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Question 1 - BINARY OPTIONS Binary options are also known ans cash or nothing call or digital options. This option has the following payoff payoff = { A if ST > K 0 if ST < K Here A is some amount, K is the strike/exercise price, T is the time of maturity. Assume that we have a Black-Scholes model, that is, a non-dividend paying stock if St satisfies the stochastic differential equation (SDE) dSt = ?Stdt + ?StdBt for constants ?, ? and B is a ?-Brownian motion. Here ? is the physical probability measure and we know that the solution to the above SDE is St = S0e^(?-1/2?²)t+?Bt where S0 is the stock price at time t = 0. 1. Show that ??[ST] = S0e??, where ? is the expected growth rate for S. 2. Show that under the risk-neutral probability measure ?, then ??[ST] = S0e??, where r is the risk free interest rate. 3. We can rewrite the payoff of the binary option as A?{ST>K}, where ?{ST>K} = 1 if ST > K and zero otherwise. Determine the price of the binary option ie. e?????????[A?{ST>K}] 4. Suppose S is lognormal process under some probability measure ? (it could be the risk-neutral measure, forward measure, etc). Let ?? = sd(ln(ST/S0)), where sd means the standard deviation. Show that ??[ST?{ST>K}] = ??[ST]N(d), where d = (ln(??[ST]/K) + 1/2??²) / ?? Here N represents the standard normal cumulative distribution function.

          Question 1 - BINARY OPTIONS Binary options are also known ans cash or nothing call or digital options. This option has the following payoff
payoff = { A if ST > K
           0 if ST < K
Here A is some amount, K is the strike/exercise price, T is the time of maturity.
Assume that we have a Black-Scholes model, that is, a non-dividend paying stock if St satisfies the stochastic differential equation (SDE)
dSt = ?Stdt + ?StdBt
for constants ?, ? and B is a ?-Brownian motion. Here ? is the physical probability measure and we know that the solution to the above SDE is
St = S0e^(?-1/2?²)t+?Bt
where S0 is the stock price at time t = 0.
1. Show that ??[ST] = S0e??, where ? is the expected growth rate for S.
2. Show that under the risk-neutral probability measure ?, then ??[ST] = S0e??, where r is the risk free interest rate.
3. We can rewrite the payoff of the binary option as A?{ST>K}, where ?{ST>K} = 1 if ST > K and zero otherwise.
Determine the price of the binary option ie. e?????????[A?{ST>K}]
4. Suppose S is lognormal process under some probability measure ? (it could be the risk-neutral measure, forward measure, etc). Let ?? = sd(ln(ST/S0)), where sd means the standard deviation. Show that ??[ST?{ST>K}] = ??[ST]N(d*), where
d* = (ln(??[ST]/K) + 1/2??²) / ??
Here N represents the standard normal cumulative distribution function.

Question 1 - BINARY OPTIONS Binary options are also known ans cash or nothing call or digital options. This option has the following payoff
payoff = A if ST > K
0 if ST < K
Here A is some amount, K is the strike/exercise price, T is the time of maturity.
Assume that we have a Black-Scholes model, that is, a non-dividend paying stock if St satisfies the stochastic differential equation (SDE)
dSt = ?Stdt + ?StdBt
for constants ?, ? and B is a ?-Brownian motion. Here ? is the physical probability measure and we know that the solution to the above SDE is
St = S0e^(?-1/2?²)t+?Bt
where S0 is the stock price at time t = 0.
1. Show that ??[ST] = S0e??, where ? is the expected growth rate for S.
2. Show that under the risk-neutral probability measure ?, then ??[ST] = S0e??, where r is the risk free interest rate.
3. We can rewrite the payoff of the binary option as A?ST>K, where ?ST>K = 1 if ST > K and zero otherwise.
Determine the price of the binary option ie. e?????????[A?ST>K]
4. Suppose S is lognormal process under some probability measure ? (it could be the risk-neutral measure, forward measure, etc). Let ?? = sd(ln(ST/S0)), where sd means the standard deviation. Show that ??[ST?ST>K] = ??[ST]N(d*), where
d* = (ln(??[ST]/K) + 1/2??²) / ??
Here N represents the standard normal cumulative distribution function.

Added by Maria M.

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