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Pricing an option via binomial one-step method 2 different ways


We will use 2 methods to hopefully get the same result of the price of a call and put option


First of all here are the assumptions and inputs.


S(0) is the Stock Price as of  today   = 100              

K is the Strike Price  = 110    

T is the option expiry  = 6 months = 0.5  year       

r is the Interest rate  = 5% = .05

u is the stock uptick price move  = 50%                  

d is the stock downtick price move = 50%

S(u) is the price if the stock increase = 150            

S(d) is the price if the stock decreases = 50

payoff(u) = MAX(0,S(u) – K) = 40                                              

payoff (d) = MAX(0,K-S(d)) = 0

q is the probability of the stock increasing            


C  is the call option price we want to calculate

P  is the put  option price we want to calculate


NB Often if we are given the drift - μ -, and volatility - σ, of an asset we can estimate the up/down tick and probability as


q =  .5 + (µ*sqrt(δt))/2*σ

​u = 1 + σ * sqrt(δt),    

d = 1 - σ * sqrt(δt)



But we will just use the values as shown.


Pricing a Call option Using Risk Neutral Valuation


This says that the option price is simply the present value of the expected return at option expiry  T.


First of all we have to determine what the probability  q is of the price going up to 150 (that’s the initial stock price + uptick %)


q = S(0)ert – S(d)/(S(u) – S(d)

  = (100*1.0253 - 50)/(150-50) = 0.525315


The probability of the stock going down is ( 1-q)


So the expected return =  

(q * payoff(u) ) + ( ( 1-q) * payoff(d) )

= 0.525315*40 + (0.474685*0) = 21.0126



Option Price C  = The present value of the expected return=>



C= 21.0126 * e-rT = 21.0126* e (-0.05*.5) = 20.4938



Call Pricing Using Portfolio Replication


In portfolio replication  we construct a synthetic (i.e a pretend) portfolio consisting of - for the call side - long a fraction of a share , D, and short 1 option such that this portfolio is worth the same  whether or not the stock increases or decreases. If we can do that then we can say that the price of our synthetic portfolio must be the same price as the option. Let’s represent this fraction of a share by the letter D and our option as B where B is the Present Value ( PV ) of the portfolio value at expiry. So we have

    

       (1) C = D*S(0) – B    = Call Option Value


At expiry the option payoff is worth 40 on the up and 0 on the down, so we can also write the following values for our synthetic portfolio at expiry in the up and down position.

   

       (2)  150D – 40  

       (3)  50D


(2) and (3) must be equal whether the stock rises or falls so


(4)  150D - 40 = 50D    


And solving (4)   we get D = 2/5.


Putting this back into (3) we get the portfolio value at expiry = 20

But we need to  PV it due to there being an interest rate r, so


B = 20*e-rT = 20*0.97531 = 19.5062


Finally, plugging in all our values into equation (1) above we get


C = D* S(0) – B = >  2/5 *100 + (-19.5062) = 20.4938


What about the PUT side of things ?


Ok, let’s look at pricing the put option using the Risk-Neutral Valuation method. The only change to our input/assumptions is  that our expected payoffs now become:-



payoff(u) = MAX(0,K- S(u)) = 0                                  

payoff (d) = MAX(0,K-S(d)) = 60

So our new expected return =


(q * payoff(u) ) + (  (1-q) * payoff(d) ) = 0.525315*0 + (0.474685*60) = 28.4811


Option Price P  = The present value of the expected return=>


P = 28.4811 * e-rT = 28.4811* e (-0.05*.5) = 27.77789


And now via portfolio replication


For the PUT side of things we consider our synthetic portfolio as being short a fraction of a share and long our option payoff i.e


(5) P = B -D*S(0)     = PUT Option Value


Where D is our fraction of the asset and B is the PV of our option payoff at expiry


We take into account our different payoffs and our option values on expiry are


(6)   150D

(7)   50D + 60


Again these must be equal so solving for the above gives


D = 3/5


And putting this back into (6) or (7) gives an option payoff value on expiry as 90. Again we must PV this back to option start and plug back into equation (5) and we get


PV(90) = 90 *e-rT = 87.77789 and putting this into (5) gives us


P =  87.77 - 3/5*100 = 27.77789


We can also check our results by using the Put-Call parity rule.


Recall, this rule states that the call option price - put option price = stock price – present value of the strike price


i.e         C – P = S - K e-rT


Plugging our values into the Put-Call parity equation gives us:-


20.4938 - 27.77789    = - 7.28409     =>  C - P

​100 - 110 e(-0.05*.5)        = - 7.28409     => S - K e-rT

Phew !!, thank goodness for that