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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.
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