Now, let's not assume that the dividend rate (q) is 0%. Our second approach for pricing...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Now, let's not assume that the dividend rate (q) is 0%. Our second approach for pricing derivatives was the "martingale method." We proceeded in the following way: Under certain assumptions about the market there's a way to "shift the probabilities" for "outcomes" for the stock-price path. Select a traded asset, called the numeraire (N). We're given the stochastic process for the numeraire. Now, there's a way to adjust the evolution of S so that, under this new process, the price of any traded asset divided by the numeraire's price, V(t)/N(t), is a martingale. That means that the SDE for V(t)/N(t) has no "dt" term. We call this new probability rule the "martingale measure" associated to N, and we denote it QN. [a] Suppose we are in the Black-Scholes framework (as in Problem 1), and we choose as our numeraire the "bank account": dB = r B dt, B(0) = 1. As given, in the "real world" (P-measure) the stock propagates as: dS = a S dt+s SdW (P-world) In the "Q = QB world" S(t)/B(t) must be a martingale. It can be shown that in the Q-world the process for the stock is: ds = (r - q) S dt + s S dw (Q-world) Why is this result very plausible? [b] Show how this result, and the fact that, for any derivative V. V(t)/B(t) must be a martingale (under Q), leads immediately to our famous pricing formula. V(S,t) = exp(-t(T-t))E[Payoff formula (function of S(T))] or, equivalently, V(S.t) = exp(-t(T-t)) EQ[Payoff formula (function of S(T))] Now, let's not assume that the dividend rate (q) is 0%. Our second approach for pricing derivatives was the "martingale method." We proceeded in the following way: Under certain assumptions about the market there's a way to "shift the probabilities" for "outcomes" for the stock-price path. Select a traded asset, called the numeraire (N). We're given the stochastic process for the numeraire. Now, there's a way to adjust the evolution of S so that, under this new process, the price of any traded asset divided by the numeraire's price, V(t)/N(t), is a martingale. That means that the SDE for V(t)/N(t) has no "dt" term. We call this new probability rule the "martingale measure" associated to N, and we denote it QN. [a] Suppose we are in the Black-Scholes framework (as in Problem 1), and we choose as our numeraire the "bank account": dB = r B dt, B(0) = 1. As given, in the "real world" (P-measure) the stock propagates as: dS = a S dt+s SdW (P-world) In the "Q = QB world" S(t)/B(t) must be a martingale. It can be shown that in the Q-world the process for the stock is: ds = (r - q) S dt + s S dw (Q-world) Why is this result very plausible? [b] Show how this result, and the fact that, for any derivative V. V(t)/B(t) must be a martingale (under Q), leads immediately to our famous pricing formula. V(S,t) = exp(-t(T-t))E[Payoff formula (function of S(T))] or, equivalently, V(S.t) = exp(-t(T-t)) EQ[Payoff formula (function of S(T))]
Expert Answer:
Answer rating: 100% (QA)
a ANSWER This result is very plausible because it makes sense that the rate of return of the stock ... View the full answer
Posted Date:
Students also viewed these finance questions
-
"To close a partner s drawing account, an entry must be made that." debits that Partners Current Account and credits that Partners Drawing Account debits that Partners Drawing Account and credits...
-
Assume that the dividend payout ratio will be 75 percent when the rate on long-term government bonds falls to 8 percent. Since investors are becoming more risk averse, the equity risk premium will...
-
Find the quote for Duke Energy. Assume that the dividend is constant. What was the highest dividend yield over the past year? What was the lowest dividend yield over the past year? You?ve collected...
-
In Table 4. 4, Professor Kip Viscusi estimates that the cost per life saved by current government risk-reducing programs ranges from $100,000 for unvented space heaters to $72 billion for a proposed...
-
What is strategy and why is marketing research important to strategy makers?
-
What control or controls would you recommend in a computer processing system to prevent the following situations? a. Working through the main control console, the night- shift computer operator made...
-
What is transfer of training? What role does transfer of training play in e-learning?
-
On December 31, 2016, Jons Company had 1,300,000 shares of $5 par common stock issued and outstanding. At December 31, 2016, stockholders' equity had the amounts listed here. Common Stock...
-
A customer walks into the store, and a sales representative welcomes the customer. The customer arrives at the cash register and places the items on the counter. The cashier asks for the customer s...
-
You have just been hired as a management trainee by Toronto-based Capri Fashions Inc., u nationwide distributor of designer Caps. The company has exclusive distribution of the Caps, and sales have...
-
3) At what quoted-annual interest rate must-$190,000-be invested so that it will grow-to-be- $850,000 in 15 years if interest is compounded monthly? NE H I/Y H NE 3 PV I/Y H H PMT- H 4) You plan to...
-
Explain how one would find the value of a nonconstant growth stock.
-
Think of a past job you have held. List three areas in which you, or some other person in the organization, could have benefited from having information generated by research. What would be the most...
-
Explain what is meant by horizon (terminal) date and horizon (continuing) value.
-
Provide an example of each of the following: Business Source Directories, Articles, Dictionaries and Encyclopedias, Marketing Directories, and Statistics and Reports.
-
What steps are taken to find a stock price using the corporate valuation model?
-
Discuss Jim's story and how perceptions transformed with advocacy eventually leading to Jim's independence. How does this connect with experiences you have had? Or how would this impact your future...
-
A police officer pulls you over and asks to search your vehicle because he suspects you have illegal drugs inside your car. Since he doesn't have reasonable suspicion to search your car, legally he...
-
Show that the relations (14.23) follow from Eqs. (14.21) and (14.22). Data from Eq. 14.21 Data from Eq. 14.22 Data from Eq. 14.23 VR (P) = E+m+o.p VR (0) 2m(E+m)
-
Verify that the Pauli-Dirac representation (14.43) satisfies Eq. (14.27). Data from Eq. 14.27 Data from Eq. 14.43 {,} = + = 2
-
Verify the projection characteristics implied by equations (14.51). Data from Eq.14.51 2 In = (s + I) Da 1- 24(1-y's) = UR
Study smarter with the SolutionInn App