Binomial Method to Price Options

Assume that the randomness of the movement of the security price follows a binomial movement on each time interval with probability $p$ that the current share price $S_t$ either goes up to $u S_t$ (u>1) or down to $l S_t$ ( l <1) with probabilities $p$ and $1-p$ respectively. If $S_t$ is known, we can then find the first two statistical moment of $S_{t+\Delta t}$ as follows $$\mathrm{E}[S_{t+\Delta t}]=p u S_t+ (1-p) l S_t, ~~~(1)$$ $$\mathrm{E}[S_{t+\Delta t}^2]=p u^2 S_t^2+ (1-p) l^2 S_t^2. ~~~(2)$$ The variance of the random variable $S_{t+\Delta t}$ is therefore $$\text{Var}[S_{t+\Delta t}]=\mathrm{E}[S_{t+\Delta t}^2]-\left(\mathrm{E}[S_{t+\Delta t}]\right)^2 ~= p (1-p) (u-l)^2 S_t^2.$$ The binomial tree is Let the empirical average growth rate of the security price be $\mu$ and the empirical volatility of the security price be $\sigma$, then $$\mathrm{E} \left[\frac{S_{t+\Delta t}-S_{t}}{S_{t}}\right]=\mu \Delta t$$ $$\text{Var} \left[\frac{S_{t+\Delta t}-S_{t}}{S_{t}} \right]=\sigma^2 \Delta t.$$ This gives $$\mathrm{E}[S_{t+\Delta t}]= S_t (1 + \mu \Delta t)~~~(3)$$ $$\text{Var}[S_{t+\Delta t}]= S_t^2 \sigma^2 \Delta t. ~~~(4)$$ By comparing the empirical statistical moments of equations (1) and (2) with the theoretical ones of equations (3) and (4), we get $$\begin{array}{lll} u=& 1 + \mu \Delta t + \sigma \sqrt{\Delta t} \sqrt{\frac{1-p}{p}}&~~~(5) \end{array}$$ $$\begin{array}{lll} l=& 1 + \mu \Delta t - \sigma \sqrt{\Delta t} \sqrt{\frac{p}{1-p}}&~~~(6) \end{array}$$ We then make the following assumptions:

1. The security prices follows the process above.
2. Short selling is permitted.
3. Securities are perfectly divisible.
4. There are no transaction fees and no dividends.
5. There are no risk-free arbitrage opportunities.

The correct price for the option at time $t$ is when $$V_t-\Pi S_t=\text{Present Value of }~(V^+-\Pi u S_t),$$ which gives $$\begin{array}{lll} V_t&=\Pi S_t+e^{-r \Delta t}~(V^+-\Pi u S_t)&~~~(7)\\ &&\\ &=\frac{V^+-V^-}{u-l} +e^{-r \Delta t}~\frac{u V^-- l V^+ }{u-l}&~~~(8), \end{array}$$ where $r$ is the risk-free interest rate. Note that $$e^{r \Delta t}(V_t-\Pi S_t)=V^+-\Pi u S_t=V^--\Pi l S_t$$

• Assume that $V_t$ is less than the terms in equation (7), then at at time $t$
  • We sell short $\Pi$ securities for $\Pi S_t$ and deposit the money in the bank.
  • We buy a put option for $V_t$ borring this sum from the bank.
• If the security price goes up (down) to $u S_t$ ( $l S_t)$ at time $t+\Delta t$.
  • We buy $\Pi$ security for $\Pi uS_t$ ( $\Pi lS_t$) and sell the Put Option for $V^+$ ( $V^-$).
  • We then have a credit of $V^++e^{r\Delta t }\Pi S_t$ ( $V^-+e^{r\Delta t }\Pi S_t$) and a debit of $e^{r \Delta t} V_t+\Pi u S_t$ and can make a riskless profit, which contradict with the assumption that there is no risk-free arbitrage opportunities.

• Assume that $V_t$ is greater than the terms in equation (7), then at at time $t$
  • We borrow the money from the bank to buy $\Pi$ securities for $\Pi S_t$.
  • We issue and sell a put option for $V_t$ and deposit this money in the bank.
• If the security price goes up (down) to $u S_t$ ( $l S_t)$ at time $t+\Delta t$.
  • We sell $\Pi$ security for $\Pi uS_t$ ( $\Pi lS_t$) and buy the Put Option for $V^+$ ( $V^-$).
  • We then have a credit of $e^{r \Delta t} V_t+\Pi u S_t$ and a debit of $V^++e^{r\Delta t }\Pi S_t$ ( $V^-+e^{r\Delta t }\Pi S_t$) and can make a riskless profit, which contradict with the assumption that there is no risk-free arbitrage opportunities.

Therefore $$V_t=\frac{V^+-V^-}{u-l} +e^{-r \Delta t}~\frac{u V^-- l V^+ }{u-l},$$ with $V(T)=\max(K-S(T)-K,0)$. The equivalent equations for an American option will be $$V_t=\max \left( \frac{V^+-V^-}{u-l} +e^{-r \Delta t}~\frac{u V^-- l V^+ }{u-l}, K-S_t \right),$$ with $V(T)=\max(K-S(T),0)$.

Three-step binomial tree

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References

1. Cox, J. C., Ross, S. A., and Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of financial Economics, 7(3), 229-263.
   
2. Tang, Q. Mathematical Models in Industry and Finance, Lecture Notes University of Sussex (2008).