INDUSTRIAL FORECASTING E4307
FINANCIAL MODELS
Homework 7, due Thursday Oct. 25
Use of Excel spreadsheet or MATHLAB is recommended.
Knock-in and Knock-out call options:
Current price of a non-dividend paying stock is S(0) = 50. Stock’s annual volatility is σ = 30% and the continuously compounded interest rate is 5%. Consider a European call with strike of 51 that matures in one year. Let Δt = 1/250 (there are approximately 250 trading days in a year) and assume incremental daily price changes have the following process
ΔS/S = ( 0.05 – 0.5 σ2 ) Δt + (σ √Δt ) dW(t) in other words at each increment of time
[S(t+ Δt) – S(t)]/S(t) has a normal distribution with mean = ( 0.05 – 0.5 σ2 ) Δt and Variance = σ2 Δt
a)- The price of the call is PV E[Max (S(1) – 51 , 0)], where PV is based on continuously compounded interest rate of 5%. Use a 1000 realization of the process to estimate the price of the call
b)- Knock-in Call: Suppose we have this call if at least one day during the year the stock price is at or below 48. Price this “Knock-in” call.
c )- Knock-out Call: Suppose we have this call provided during the year the stock price never falls below 48. Price this “Knock-out” call.
d)- Show that the price of the call is approximately the sum of the prices of part b and c.
e)- Prove that in general the price of a call is the sum of a knock-in and a knock-out call with the same trigger level (in this case 48).
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