2 Clearified the nature of BSM(S_t)

I want to know the joint dynamics of a stock and it's option for a finite number of moments between now and $$T$$ the expiration date of the option for a number of possible paths.

Let $$r_{\mathrm{s}}$$ and $$r_{\mathrm{o}}$$ denote the return on the stock and the option. Then I'm interested in knowing $$\mathrm{E}_t([r_{\mathrm{s}}; r_{\mathrm{o}}])$$ the expectation of the return from $$t$$ to $$t+1$$ and $$\mathrm{Var}_t(r_{\mathrm{s}}, r_{\mathrm{o}})$$ the variance from $$t$$ to $$t+1$$. The expected return and volatility of the stock are known.

My first idea is to use Monte Carlo with the following pseudocode:

N <- number of paths
T <- number of moments
M <- number of subpaths
S <- current stock price
for i = 1 to N:
S_0 <- S
for t = 0 to T-1:
for j = 1 to M:
S_{t+1,j} = f(S_t)
O_{t+1,j} = BSM(S_{t,j})
S_{t+1} <- mean(S_{t+1,j})
E_{i,j} <- mean(S_{t+1,j}, O_{t+1,j})
V_{i,j} <- var(S_{t+1,j}, O_{t+1,j})
return (E, V)


where $$S_t$$ is the current stock price, $$f(S_t)$$ gives a realization of stock price at the next moment given the current stock price, let's assume geometric Brownian Motion, $$\textrm{BSM}(S_t)$$ gives the option price given the current stock price and some arbitrary parameters using the BSM formula and E and V the values I'm interested in.

This can probably be optimized by a number of clever ideas such as reusing the draws from the probability distribution and discretizing the state space and use a memoized BSM. This is, however, not what I'm looking for. I rather calculate the mean and variance directly, the question is: how?

I want to know the joint dynamics of a stock and it's option for a finite number of moments between now and $$T$$ the expiration date of the option for a number of possible paths.

Let $$r_{\mathrm{s}}$$ and $$r_{\mathrm{o}}$$ denote the return on the stock and the option. Then I'm interested in knowing $$\mathrm{E}_t([r_{\mathrm{s}}; r_{\mathrm{o}}])$$ the expectation of the return from $$t$$ to $$t+1$$ and $$\mathrm{Var}_t(r_{\mathrm{s}}, r_{\mathrm{o}})$$ the variance from $$t$$ to $$t+1$$. The expected return and volatility of the stock are known.

My first idea is to use Monte Carlo with the following pseudocode:

N <- number of paths
T <- number of moments
M <- number of subpaths
S <- current stock price
for i = 1 to N:
S_0 <- S
for t = 0 to T-1:
for j = 1 to M:
S_{t+1,j} = f(S_t)
O_{t+1,j} = BSM(S_{t,j})
S_{t+1} <- mean(S_{t+1,j})
E_{i,j} <- mean(S_{t+1,j}, O_{t+1,j})
V_{i,j} <- var(S_{t+1,j}, O_{t+1,j})
return (E, V)


where $$S_t$$ is the current stock price, $$f(S_t)$$ gives a realization of stock price at the next moment given the current stock price, let's assume geometric Brownian Motion, $$\textrm{BSM}(S_t)$$ gives the option price given the current stock price and E and V the values I'm interested in.

This can probably be optimized by a number of clever ideas such as reusing the draws from the probability distribution and discretizing the state space and use a memoized BSM. This is, however, not what I'm looking for. I rather calculate the mean and variance directly, the question is: how?

I want to know the joint dynamics of a stock and it's option for a finite number of moments between now and $$T$$ the expiration date of the option for a number of possible paths.

Let $$r_{\mathrm{s}}$$ and $$r_{\mathrm{o}}$$ denote the return on the stock and the option. Then I'm interested in knowing $$\mathrm{E}_t([r_{\mathrm{s}}; r_{\mathrm{o}}])$$ the expectation of the return from $$t$$ to $$t+1$$ and $$\mathrm{Var}_t(r_{\mathrm{s}}, r_{\mathrm{o}})$$ the variance from $$t$$ to $$t+1$$. The expected return and volatility of the stock are known.

My first idea is to use Monte Carlo with the following pseudocode:

N <- number of paths
T <- number of moments
M <- number of subpaths
S <- current stock price
for i = 1 to N:
S_0 <- S
for t = 0 to T-1:
for j = 1 to M:
S_{t+1,j} = f(S_t)
O_{t+1,j} = BSM(S_{t,j})
S_{t+1} <- mean(S_{t+1,j})
E_{i,j} <- mean(S_{t+1,j}, O_{t+1,j})
V_{i,j} <- var(S_{t+1,j}, O_{t+1,j})
return (E, V)


where $$S_t$$ is the current stock price, $$f(S_t)$$ gives a realization of stock price at the next moment given the current stock price, let's assume geometric Brownian Motion, $$\textrm{BSM}(S_t)$$ gives the option price given the current stock price and some arbitrary parameters using the BSM formula and E and V the values I'm interested in.

This can probably be optimized by a number of clever ideas such as reusing the draws from the probability distribution and discretizing the state space and use a memoized BSM. This is, however, not what I'm looking for. I rather calculate the mean and variance directly, the question is: how?

1

# Simulating the joint dynamics of a stock and an option

I want to know the joint dynamics of a stock and it's option for a finite number of moments between now and $$T$$ the expiration date of the option for a number of possible paths.

Let $$r_{\mathrm{s}}$$ and $$r_{\mathrm{o}}$$ denote the return on the stock and the option. Then I'm interested in knowing $$\mathrm{E}_t([r_{\mathrm{s}}; r_{\mathrm{o}}])$$ the expectation of the return from $$t$$ to $$t+1$$ and $$\mathrm{Var}_t(r_{\mathrm{s}}, r_{\mathrm{o}})$$ the variance from $$t$$ to $$t+1$$. The expected return and volatility of the stock are known.

My first idea is to use Monte Carlo with the following pseudocode:

N <- number of paths
T <- number of moments
M <- number of subpaths
S <- current stock price
for i = 1 to N:
S_0 <- S
for t = 0 to T-1:
for j = 1 to M:
S_{t+1,j} = f(S_t)
O_{t+1,j} = BSM(S_{t,j})
S_{t+1} <- mean(S_{t+1,j})
E_{i,j} <- mean(S_{t+1,j}, O_{t+1,j})
V_{i,j} <- var(S_{t+1,j}, O_{t+1,j})
return (E, V)


where $$S_t$$ is the current stock price, $$f(S_t)$$ gives a realization of stock price at the next moment given the current stock price, let's assume geometric Brownian Motion, $$\textrm{BSM}(S_t)$$ gives the option price given the current stock price and E and V the values I'm interested in.

This can probably be optimized by a number of clever ideas such as reusing the draws from the probability distribution and discretizing the state space and use a memoized BSM. This is, however, not what I'm looking for. I rather calculate the mean and variance directly, the question is: how?