# How to price an European put option using binomial model with dividend yield?

The initial stock price (S0) is 45, the stock volatility is 0.20 (20% per annum), and the risk-free rate is 0.02 (2% per annum). Consider a European put option whose strike price is equal to 30, with a time-to-maturity of two years. The dividend yield is 0.04 (4% per annum)

Is it right if I draw a binomial tree with ex-dividend model, but add 45 x 0.04 x e^(-0.02 x 2) to the option price?

Since dividends and interest rates mature annually and the time of expiry is two years, we can model the option as a two-step binomial tree, structured as the one in figure:

Data from OP question:

$$S_0 = 45$$ current price of the underlying $$S$$; $$\hspace{2.5cm}$$ $$\sigma = 0.2\;\text{per annum}$$ volatility of $$S$$;

$$r = 0.02\;\text{per annum}$$ interest rate; $$\hspace{3.5cm}$$ $$d = 0.04\;\text{per annum}$$ dividend from $$S$$;

$$T = 2\ \text{years}$$ expire date of option on $$S$$; $$\hspace{2.7cm}$$ $$K = 30$$ strike price of the option;

$$P(T) = \max(K - S(T), 0)$$ we are trying to price a put option.

One can solve this in various ways. Let's proceed here with risk neutral valuation, that implies following the steps below:

1. compute how much $$S$$ can shift up or down when moving from step to step
2. compute risk neutral probability of an upmove, $$p_{RN}$$
3. evaluate $$S$$ at each node $$B, C, D, E, F$$
4. obtain value of the option $$P(T)$$ at final possible nodes $$D, E, F$$
5. calculate the current value of the option, $$P(0)$$, as the expected value from $$P(T) \sim \{P_D, P_E, P_F \}$$

1. $$S$$ can move one standard deviation up or down, that is $$S_{i} - S_{i - 1} = \Delta S = \sigma \, S_{i - 1} \qquad i = 1,2$$

2. The risk neutral probability $$p_{RN}$$ of an upmove is obtained considering that a safe investment of the same sum $$S_0$$ at interest rate $$r$$ yields $$S_0 e^{rT}$$ at option expiry $$T$$. In order to exclude arbitrage opportunities, $$S$$ has to grow on average as the safe (risk-free) rate. However, dividends yielded by the stock while held in our portfolio mean that we should consider a lower risk-free sum of $$S_0 e^{(r - d)T}$$ for our risk-neutral probability calculation: $$p_{RN} S_{up} + \left(1 - p_{RN} \right) S_{down} = S_0 e^{(r - d)\frac{T}{2}}$$ where length of time step is $$\Delta t = T / 2$$. So $$p_{RN}$$ is: $$p_{RN} = \frac{S_0 e^{(r-d)\frac{T}{2}} - S_{down}}{S_{up} - S_{down}} = \frac{\require{cancel}\cancel{S_0} e^{(r-d)\frac{T}{2}} - \cancel{S_0} (1 - \sigma)}{2 \sigma \cancel{S_0}} = 0.4505$$

3. To obtain the values of $$S$$ at each node, traverse it from the root $$S_A = S_0$$ adding or subtracting $$\Delta S$$ (that assumes different values for each of the nodes): \begin{aligned} S_A &= 45\\ S_B &= S_A - \Delta S = 36 \hspace{1.35cm} S_C = S_A + \Delta S = 54 \qquad\\ S_D &= S_B - \Delta S = 28.8 \qquad S_E = S_B + \Delta S = S_C - \Delta S = 43.2\\ S_F &= S_C + \Delta S = 64.8 \end{aligned}

4. Quickly evaluate the option payoffs at expiry, $$P_D, P_E, P_F$$: \begin{align} P_D &= \max(K - S_D, 0) = 1.2\\ P_E &= \max(K - S_E, 0) = 0\\ P_F &= \max(K - S_F, 0) = 0 \end{align}

5. The expected value of $$P(0)$$ is the present money value (multiply by $$e^{-(r - d) T}$$) of the expected value of the option at expiry, $$E[P(T)]$$: $$P(0) = e^{-(r - d) T} E[P(T)] = e^{-(r - d) T} \left[ \left( 1 - p_{RN} \right)^2 P_D + 2 \cdot p_{RN} \left( 1 - p_{RN} \right) P_E + p_{RN}^2 P_F \right] = 1.0408 \left[ 0 + 0 + 0.302 \cdot 1.2 \right] = 0.3771$$

Thus the answer to the question is $$P(0) = 0.3771$$.