# INTRODUCTION: also be written as: Sn = S0

INTRODUCTION:A derivative is simply a financialinstrument whose value is dependent on the value of some other underlyingasset. Some of them are options, future contracts, swaps, etc…Options are the type of contract issuedby firm, bank or any other financial company giving the buyer (owner) the rightto buy/sell something of value on pre-determined rates at a certain period oftime in the future.Our discussionswill be built around options .

Two main financial models are used for optionpricing – Binomial model and Black-Scholes model. It was introduced by John Cox, Mark Rubinstein, and Stephen Ross in 1979. The CRR binomial model is similarto Bernoulli scheme in probability theory which helps in calculating optionprices and provide hedging strategies.               LITERATURE REVIEW:Binomial model works on discrete time and Black-Scholes model oncontinuous time market.

Black-Scholesmodel defines the share price by using a stochastic equation: dSt = ?St dBt + µStdt , which can be replaced by binomial model (more flexible by allowing us to change the input values tosee how it affects the option price) which breaks down the time intomany intervals and calculate the share price at each level by assuming , iteither increased by an amount ‘u’ or decreased by an amount ‘d’. It produces abinomial distribution of the stock prices and also provide a visualrepresentation of the model in the form of a binomial tree to show the possiblepaths taken by a share during the course of the option period. The option priceis calculated at each level by working back from date to maturity to time 0. Let value of stock S= (Sn)for n ? 0 be Sn = (1+ ?n) Sn-1 or equivalently, ?Sn = ?n Sn-1where the market interest on thisstock is ?n for n? 1 is the bernoulli sequence of iid random variables taking any of the twovalue x or y, (x < y) , then?n =  +(  ) ?n where ?n= {-1,+1}Sn is given by S01+ ?t) which can also be written as:Sn = S0 exp(h1+h2 +…..+ hn ) where hn = ln(1+ ?t )Let x = ?-1-1 , y = ?-1Then Sn = ? Sn-1when ?n= y and Sn = ?-1Sn-1 when ?n = xHence we can write Sn as S0 ?(?1+ ?2+….

. ?n)where ht = ?tln(?)This random sequence of Sn is called the geometricrandom walk.To obtain adiscrete-time model of the evolution of the interest rates ri(n)ranging in the sets { ri(n)(j), j = 0,±1,….±i },let r0(n) = r0 and assume that the state ri(n)(j),j = 0,±1,….±i  changes either to ri+1(n)(j+1)or to state ri+1(n)(j-1) , each with probability ½ .Hence, thebinomial model of the evolution of the interest rates can be depicted asfollows,                                       r2(n)(-2)                                                                  r1(n)(-1)       ro                                                     r2(n)(0)                                      r1(n)(+1)                                                                                                                                                                                                                                                                                r2(n)(+2)                                                         INTEREST RATE MODELS:Short-rateinterests (r(t)) is the instantaneous risk-free rate of interest at time t .

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Short-ratemodels such as the Vasicek model usean Ito process to model the short-rate.It shows how the interest rates varyover time interval T-t , where ‘t’ is the current time and T is the time tomaturity.This model is a time-homogeneous model, where future dynamics(movement) of the short-rate will depend only on the current value of r(t) andnot on the time’t’.Ito process (Xt) isdefined using an Ito differential as follows:dXt = ?t dBt + µtdtwhere ?t isthe volatility and µt is the drift.{ Bt ,t ? 0 } is the standard Brownian motion , which is continuous random processthat has homogeneous independent increments where,B0 =0 , E(Bt) = 0 and E(Bt2) = t.

The Vasicek model :d r(t) =?(µ- r(t))dt + ? dBtwhere the driftcoefficient is ?(µ- r(t)) which depend on the current rate of interest. Since ?is always said to take positive values, the drift will direct all the movementtowards µ. That means, thisprocess is mean reverting towards the µ which is the constant mean value.? is thevolatility of the movements in the short rate.

Since this model is basedon the brownian motion, the short rate movements are said to follow normaldistribution.For example,If we simulatethe short rate movements using the Vasicek model , we get the following graph:( using values of parameter:  volatility? = 0.02, mean µ = 0.

06 and ? = 0.1)     The interestrates can sometimes be negative which is a main limitation of this modelbecause one of the desirable charateristics of modelling interest rates arethat it should be positive, however it is not that much significant becauseprobability of interest rates being negative are comparatively very small. The yields fromthe graph are fitted by the Vasicek model which gives volatility ? = 0.037,mean µ = 0.083 and ? = 0.131 , and the equation becomes:d r(t) = -0.131(r(t)-0.083)dt+ 0.037dBtand this processis mean-reverting towards 0.083The short rater(t) can be derived using the Ito process mentioned above:d r(t) = -?(r(t))-µ)dt+ ? dBtCe-?t  is the general solution of d r(t) =-?(r(t))-µ)dt Hence, r(t) = e-?t  U(t) U(t) = r(t)e?t  dU(t) =d(r(t)e?t) = e?td(r(t)) + r(t)d(e?t)Expanding willgive:dU(t) = ?µe?tdt + ?e?t dBtIntegral  form of Ito process is given by Xt= X0 + s dBs +s dsApplying thisform to the above equation, gives us U(t) = U(0)+ µe?s ds + e?s dBsSubstituting ourinitial condition , r(t) = e-?t  U(t) ,

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