Two-Period Binomial Pricing and Delta Hedging for European Options

# Two step binomial model to price European option


# Two step binomial model to price European option












Given the dynamics of stock prices described by a two-period binomial model, the stock today, at time t=0, has a price of €30. Assuming that the stock does not pay dividends, its volatility is 25%, and the risk-free rate is 2%, determine:

a) the price of a plain vanilla European call option with a strike price of €32 and a maturity of 1 year;

b) the price of a plain vanilla European put option with a strike price of €32 and a maturity of 1 year.

c) the delta hedging strategy in case of the put and call option

My attempt is the following:

Each period is given by $\Delta_t=0.5$ I computed the growth rate($U$) and decrease rate($D$).

$U = exp^{\sigma \sqrt{\Delta_t}}=1,19$

$D= \frac{1}{U} = 0,84$

The probability of going up in the first period ($p_{u})$ is given by: $$p_{u}=\frac{exp^{r_f\Delta_t}-d}{u-d}= 0.484$$ consequently the probability of the stock going down is given by $p_{d}=1-p_{u}=1-0.484=0,516$ Since we are dealing with an European option we can only exercise the option at T=1 and we can have 4 possible scenarios:

- the stock goes up at $\Delta_t=0.5$ and then again at t=1,that is $30*U^2=42.72$

- the stock goes up at $\Delta_t=0.5$ and then down at t=1, that is $30*U*D= 30$

- the stock goes down at $\Delta_t=0.5$ and then up at t=1, that is $30*D*U= 30$

- the stock goes down at $\Delta_t=0.5$ and then again at t=1, that is $30*D^2= 21.07$

In the first case the payoff of the option is equal to $MAX(0;42.72-32)=10.72$,in the second and third case the payoff of the option is equal to $MAX(0;30-32)=0$,in the last case the payoff of the option is equal to $MAX(0;21.07-32)=0$

The value of the call option is $VC_0= exp^{-0.02*1}[p_{u}^2*10.72+2*p_{ud}*0+p_{d}^2*0]= 2.46$

where $p_{ud} = p_u*p_d=0.25$

To double check my result I used in MATLAB the built-in function to price option through the black-scholes formula and I got a really close result ( 2.41)

b) in the case of the put option the procedure is similar: the payoff of the put option in the four possible scenarios is:

- $MAX(0;32-42.72)=0$

- $MAX(0;32-30)=2$

- $MAX(0;32-30)=2$

- $MAX(0;32-21.07)=10.93$

the value of the put option is $VP_{0}= exp^{-0.02*1}[p_{u}^2*0+2*p_{ud}*2+p_d^2*10.93]=3.83$

Using the built-in black-scholes formula in MATLAB I get a similar result(3.78)

Are there any mistakes in my calculations?

If not how can I proceed with point c?

## Answer by welcra (score 0, accepted)

https://quant.stackexchange.com/a/81704

Your calculations for parts a and b look correct. The number of shares to hedge (sell for calls and buy for puts) in a delta hedging strategy is $number \space of \space shares = \Delta \cdot \# \space of \space contracts \cdot 100$. Keep in mind that if you get a negative number of shares, it means to buy that amount, not sell (because put options have negative deltas). The calculation for delta in a binomial model at each node is $\Delta = \frac{C_u-C_d}{S_u=S_d}$, where $C_u$ is the option price at the up node, $C_d$ is the option price at the down mode, $S_u$ is the price of the underlying asset at the up node, and $S_d$ is the price of the underlying asset at the down node. You can use these formulas to calculate the delta hedging strategy at each node.