Idea of pricing method: Take historical stock prices, estimate $\hat \mu$ and $\hat \sigma$. Specify strike and 3 years maturity for a plain vanilla call option to get the price $P_0$. Put these parameters in a Geometric Brownian Motion (GBM) to get $M$ different stock paths. For example M=1000. Maturity is 3 years. Put these price paths into the matrix S.mat
which has M rows and T=250*3=750 columns.
Let's split up time intwo three steps: year 1, 2 and 3 which stops at T=250, T=500 and T=750 respectively.
There are M different paths and each one has their own list of stock prices. For example path $p=1$ has the following list: $(S^1_{t=0}, S^1_{t=1}, ..., S^1_{t=750})$ and it can be found in the first row of S.mat
. Taking the stdev()
of each path -- each row in the matrix -- gives a list of volatilities: $(\hat \sigma^1, \hat \sigma^2, ...., \hat \sigma^M)$.
After 1 year we have a list of stock prices that day: $(S^1_{t=250}, S^2_{t=250}, ..., S^M_{t=250})$. This is the 250th column of S.mat
.
Now we can use black scholes:
- We have a list of current stock prices (the 250th column of
S.mat
).
- We have a list of volatilities (the stdev of each row in
S.mat
).
- So we can get a list of M number of black scholes prices out of these two lists. All we need to do is to specify $K$ and time left (it is 2.25 years left). Each volatility $\hat \sigma^p$ in that list gives an option price at day 250: $P^1_{t=250}, P^2_{t=250}, ... P^p_{t=250}$. The mean of this list is $\bar P_{t=250}$.
The rule can be stated as: if an event U occurs, the owner of the option is forces to sell it for the least of $P_0$ and whatever the price of the option is that day, which in expectation is $\bar P_{t=250}$. This option has some real value and some time value.