE RADR OF THE EUROPEAN CALL OPTION ?

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•BINOMIAL OPTION PRICING MODEL

• Shift from 1 period to 2‐period

• Spot = 200, Strike = 200, rf = 7%, Volatility = 30%

•

Assume the

 Option

 has

 a life

 = 1 year

 (T)

• If  the number of  periods (time‐steps/steps) in this 1 year = 2 (N)

• Then each step = ½ = 0.5 years

•Hence dt = (T/N) = 0.5

• The more number of  steps, the better the results.

•Accordingly up‐state factor is scaled as follows : u = exp(volality X √dt)

• The Dn‐state factor = d = 1/u

•And the risk free FVF : a = exp(rf X dt)

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•CALCULATIONS  – AMERCIAN & EUROPEAN

• = Value of  Derivative in the Up‐state in the next period

• = Value of  Derivative in the Down‐state in the next period

•PV = exp(‐rf * dt)

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NODES EUROPEAN AMERICAN

Terminal Call : Max (S‐K,0)

Put : Max (K‐S,0)

Intermediate

First

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•Valuing European Options  – CALL & PUT

•As we increase the number of  time steps, the Value of  the Option fluctuates initially and then converges.

•Here we take 4 different time steps  – 2, 5, 50, 100.

•And the results are as follows (for both Puts and Calls)

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•DELTA

•Did you notice how the Delta changed for both the European Call and the European Put ?

• Putting it in a table form for European Call options

•Can you create one for European Put Options?

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Moneyness Delta

Out‐of ‐The‐Money Between 0 and 0.5

At‐The‐Money 0.5 (approx)

In‐The‐Money Between 0.5 and 1

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•Binomial Option Pricing Model 

• Incorporating the Cost of  Carry

• Till now the only element in the Cost of  Carry = risk free rate

• Suppose the underlying has a yield like foreign currencies (remember Forwards & 

Futures)• Then the Cost of  Carry = c = rf – q

•And hence a = exp(c * dt)

• The rest of  the calculation is as before.

• In the case of Options on Futures, since it does not cost anything to enter into Futurescontract (keeping margins aside), hence in the Risk Neutral (RN) world, the expectedgrowth rate = 0.

• So we set a = 1 and thus RN probability(up) = (1‐d)/(u‐d)

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•Binomial Option Pricing Model is the starting point for pricing Options.

• If  we wish to value a Bermudan Put Option, it is very simple using the BOPM.

• Going back to the previous numerical.

• Spot = 200 , Strike = 200• Rf = 7% and Volatility = 30%.

• T = 1 year.

• If the Put Option can be exercised on completion of 0.5 years and thereafter at the end of the

life.• This is a Bermudan Put Option.

• We take 4 time steps and check for exercise at the 2nd and 4th Time Step.

•

Do as

 Excel

 Exercise.

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•Binomial Option Pricing Model is the starting point for pricing Options.

• As we increase the time steps, the BOPM leads us to the Black Scholes Option Pricing Model (Next Session).

• For your information, the following lists the popular option pricing methods

• Binomial Option Pricing Model

• Trinomial Option Pricing Model

•

Black 

Scholes 

Option 

Pricing 

Model• Monte Carlo Simulation

• Partial Differential Equation  – Finite Difference Method  – (1) Explicit and (2) Implicit

• Partial Differential Equation  – Finite Element Method 

•

Of  

course 

 – you 

can 

also 

do 

some 

innovation 

to 

arrive 

at 

a 

new 

way 

to 

price 

an 

option 

….

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•A List of  the Pricing Methods and the instruments that can be priced. 

• Binomial Option Pricing Model  – European, American, Bermudan

• Trinomial Option Pricing Model  – European, American, Bermudan

• Black Scholes Option Pricing Model  – European• Monte Carlo Simulation  – All

• The Partial Differential Equation route requires higher level mathematics – hence we shalldemonstrate the pricing of European and American Options using the Explicit Finite DifferenceMethod – but this is not a part of the syllabus.

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•SUMMARY

• Understand how Leveraged Options are (using a 1‐period Call Option) 

•Understand Option Pricing using the most basic yet popular method  – Binomial Option Pricing Model

• Valuing both European and American Call & Put Options

• Demonstrating the Put‐Call Parity of  the European Options valued by the BOPM.

• Next Set of  Sessions …..

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DERIVATIVES– KAUSHIKDESARKAR,GOAINSTITUTEOFMANAGEMENT 

•SUMMARY

• Next Set of  Sessions

•Session 10 : BSM Pricing Model  – incorporating the Cost‐of ‐Carry and Excel Spreadsheet calc.

• Session 11 : Mid‐Term  – Linear Derivatives

• Session 12 : Option Strategies (along with a Case)

• Session 13 : The Greeks & introduction to Volatility

• Session 14 : Implied Volatility & Basic Volatility Modelling (inc. Group Assignment)• Session 15‐17 : Monte Carlo Simulation and Exotic Options

• Session 18‐19 : Credit Derivatives

• Session 20 : Further Material/ Complete Review

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