Black scholes PDE
Use the black scholes methodology to find an explicit formula for the fair price at time t of a contingent claim of which payoff at maturity T is sqrt(St), where St denotes the price of underlying at Time T
Homework Answers
Formula for a regular call option = c= S(t) N(d1) - e-rT K N(d2)
N(d1) & N(d2) are the standard normal cumulative distributive functions wherein N (d1) is the probability linked to the stock price and N(d2) is the probability that the call will end in money.
Black scholes Merton Model of a regular call option can viewed as Leveraged stock investment wherein
We buy N(d1) units of the stock with borrrowed funds e-rT K N(d2).
The pay off therefore is the value of the underlying less exercise price and the premium paid for the option.
(a)
Stock or nothing option can be better understood with understanding the payoff received on this option.
Here the payoff is the underlying price once the exercise price exceeds the underlying price by paying a premium for the option. If the underlying price does not exceed the exercise price then payoff is zero. So, any given point time it is the underlying price or zero.
So the fair price of such an option is simply S(t) N(d1) if the underlying price exceeds the exercise price else zero.
The only payoff is the stock price.
5. Use the Black-Scholes methodology to find, by direct calculation, an explicit formula for the fair...
5. Use the Black-Scholes methodology to find, by direct calculation, an explicit formula for the fair price (at time t) of the following contingent claims (European type options). The price of the underlying (stock) at time t is denoted by S(0); the time of maturity by T; the risk-free interest rate by r; the volatility of the underlying by o (a) The stock or nothing call option: This is a claim that will pay exactly the price of the underlying...
8. (10 points) Recall the Black-Scholes PDE and Euro call payoff C(S, T) = =0, S<1. if 0 C(S, t) satisfies a) By differentiating with respect to S, show that u b) The v above is a Greek. Which...
8. (10 points) Recall the Black-Scholes PDE and Euro call payoff C(S, T) = =0, S<1. if 0 C(S, t) satisfies a) By differentiating with respect to S, show that u b) The v above is a Greek. Which Greek is it, and does it satisfy the Black-Scholes PDE c) Find the formula for u (s,t) by differentiating the Black-Scholes formula dx, Ceuro(S,t) = SN(d.)-Ke-r(T-t) N(da) where N(z)= 0e Assume without proof thatsw(h)-Ke-r(T-t)N,(d) = 0.
8. (10 points) Recall the...
Question 1 Consider the derivation of the Black-Scholes model of option pricing. Let S=S(t) be the...
Question 1 Consider the derivation of the Black-Scholes model of option pricing. Let S=S(t) be the underlying stock price at time t and let f=f(S, t) be the option price at time t. a) Write down the value P of the portfolio defined in the Black-Scholes model. [2 marks] b) Use Itô’s lemma to find an expression for the change Δf in the discrete time Δt. [5 marks] c) Use the expression you have found in point b) to find...
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Use the Black-Scholes formula to price a call option for a stock whose share price today is $16 when the interest rate is 4%, the maturity date is 6 month, the strike price is $17.5 and the volatility is 20%. Find the price of the same option half way to maturity if the share price at that time is $17.
In this question we assume the Black-Scholes model. We denote interest rate by r, drift rate...
In this question we assume the Black-Scholes model. We denote interest rate by r, drift rate pi and volatility by o. A European power put option is an option with the payoff function below, Ka – rº, ha if x <K, 0, if x > K, for some a > 0. In particular, it will be a standard European put option when a = 1. (a) Derive the pricing formula for the time t, 0 <t< T, price of a...
Traders in major financial institutions use the Black-Scholes formula in a backward fashion to infer other...
Traders in major financial institutions use the Black-Scholes
formula in a backward fashion to infer other traders' estimates of
from option prices. In fact, traders frequently quote sigmas to
each other, rather than prices, to arrange trades. Suppose a call
option on a stock that pays no dividenf for 6 months has a strike
price of $35, a premium of $2.15, and time to myurity of 7 weeks.
The current short-term T-bill rate is 7%, and the price of the...
The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price...
The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price at time t = 0. At time t = 1,2,-. ., the stock price is S,ett+σ Σ. 2. the drift where a 0 is known as the volatility and the independently and identically distributed standard Normal N(0,1) random 0 is known as Zi variables are (a) Show that S, = S¢_1e#+oZ¢ _ St St-1 (b) What is the distribution of ln (c) What is...
3.5 In the Black-Scholes option pricing model, value of an option decreases, all else equal, as...
3.5 In the Black-Scholes option pricing model, value of an option decreases, all else equal, as it nears expiration. (True / False) 3.6 The Black-Scholes option pricing model assumes which of the following? a. Jumps in the underlying price b. Constant volatility of the underlying c. Possibility of negative underlying price d. Interest rate increasing as option nears expiration 3.7 Which Greek shows how sensitive option delta is to the price of the underlying asset? a. Vega b. Gamma c....
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12. The black-scholes OPM is dependent on which five parameters? a. stock price, exercise price, risk...
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5. Use the Black-Scholes methodology to find, by direct calculation, an explicit formula for the fair price (at time t) of the following contingent claims (European type options). The price of the underlying (stock) at time t is denoted by S(0); the time of maturity by T; the risk-free interest rate by r; the volatility of the underlying by o (a) The stock or nothing call option: This is a claim that will pay exactly the price of the underlying...
8. (10 points) Recall the Black-Scholes PDE and Euro call payoff C(S, T) = =0, S<1. if 0 C(S, t) satisfies a) By differentiating with respect to S, show that u b) The v above is a Greek. Which Greek is it, and does it satisfy the Black-Scholes PDE c) Find the formula for u (s,t) by differentiating the Black-Scholes formula dx, Ceuro(S,t) = SN(d.)-Ke-r(T-t) N(da) where N(z)= 0e Assume without proof thatsw(h)-Ke-r(T-t)N,(d) = 0.
8. (10 points) Recall the...
In this question we assume the Black-Scholes model. We denote interest rate by r, drift rate pi and volatility by o. A European power put option is an option with the payoff function below, Ka – rº, ha if x <K, 0, if x > K, for some a > 0. In particular, it will be a standard European put option when a = 1. (a) Derive the pricing formula for the time t, 0 <t< T, price of a...
Traders in major financial institutions use the Black-Scholes
formula in a backward fashion to infer other traders' estimates of
from option prices. In fact, traders frequently quote sigmas to
each other, rather than prices, to arrange trades. Suppose a call
option on a stock that pays no dividenf for 6 months has a strike
price of $35, a premium of $2.15, and time to myurity of 7 weeks.
The current short-term T-bill rate is 7%, and the price of the...
The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price at time t = 0. At time t = 1,2,-. ., the stock price is S,ett+σ Σ. 2. the drift where a 0 is known as the volatility and the independently and identically distributed standard Normal N(0,1) random 0 is known as Zi variables are (a) Show that S, = S¢_1e#+oZ¢ _ St St-1 (b) What is the distribution of ln (c) What is...
3.5 In the Black-Scholes option pricing model, value of an option decreases, all else equal, as it nears expiration. (True / False) 3.6 The Black-Scholes option pricing model assumes which of the following? a. Jumps in the underlying price b. Constant volatility of the underlying c. Possibility of negative underlying price d. Interest rate increasing as option nears expiration 3.7 Which Greek shows how sensitive option delta is to the price of the underlying asset? a. Vega b. Gamma c....
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