Binomial Options Pricing

7/31/2019 Binomial Options Pricing

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BINOMIAL OPTIONSPRICING

ByGroup 6Garvit Agarwal

Gyan Prakash

Karan Gupta

Ravikumar Soni

Sahil Singla


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Options and Futures

Futures contracts are an obligation

Must deliver or offset

Liable for margin calls

Locked into a price

Options on futures contracts are the right to take a position in thefutures market at a given price called the strike price, but beyondthe initial premium, the option holder has no obligation to act on thecontract

Lock-in a price but can still participate in the market if pricesmove favorably

No margin calls

Pay a premium for the option (similar to price insurance)


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Put and Call Options

Put option: the right to sell a futures contract at a givenprice

Call option: the right to buy a futures contract at a given

price Call and put options are separate contracts and not

opposite sides of the same transaction. They are linkedto the Futures

Put OptionBuyer Seller

Call OptionBuyer Seller

Buyer Futures Seller


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What can one do with an optiononce you buy it

Let it expire Lose the premium that was paid

Offset it: If one April Long Call is purchased then canoffset by selling one April Short Call

Exercise it (places in a short position (put) or a long

position (call) in the futures market. The holder then hasthe same obligations as if a futures contract hadoriginally been bought or sold)


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Strike Price Relationship to CurrentFutures Price

Condition Put Option Call Option

SP < futures Out-of-the money In-the money

SP = futures At-the money At-the money

SP > futures In-the money Out-of-the money


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History of Binomial OptionsPricing

The binomial options pricing model was developed by Cox, Ross,and Rubenstein in 1979 and has subsequently been usedthroughout the marketplace to price financial derivatives.

Instead of treating the underlying as if it follows a log normal

distribution like the Black-Scholes model, the binomial pricing modelassumes the stock price follows a simple binomial process brokeninto discrete time steps.

Thus instead of a continuous distribution of stock prices, the stock isconsidered to undergo a particular percentage change, up or down,at each time step.

Pricing is then accomplished using the no-arbitrage assumption andthe creation of a risk-free portfolio that replicates option prices byusing the theoretical combination of stocks and risk-free bonds.


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Assumptions of the BOPM

There are two (and only two) possible prices for the underlyingasset on the next date. The underlying price will either:

Increase by a factor of u% (an uptick) Decrease by a factor of d% (a downtick)

The uncertainty is that we do not know which of the two priceswill be realized.

No dividends.

The one-period interest rate, r, is constant over the life of theoption (r% per period).

Markets are perfect (no commissions, bid-ask spreads, taxes,price pressure, etc.)


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Replicating Portfolio

P= $100

X= $125

T= 1 year

i= 8%

Su= $200

Sd= $50


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Stock CallOption

Su= $200

Sd= $50P= $100

Cu= max($0,

$200-$125)=$75

Cd= max( $0,$50-$125)= $0

C=??


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Step 1: Construct Bankruptcy free portfolio

Stocks minimum value is $50

So, borrow $50 at PV = $50/1.08= $46.3 Therefore, you invest $53.70 of your own money

Step 2: Replicate future Returns

We want net returns from option to equal $0, or $150

Therefore, you have to buy 2 call options Step 3: Align the dollar cost of option and the

portfolio

As dollar outlay of bankruptcy free portfolio is $53.70, this

should be same for two call option contracts

Step 4: Value the Option

Cost of one contract comes out to be C = $53.7/2= $26.85


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Using Hedge Ratio to develop ReplicatingPortfolio

To eliminate the price variation through short sale of an assetexhibiting the same price volatility as asset to be hedged.Perfect hedge creates a riskless position

Hedge ratio indicates number of asset units needed toeliminate price volatility of one call option

In previous example, a perfect hedge can be created byselling one share short at $100 and buying two call options Here no matter which way stock price moves, net value of hedged

portfolio will be $50

Therefore, PV of this strategy is $50/1.08= 46.3

C= 26.85 Hedge Ratio= (Cu-Cd)/(Su-Sd)

Synthetic Call Replication: C= Hedge ratio X [Stock Price PV (borrowing)]


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Risk Neutral Valuation

One step binomial model can also be expressedin terms of probabilities and call prices

The sizes of upward and downward movements

are defined as functions of the volatility U= size of up-move factor= e^(*sqrt t)

D= size of down move factor= 1/U

Risk Neutral probabilities

u = (ert D)/(U-D) d = 1- u ; r= continuously compounded annual risk free rate


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Example

P = $20 = 14%

r = 4%

Dividends = Nil Strike price = $20

Calculate the value of 1-year European call

option


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U = e 0.14*1 = 1.15

D= 1/U= 0.87

u = (e0.04*1 D)/(U-D) = 0.61

d = 1-0.61 = 0.39

Binomial tree for stock

Binomial Tree for Options

$20*1.15= $23

$20*0.87= $17.40$20

$3= max(0, 23-20)

$0= max(0, 17.4-20)Co


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Expected value of option in one year iscalculated as:

Cu X u + Cd Xd($3*0.61) + ($0*0.39) = $1.83

Present Value = Co = $1.83/(e0.04*1)=

$1.76 To calculate value of put option, use put

call parity


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Two Periods

Suppose two price changes are possible duringthe life of the option

At each change point, the stock may go up by

Ru% or down by Rd%


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Two-Period Stock PriceDynamics

For example, suppose that in each of twoperiods, a stocks price may rise by 3.25% orfall by 2.5%

The stock is currently trading at $47At the end of two periods it may be worth as

much as $50.10 or as little as $44.68


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Two-Period Stock PriceDynamics

$47

$48.53

$45.83

$50.10

$47.31

$44.68


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Terminal Call Values

$C0

$Cu

$Cd

Cuu =$5.10

Cud =$2.31

Cdd =$0

At expiration, a call with a strikeprice of $45 will be worth:


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Two Periods

The two-period Binomial model formula for aEuropean call is

Cp2CUU 2p(1 p)CUD (1 p)

2CDD

1 r 2

P=47.1%C=2.28


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The n Period BinomialFormula:

In general, the n-period model is:

.K]Sd)(1u)[(1p)(1pj

n

r)(11C

n

aj

nTjnjjnj

n

Where a in the summation is the minimum number ofup-ticks so that the call finishes in-the-money.


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Conclusion

Pricing by arbitrage is the main theory behind the binomialpricing model.

In order to price an option, the model generates a portfolio ofstocks and risk-free bonds that exactly replicates the payoff ofthe option.

For a stock that moves up or down a given amount in aparticular time step, a linear combination of stocks and risk-free bonds can be put together in a portfolio that uniquelyrepresents this option price.

The uniqueness of this price is guaranteed by the arbitrage

theorem: if two assets or sets of assets have the samepayoffs, they must have the same market price. Thus, we are given a procedure to determine the price of

options by dealing with portfolios that accurately replicatetheir payoffs.

Binomial Options Pricing

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