FE520 – MT1

One-Period Binomial Model

If the probability of head is $p$$p$, then the probability of tail is $q=1-p$$q=1-p$.

No arbitrage rule:

Europian Call Option:

which confers on its owner the right but not the obligation to buy one share of the stock at time maturity for the strike price K.

General One Period Binomial Model:

• Europian Call : $payoff=(S_1-K{)}^{+}$$payoff = (S_1 - K)^+$

• Europian Putt: $payoff=(K-S_1{)}^{+}$$payoff = (K -S_1)^+$

• Forward Contract: $payoff=(S_1-K)$$payoff = (S_1 - K)$

Some formulations for single period model:

• $X1 = \Delta_0S_1+(1+r)(X_0 - \Delta_oS_0)= (1+r)X_0 +\Delta_o(S_1 - (1+r)S_0) $

• $S_0=\frac{1}{1+r}[\tilde{p}{S}_1(H)+\tilde{q}{S}_1(T)]$$S_0 = \frac{1}{1+r}[\tilde p S_1(H) + \tilde q S_1(T)]$

• $X_0=\frac{1}{1+r}[\tilde{p}{V}_1(H)+\tilde{q}{V}_1(T)]$$X_0 = \frac{1}{1+r}[\tilde p V_1(H) + \tilde q V_1(T)]$

• $S1(H)=u{S}_0$$S1(H) = uS_0$

• $S1(T)=d{S}_0$$S1(T) = dS_0$

• $\tilde{p}=\frac{1+r-d}{u-d}$$\tilde p = \frac{1+r-d}{u-d}$

• $\tilde{q}=\frac{u-(1+r)}{u-d}$$\tilde q = \frac{u-(1+r)}{u-d}$

• $\tilde{p}+\tilde{q}=1$$\tilde p + \tilde q = 1$

Multi-period Binomial Tree

• $V_2=(S_2-K{)}^{+}$$V_2 = (S_2 - K)^+$

• $X1={\mathrm{\Delta }}_0{S}_1+(1+r)(V_0-{\mathrm{\Delta }}_o{S}_0)$$X1 = \Delta_0S_1+(1+r)(V_0 - \Delta_oS_0)$

• $X1(H)={\mathrm{\Delta }}_0{S}_1(H)+(1+r)(V_0-{\mathrm{\Delta }}_o{S}_0)$$X1(H) = \Delta_0S_1(H)+(1+r)(V_0 - \Delta_oS_0)$

• $X1(T)={\mathrm{\Delta }}_0{S}_1(T)+(1+r)(V_0-{\mathrm{\Delta }}_o{S}_0)$$X1(T) = \Delta_0S_1(T)+(1+r)(V_0 - \Delta_oS_0)$

• $V_2={\mathrm{\Delta }}_1{S}_2+(1+r)(X_1-{\mathrm{\Delta }}_1{S}_1)$$V_2 = \Delta_1S_2+(1+r)(X_1 - \Delta_1S_1)$

• $\tilde{p}{S}_2(HH)+\tilde{q}{S}_2(HT)=(1+r)S_1(H)$$\tilde pS_2(HH) + \tilde qS_2(HT) = (1+r)S_1(H)$

• ${\mathrm{\Delta }}_1(H)=\frac{V_2(HH)-V_2(HT)}{S_2(HH)-S_2(HT)}$$\Delta_1(H) = \frac{V_2(HH) - V_2(HT)}{S_2(HH) - S_2(HT)}$

• ${\mathrm{\Delta }}_1(T)=\frac{V_2(TH)-V_2(TT)}{S_2(TH)-S_2(TT)}$$\Delta_1(T) = \frac{V_2(TH) - V_2(TT)}{S_2(TH) - S_2(TT)}$

• $X_1(H)=\frac{1}{1+r}[\tilde{p}{V}_2(HH)+\tilde{q}{V}_2(HT)]=V_1(H)$$X_1(H) = \frac{1}{1+r}[\tilde p V_2(HH) + \tilde q V_2(HT)] = V_1(H)$

• $X_1(T)=\frac{1}{1+r}[\tilde{p}{V}_2(TH)+\tilde{q}{V}_2(TT)]=V_1(T)$$X_1(T) = \frac{1}{1+r}[\tilde p V_2(TH) + \tilde q V_2(TT)] = V_1(T)$

• $X_{n+1}={\mathrm{\Delta }}_n{S}_{n+1}+(1+r)(X_n-{\mathrm{\Delta }}_n{S}_n)=V_{n+1}$$X_{n+1} = \Delta_n S_{n+1} + (1+r)(X_n - \Delta_n S_n) = V_{n+1}$

Replication in Multi-Period Model

Algorithm

1. Find all payoffs at maturity

2. Calculate $\tilde{p}$$\tilde p$ and $\tilde{q}$$\tilde q$

3. Find all $V_n$$V_n$ and ${\mathrm{\Delta }}_n$$\Delta_n$

Europian Options

American Call

American Put