Option Valuation Model

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Option Valuation Models


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Section 1:

Put Call Parity

3


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3

What is Put Call Parity ?

Put Call Parity

Relationship between

The price of the European call option (C)

the price of the European put option (P)

of the same strike price (K)

At Maturity Date (T)

Mathematical Equation

C-P = Spot Price (S)- Present Value of Strike Price (PV of K)


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4

Logic behind Put Call Parity

For a given underlying,

Consider a portfolio of Long Future, Long Put & Short Call (only conditionbeing Call & Put must of the same strike price)

Such a portfolio will assure the holder pay in equal to the strike priceirrespective of the level of expiry.

e.g. Investor holds Nifty Future @ 5200, Nifty Put of the strike price 5200 &has sold call of Nifty 5200 strike. (Assured Pay in equal to 5200)

Nifty on

Expiry

Payoff on Long

Put

Payoff on Short

Call

Total

Assured Pay

inCase 1 (Expiry=

CMP) 5200 0 0 5200

Case 2 (Expiry >

CMP) 5500 0 -300 5200

Case 3 (Expiry

CMP) -5500 0 +300 -5200

Case 3 (Expiry 1 & d


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Generalisation

Value of the portfolio in case Price moves up to Su, Su- Cu

Value of the portfolio in case Price moves down to Sd,

Sd- Cd

(Delta)= (Cu-Cd)/ (Su-Sd), i.e. Delta is rate or the ratio of change in

option price as a function of change in Spot Price.

Present value of the Portfolio ( I )

(Su- Cu)* e( - rT)

Cost of Setting up Portfolio (II)

S- C


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Generalisation

Equating (I) & (II) we get, (Su- Cu)* e( - rT) = S- C

i.e. C= S- (Su-Cu) )* e( - rT)

Substituting for we get,C = e(-rT) *(p Cu+(1-p)*Cd) (III)

Where P= (e^(rT)- d)/ (u-d) (IV)

Where, u= e^( *sqrt (T ))

d = e^(- *sqrt (T ))

Also, T = Total time T

No. of Stages


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Important Learning's from the expression

Observations

The option pricing formula does not involve the probabilities of the stock prices

moving up or down as per their risky returns.

Rate of return used for all the calculations is risk free rate.

Conclusion

Options are always valued in terms of the price of the underlying stock & not in

absolute terms.

Future up or down price movements are already incorporated in the price of the

option.


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Risk Neutral Valuation

Risk Neutral World

A world where investors are assumed to require no extra return on average for bearing

risk.


The valuation of an option assuming the world is risk neutral. Risk Neutral valuation

gives the correct price for the option in all worlds, not just in risk neutral world.


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Explanation

From expression (III) in previous slide value of the call option is nothing but the

present value of expression,

p Cu+(1-p)*Cd

Where p= probability of up movement

1-p = probability of down movement

Now let us look at expected returns from the stock.

E(S)= p Su+ (1-p) Sd


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i.e. E(S)= p S(u-d)+Sd

Substituting for p, we get

E(S)= S* e^(rT)

The above expression shows on an average return on stock equals risk free rate.


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Real world vs. Risk Neutral world

It is not easy to know the correct discount rate to apply to the expected

payoff in the real world.

A position in Call option is riskier than position in stock.

So the discounting for the option needs to be more than what it is

for underlying.

However. Risk Neutral Valuation solves this problem so that all

securities are discounted at Risk free rate.


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Two Step Binomial Trees

Here We extent Binomial Tree to 2 steps

Objective is to calculate option price at initial node.

This can be done by repeatedly doing exercise we did in one step binomial tree.

More the number of iterations we do we get an answer close to the answer we get

using Black- Scholes Model.


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Section 3:BlackScholes Model

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Behaviour of Stock Prices

Any variable whose value changes over time in an uncertain way is said to follow a

Stochastic Process.

Stochastic Process can be Continuous Time or Discrete Time.

Discrete time- Variable can change only at fixed points of time.

Continuous timeVariable can change at any time

Stochastic Process can be Continuous Variable or Discrete Variable.

Discrete Variable- Variable can only take value amongst fixed set of values.

Continuous VariableVariable can take any value.


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Stock Prices follow Continuous Time & Continuous Variable process.

In realty, stock prices are restricted to discrete values (i.e. multiple of 5 paisa)

Prices change only when the exchanges are open.


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Markov Process

A Markov process is a particular type of stochastic process where only

the present value is relevant for predicting the future.

The Markov Process implies that the probability distribution of the price at

any particular future time is not dependent on the particular path

followed by the price in the past.


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Weak Form of Market (WFME) Efficiency

As per WFME the present price of stock contains all the information carried in

the record of the past prices.

As per WFME , Technical analyst can not make above average returns

just by observing historical price charts because there is competition in the

market place & there are many investors & traders who observe the

price movements closely.

So whatever price movement that has to happen will happen immediately.


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Normal Distribution

The standard Bell-Shaped distribution of Statistics.

It can be represented by (Mean, Standard Deviation) i.e. (,)

Where = Mean, = Standard Deviation about the Mean

Key Property of the Normal distribution,

+/- 1 Covers 68.3% Of The Curve

+/- 2 Covers 95.4% Of The Curve

+/- 3 Covers 99.7% of the Curve

The function N(x) is the cumulative probability distribution function for

standardized normal distribution. i.e. for =0, =1


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Lognormal Distribution

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A Variable has a Log-normal distribution when the logarithm of the variable

has a normal distribution.

A variable that has a lognormal distribution can take any value from 0to .

Even though stock returns can be negative Stock prices can never be negative

hence stock prices are said to follow a lognormal distribution.


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Black-Scholes Pricing Formulas

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C= S * N(d1) K e^(-rT) * N (d2)

P= K e^(-rT) * N (-d2) S * N(-d1)

Where,

d1 =[ In(S/K) + (r + 2/2)*T] / [* Sqrt(T)]

d2 =[ In(S/K) + (r - 2/2)*T] / [* Sqrt(T)] = d1- * Sqrt(T)


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Variables in the Formula

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C= European Call option Price

P= European Put option Price

S= Spot Price at time t=0

K= Strike Price

r= Continuously compounded risk-free rate

= Stock Price Volatility

T= Time to Maturity of the Option


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Significance of N(d1) & N(d2)

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Let us look at the B&S formula

C= S * N(d1) K e^(-rT) * N (d2)

In a certain world investors knows that a call will finish out of money hence the value of call

will be 0.

If investors knew that a call would finish in the money,the value of the call at expiration

would be = St- K

Its value today would be =( St-K)exp(-rT)

Co= So- K exp(-RT)

Compare this with the B& S formula.

You just need to multiply this by N(d1) & N(d2) and get the B&S model. The term

N (d1)= Hedge Ratio or Delta ()

N (d2)= Probability that option would end up in the money


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Section 4:Implied Volatilities

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Implied Volatility

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The only variable that can not be directly observed in the Black- Scholes Model isthe volatility of the stock price ().

So, is the volatility which is implied by the option price hence it is known as

Implied Volatility.

It is not possible to invert the Black-Scholes Model so as to calculate as a function

of other variables.

Traders typically take observations for 20-24 previous trading days estimate

historical volatility of the daily returns from the underlying.

Typical volatilities of the stocks are in the range of 20% to 40% per annum


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Volatility Smile

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The variation of Implied Volatility as a function of Strike price is known as Volatility

Smile.

Volatility Smile is also referred as Volatility Skew in case of Equity Options.

The volatility decreases as the Strike Price Increases.

The volatility for low strike price options is significantly higher as compared to high

strike price options.

i.e. Deeply OTM puts or Deeply ITM calls will have more as compared to Deeply

ITM puts or Deeply OTM calls.


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Explanation for Volatility Smile

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The only logical explanation given for the smile is leverage.

At Lower Strike Prices companysequity declines in value, the companys leverage

increases & hence the Risk & consequently the volatility

It is natural for people to assume that after one crash there will be another crash. It

gets reflected in option prices.


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Option Valuation Model