In the following discussion we find the Δ t that yields such a portfolio If P t

In the following discussion we find the δ t that

This preview shows page 7 - 9 out of 9 pages.

). In the following discussion we find the Δ t that yields such a portfolio. If P ( t, S ) is a function such that P t = P ( t, S t ), Itˆo’s formula gives us t = parenleftbigg ∂P ∂t + 1 2 σ 2 S 2 t 2 P ∂S 2 parenrightbigg d t + ∂P ∂S d S t - Δ t d S t - Δ t D 0 S t d t = parenleftbigg ∂P ∂t + 1 2 σ 2 S 2 t 2 P ∂S 2 - Δ t D 0 S t parenrightbigg d t + parenleftbigg ∂P ∂S - Δ t parenrightbigg d S t . 62
Image of page 7

Subscribe to view the full document.

with ∂P ∂t , 2 P ∂S 2 , and ∂P ∂S each evaluated at t and S = S t . Choosing Δ t = ∂P ∂S ( t, S t ) yields t = parenleftbigg ∂P ∂t + 1 2 σ 2 S 2 t 2 P ∂S 2 - Δ t D 0 S t parenrightbigg d t. (43) For the portfolio Π t to be hedged with respect to the underlying S t t is unaffected by changes in S t ), it must provide the risk-free rate of return r in the risk-neutral world. And thus it should satisfy: t = r Π t d t = r ( P t - Δ t )d t = r ( P t - ∂P ∂S S t )d t. (44) The equations (43) and (44) will be equivalent only when the drift terms are equal. Thus, matching the drift terms in those equations and rearranging implies that the function P ( t, S ) satisfies the PDE: ∂P ∂t + 1 2 σ 2 S 2 2 P ∂S 2 + ( r - D 0 ) S ∂P ∂S - rP = 0 , for all t T , and S > 0, with a known boundary condition P ( T, S T ). Thus, in the risk neutral world, the dynamics of S t are given by: d S t = ( r - D 0 ) S t d t + σS t d W Q t , where W Q t a Brownian motion under the risk neutral measure Q . European call option on dividend paying stock Consider a stock S t that moves like a Geometric Brownian Motion (GBM) and pays constant dividend rate D 0 , and suppose C 0 is the price of a European call option on the stock, with maturity T , and strike K . We showed above that GBM associated with dividend-paying stock S t in the risk-neutral world has a drift parameter r - D 0 and volatility σ . Pricing a call option on S t requires evaluating the expectation: C 0 := e - rT E Q { ( S T - K ) + } . Recall that the Black-Scholes formula for a European call option, assuming no dividend payments, is just is given by C BS (0 , S 0 ; K, T, r, σ ) = E Q { e - rT ( S T - K ) + } , where S t is a GBM with drift r and volatility σ in the risk-neutral world. Hence, we can express the price of our call option on a dividend-paying stock as: C 0 := E Q { e - rT ( S T - K ) + } = E Q { e - ( r - D 0 + D 0 ) T ( S T - K ) + } = e - D 0 T E Q { e - ( r - D 0 ) T ( S T - K ) + } = e - D 0 T C BS (0 , S 0 ; K, T, r - D 0 , σ ) . Simply put, we replace r by r - D 0 in the Black-Scholes formula but add a multiplicative term of e - D 0 T to ensure we discount the expectation (in risk-neutral world) the option’s payout at maturity using only the risk-free rate r . 63
Image of page 8
Foreign Exchange Options Every major currency has an associated prevailing interest rate that corresponds to the return on a “risk-free” asset denominated in that currency (in quotes because, in practice, US Treasury securities are the closest to being risk-free). Speculation in foreign exchange markets entails purchasing a foreign currency X t and placing it in that currency’s risk-free asset paying an interest rate r f . Such a scenario is akin to buying a stock with constant dividend rate r f . For example, X t can represent the number of US dollars to buy one British pound, with r and r f being the respective risk-free rates in the US and Britain. Suppose
Image of page 9
  • Fall '11
  • COULON
  • Dividend, Mathematical finance, Black–Scholes

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes