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Financial DerivativesFinancial Derivatives

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7. Option Pricing Calculation of Option Premium

Discrete Time Continuous Time

Contract Life is converted into ‘time slice’ or ‘time interval’

Binomial Method

Contract Life is not converted into ‘time slice’

BSM

Black Shole Merton

Portfolio Replicating Approch

Risk Neutral Method

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a) Portfolio Replicating Approach

i. This method says that an asset price (underlying asset price) can move in only two directions Upward (u) or Downward (d) over a given time frame.

[the time period over which the price of an underlying is expected to move is called Binomial trial Period ]

ii. Concept behind Replicating Portfolio Approach :

Two assets having similar Cash Flow can’t have two different prices (this means they should have same prices)

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iii. What is Replicating Portfolio ?

A replicating portfolio is constructed by combining the underlying asset and risk free asset.

This portfolio is called replicating because it will have cash flow similar to option being valued. Then the value of replicating portfolio should be the value of option.

If Cash Flow of a replicating Portfolio

If Cash Flow of an Option =

Then the value of replicating Portfolio should be the value of Option

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• How to create Replicating Portfolio for Call Option

Call Option Seller Position at t = 0

• Option Seller would receive Ce Premium.

Ce = Call Option Premium (European Style) Obligation is to deliver the underlying

• Buy Δ Stock at spot price and finance this partly with Ce (Premium) and partly with borrowed amount [ and borrowing is done at Risk Free Rate] Hedge Strategy

Ce + B = ΔS i.e

Ce = ΔS - B

B = Borrowings

S = Spot Price

Liability Asset

Ce

B

ΔS

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• What ‘Δ’ (Delta) Indicates

Δ indicates that if option seller sells a call option having ‘n’ contract size than he should buy n × Δ stock to nutralise exposure for change in underlying price.

This Delta Hedging and resulting position is called Delta Hedging Position.

Cu – Cd Change in Option Price

Δ =

Su - Sd Change in Underlying Price

Δ indicates sensitivity of option to a change in underlying price.

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• How to create Replicating Portfolio for Put Option• Put Option Seller Position at t = 0

• Option Seller would receive Pe Premium.

• Pe = Put Option Premium (European Style) • Obligation is to take deliver the underlying

• Sell Δ Stock at spot price and Deposit Sale proceeds of stock along with premium amount at Risk Free Rate Hedge Strategy

• Pe + ΔS = B i.e

• Pe = B - ΔS

B = Deposits

S = Spot Price

Liability Asset

Pe

ΔS

B

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Put Call Parity or PCP Relationship

• The relationship is valid only for European Style

• Both option should have same underlying, same strike price and same maturity.

Pe = Ce – S + Xe-rt

S = Spot Price of Underlying

X = Strike Price

Pe = Put Option Premium

Ce = Call Option Premium

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S = Ce – Pe + Xe-rt

Synthetic Spot Rate

Ce = Long Call (LC)

Pe = Short Put (SP)

LCSP combo having Same Strike Price, Underlying Price and Maturity Period will purchase the underlying at maturity date because you want to or you are forced at Strike price X.

LCSP combo along with the money is required to buy the stock today

(Xe-rt) creates a synthetic long position on stock.

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b) Risk Neutral Approach

The Best method to value American Option.

It can also be used for European Option.

Investors are indifferent to Risk Expected Return of Stock is equal to Risk Free Rate of

Return

Why we assume Risk Neutral World ?

The risk of an option can be hedged by creating short or long position in the underlying.

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This method requires concept of Risk Neutral Probability which is calculated as under :

P Probability of Upward Movement

1 - P Probability of downward Movement

P = R - d

u - d

R = ert

Where t is 1 year R = 1 + r (approx)

u 1 + % increase in underlying price

d 1 - % decrease in underlying price

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When the value of ‘u’ and ‘d’ are not directly given ( in % increase or decrease form) than we estimate these values using COX, ROSS, Rubinstein.

u = 1 + σ√t d = 1 / u

We can now estimate the option price by simply discounting expected cash flow from option at risk free rate.

Ce = ?

P

1-P

Cu

Cd

t = 0 t = 1

Expected Cash Flow = P × Cu + (1- P) × Cd

Ce = Present Value of expected Cash Flow

Ce = P × Cu + (1- P) × Cd

(1 + r)

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• How to price American Option using ‘Risk Neutral Method’

• At Expiry Date, Cash Flow under American option and European Option are always same.

• The Valuation process for American option (under risk neutral) is similar to European option except that we need to test for early exercise at every node (except at t = 0)

• The value of each node is greater of two things.

a) PV expected Cash Flow at that node

(similar to European style)

b) Value option, if it is exercised at that Node (intrinsic value)

More from Right to left on Binomial tree assigning maximum of (a) and (b) at each node till you reach at t = 0