The most important principle of the risk-neutral valuation which underpins its application is the notion that the value of an option being created is completely independent of the expected rate of return of the underlying asset (Hoadley, 2017). In the application of this method, the fair value of the option on the underlying asset is affected by the volatility of the asset values and the risk-free rate. That is to say, the price of an option is independent of the risk preferences of investors (Goddard, 2015; Mun, 2006; Copeland
& Antikarov, 2003). This implies that a risky asset could be valued with the assumption that the return from
their underlying assets is the risk-free rate.
The risk-neutral valuation utilises three equations to obtain three parameters of the binomial model assuming that the model behaviour will be similar to those of the risky asset being valued over a short period. The following are the three equations:
• Matching return equation: this formula calculates the value of the expected return of the binomial model over a short period of time Δt, and it matches the calculated value to the expected return in a risk-neutral state.
𝑝𝑢 + (1 − 𝑝)𝑑 = 𝑒𝑟∆𝑡 2.4
Where 𝑝 is the probability of moving up, 𝑢 is the upside percentage and 𝑑 is the down side percentage. • Matching variance equation: this formula enables the variance of the expected return of the
binomial model to match the variance of the expected return in a risk-neutral state.
𝑝𝑢2+ (1 − 𝑝)𝑑2− (𝑒𝑟∆𝑡)2= 𝜎2∆𝑡 2.5 2.8.2.1 Cox-Ross-Rubinstein equation
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𝑢 = 1/𝑑 2.6
The reorganisation of the above three equations produces the following risk-neutral equations and provides a solution for the three parameters which are 𝑝, 𝑢 and 𝑑.
𝑝 =𝑒 𝑟∆𝑡− 𝑢 𝑢 − 𝑑 2.7 𝑢 = 𝑒𝜎√∆𝑡 2.8 𝑑 = 𝑒−𝜎√∆𝑡 2.9
The solution of these parameters is the intention of the first two equations which are to enable the expected value of the binomial model to match the mean and variance of an asset in a risk-neutral state.
2.8.2.2 Cox-Ross-Rubinstein (CRR) with drift
The CRR model with drift is a modified version of the standard CRR model with an arbitrary drift η, applied to parameters 𝑢 and 𝑑 to generate a binomial model that can hold its assumptions (Cox et al., 1979). The CRR equations with drift area:
𝑢 = 𝑒ƞ∆𝑡+𝜎√∆𝑡 2.10
𝑑 = 𝑒ƞ∆𝑡−𝜎√∆𝑡 2.11
Any increase in drift term 𝜂 moves the prices on lattice further away from the current asset value 𝑆0 and if the drift is decreased until its arbitrary value reaches zero, the model with drift term collapses to the original Cox-Ross-Rubinstein binomial model which generates a lattice of prices that is centred around the current asset price 𝑆0 (Goddard, 2015; Mun, 2006).
The Jarrow-Rudd Risk Neutral model which will be discussed in the next section is a specific case of the CRR with drift model. The drift for this model is:
𝜂 =(ln (𝑋) − ln (𝑆0))
𝑇
2.12 Where 𝑙𝑛() is the natural logarithm and 𝑇 is the time to expiry in years.
This equation balances out the tree to attain a symmetric growth as it moves up, moves down and converges in the centre. However, it should be noted that the tree works only for a specific individual strike price and any change in the strike price usually requires a regeneration of the tree.
2.8.2.3 Equal-probability model (Jarrow – Rudd model)
This model was proposed by Jarrow & Rudd (1983),and it is commonly referred to as the Jarrow-Rudd model. The central concept of this model is to provide a mathematical equation that will allow the expected
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mean of returns and variance of the binomial model (Goddard, 2015) to match those expected in a risk- neutral world over a short period. Since there are three variables in the binomial model (p, u and d), three equations are needed to calculate unique values for them. Two of these equations are the risk-neutral equations, and the third equation proposed by Jarrow and Rudd is:
𝑝 = 1/2 2.13
Eq. (2.13)implies that there are two possible outcomes with equal probabilities of the asset price rising or falling. Eqs. (2.14) and (2.15) are used to calculate the parameters for the Jarrow-Rudd binomial model. These parameters are 𝑝, 𝑢 and 𝑑 which are calculated similarly to those of the standard binomial price tree and uses it for pricing options. The Jarrow-Rudd model is not risk-neutral. The three equations of the Jarrow-Rudd model are:
𝑢 = 𝑒(𝑟−𝜎 2 2 )∆𝑡+𝜎√∆𝑡 2.14 𝑑 = 𝑒(𝑟−𝜎 2 2 )∆𝑡−𝜎√∆𝑡 2.15
To address the main limitations of the Jarrow-Rudd model, which is not risk-neutral, a small modification is applied and this results in what is referred to as the Jarrow-Rudd risk - neutral model. This model is a modified standard Jarrow-Rudd model. Apart from the substitution of the probability equation where the risk-neutral value for p is fixed, the two models are very similar in most aspects. Therefore, the resulting equation is similar to Eq. (2.7).As stated in the previous section, the Jarrow-Rudd risk-neutral model is a special case of the Cox-Ross-Rubinstein with drift model.
2.8.2.4 Tian’s model
The advantage of the model proposed by Tian is its ability to match the expected mean and variance of the binomial model to values of the risk-neutral model (Tian, 1993, 1999).The mean and variance are referred to as the first moments of a lognormal distribution. Eqs. (2.16), (2.17) and (2.18)represent Tian’s model.
𝑝𝑢 + (1 − 𝑝) = 𝑒𝑟∆𝑡 2.16
𝑝𝑢2+ (1 − 𝑝)𝑑2 = (𝑒𝑟∆𝑡)2𝑒𝜎2∆𝑡 2.17
𝑝𝑢3+ (1 − 𝑝)𝑑3 = (𝑒𝑟∆𝑡)3(𝑒𝜎2∆𝑡)3 2.18
Similar to a standard binomial model, the calculation of variables p, u and d does not change.
𝑢 = 0.5𝑒𝑟∆𝑡𝑣 (𝑣 + 1 + √𝑣2+ 2𝑣 − 3) 2.19
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𝑣 = 𝑒𝜎2∆𝑡
2.21 where 𝑣 is the option value.
2.8.2.5 Leisen-Reimer model
Leisen and Reimer developed this model to make the convergence of the binomial tree smoother (Leisen
& Reimer, 1996).As discussed by Goddard (2015), most of the binomial model ‘converge to the Black-
Scholes solution when the function approaches its limit as the size of the time step Δt is reduced to zero’
(Goddard, 2015; Mun, 2006). The Leisen-Reimer binomial tree is generated using the following equations:
𝑝 = ℎ−1(𝑑 1) 2.22 𝑝 = ℎ−1(𝑑 2) 2.23 𝑢 = 𝑒𝑟∆𝑡𝑝̅ 𝑝 2.24 𝑑 =𝑒 𝑟∆𝑡− 𝑝𝑢 1 − 𝑝 2.25
where ℎ − 1() is a discrete approximation to the cumulative distribution function for a normal distribution
(Goddard, 2015)and can be calculated using the method suggested by Leisen and Reimer which is shown
below. ℎ−1(𝑧) = 0.5 + 𝑠𝑔𝑛(𝑧) [ 0.25 − 0.25𝑒 −( 𝑧 𝑛+13+𝑛+10.1 )(𝑛+16) ] 1/2 2.26
Where 𝑑1 and 𝑑2 are the parameters of the Black-Scholes equation and n is the number of time intervals which must be odd within the model that starts from 0 to 𝑇 inclusively. These two parameters, 𝑑1 and 𝑑2 still take their usual definitions from the Black-Scholes formulation. This equation tends to be complicated and is rarely utilised in RO analysis.
Having explored the risk-neutral valuation techniques an their probabilities in Eqs. (2.4) – (2.26), it is important to highlight that those equations formed the basis of the binomial tree constructions that is commonly referred as the binomial model that is discussed in Section 2.8.3.