1 / 58

Framework for pricing derivatives

Framework for pricing derivatives. Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html. The Market Price of Risk. Observable underlying process, for example stock, interest rate, price of a commodity, etc. Here dz is a Brownian motion. We assume that m(x, t) and s(x, t).

keran
Télécharger la présentation

Framework for pricing derivatives

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Framework for pricing derivatives Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html J. Hull, Options, Futures and Other Derivatives

  2. The Market Price of Risk Observable underlying process, for example stock, interest rate, price of a commodity, etc. Here dz is a Brownian motion. We assume that m(x, t) and s(x, t). x is not necessarily an investment asset! Hull - 19

  3. The same! Suppose that f1 and f2 are prices of two derivatives dependent only on x and t. For example options. We assume that prior to maturity f1 and f2 do not provide any cashflow. Hull - 19

  4. Form an instantaneously riskless portfolio consisting of 2f2 units of the first derivative and 1f1 units of the second derivative. Hull - 19

  5. An instantaneously riskless portfolio must earn a riskless interest rate. Hull - 19

  6. This implies or  is called the market price of risk of x. Hull - 19

  7. If x is a traded asset we must also have But if x is not a financial asset this is not true. For all financial assets depending on x and time a similar relation must held. Hull - 19

  8. Volatility Note that  can be positive or negative, depending on the correlation with x. |  | is called the volatility of f. If s>0 and f and x are positively correlated, then >0, otherwise it is negative. Hull - 19

  9. Example 19.1 Consider a derivative whose price is positively related to the price of oil. Suppose that it provides an expected annual return of 12%, and has volatility of 20%. Assume that r=8%, then the market price of risk of oil is: Hull - 19

  10. Example 19.2 Consider two securities positively dependent on the 90-day IR. Suppose that the first one has an expected return of 3% and volatility of 20% (annual), and the second has volatility of 30%, assume r=6%. What is the market price of interest rate risk? What is the expected return from the second security? Hull - 19

  11. Example 19.2 The market price of IR risk is: The expected return from the second security: Hull - 19

  12. Differential Equation f is a function of x and t, we can get using Ito’s lemma: Finally leading to: Hull - 19

  13. Differential Equation Comparing to BMS equation we see that it is similar to an asset providing a continuous dividend yield q=r-m+s. Using Feynman-Kac we can say that the expected growth rate is r-q and then discount the expected payoff at the risk-free rate r. Hull - 19

  14. Risk-neutral approach True dynamics Risk-neutral dynamics Hull - 19

  15. Example 19.3 Price of copper is 80 cents/pound. Risk free r=5%. The expected growth rate in the price of copper is 2% and its volatility is 20%. The market price of risk associated with copper is 0.5. Assume that a contract is traded that allows the holder to receive 1,000 pounds of copper at no cost in 6 months. What is the price of the contract? Hull - 19

  16. Example 19.3 m=0.02, =0.5, s=0.2, r=0.05; the risk-neutral expected growth rate is The expected (r-n) payoff from the contract is Discounting for six months at 5% we get Hull - 19

  17. Derivatives dependent on several state variables State variables (risk factors): Traded security Hull - 19

  18. Multidimensional Risk Here i is the market price of risk for xi. This relation is also derived in APT (arbitrage pricing theory), Ross 1976 JET. Hull - 19

  19. Pricing of derivatives To price a derivative in the case of several risk factors we should • change the dynamics of xi to risk neutral • derive the expected (r-n) discounted payoff If r is deterministic Hull - 19

  20. Example 19.5 Consider a security that pays off $100 at time T if stock A is above XAand stock B is above XB. Assume that stocks A and B are uncorrelated. The payoff is $100 QA QB, here QA are QB are r-n probabilities of stocks to be above strikes. Hull - 19

  21. Example 19.5 Hull - 19

  22. Derivatives on Commodities The big problem is to estimate the market price of risk for non investment assets. One can use futures contracts for this. Assume that the commodity price follows (no mean reversion and constant volatility) Hull - 19

  23. Derivatives on Commodities The expected (r-n) future price of a commodity is its future price F(t). Hull - 19

  24. Example 19.6 Futures prices August 99 62.20 Oct 99 60.60 Dec 99 62.70 Feb 00 63.37 Apr 00 64.42 Jun 00 64.40 The expected (r-n) growth rate between Oct and Dec 99 is ln(62.70/60.60)=3.4%, or 20.4% annually. Hull - 19

  25. Convenience Yield y - convenience yield u - storage costs, then then r-n growth rate is m -  s = r - y + u Hull - 19

  26. Martingales and Measures A martingale is a zero drift stochastic process for example: dx = s dz an important property E[xT] = x0, fair game. Hull - 19

  27. Martingales and Measures Real world Risk-neutral world In the risk-neutral world the market price of risk is zero, while in the real world it is Hull - 19

  28. Martingales and Measures By making other assumptions we can define other “worlds” that are internally consistent. In a world with the market price of risk * the drift (expected growth rate) * must be Hull - 19

  29. Equivalent Martingale Measures Suppose that f and g are price processes of two traded securities dependent on a single source of uncertainty. Define x=f/g. This is the relative price of f with respect to g. g is the numeraire. Hull - 19

  30. Equivalent Martingale Measures The equivalent martingale measure result states that when there are no arbitrage opportunities, x is a martingale for some choice of market price of risk. For a given numeraire g the same market price of risk works for all securities f and the market price of risk is equal to the volatility of g. Hull - 19

  31. Equivalent Martingale Measures Using Ito’s lemma Hull - 19

  32. Equivalent Martingale Measures A martingale Hull - 19

  33. Forward risk neutral wrt g Since f/g is a martingale (19.19) in Hull Hull - 19

  34. Money market as a numeraire Money market account dg = rgdt zero volatility, so the market price of risk will be zero and we arrive at the standard r-n world. g0=1 and Hull - 19

  35. Zero-Coupon Bond as a Numeraire Define P(t,T) the price at time t of a zero-coupon bond maturing at T. Denote by ET the appropriate measure. gT = P(T,T)=1, g0 = P(0,T) we get Hull - 19

  36. Zero-Coupon Bond as a Numeraire Define F as the forward price of f for a contract maturing at time T. Then In a world that is forward risk neutral with respect to P(t,T) the forward price is the expected future spot price. Hull - 19

  37. Important Conclusion We can value any security that provides a payoff at time T by calculating its expected payoff in a world that is forward risk neutral with respect to a bond maturing at time T and discounting at the risk-free rate for maturity T. In this world it is correct to assume that the expected value of an asset equals its forward value. Hull - 19

  38. Interest Rates With a Numeraire Define R(t, T1, T2) as the forward interest rate as seen at time t for the period between T1 and T2 expressed with a compounding period T1- T2. The forward price of a zero coupon bond lasting between T1 and T2 is Hull - 19

  39. Interest Rates With a Numeraire A forward interest rate implied for the corresponding period is Hull - 19

  40. Interest Rates With a Numeraire Setting We get that R(t, T1, T2) is a martingale in a world that is forward risk neutral with respect to P(t,T2). Hull - 19

  41. Annuity Factor as a Numeraire Consider a swap starting at time Tn with payment dates Tn+1, Tn+2, …, TN+1. Principal $1. Denote the forward swap rate Sn,N(t). The value of the fixed side of the swap is Hull - 19

  42. Annuity Factor as a Numeraire The value of the floating side is The first term is $1 received at the next payment date and the second term corresponds to the principal payment at the end. The swap rate can be found as Hull - 19

  43. Annuity Factor as a Numeraire We can apply an equivalent martingale measure by setting P(t,Tn)-P(t,TN+1) as f and An,N(t) as g. This leads to For any security f we have Hull - 19

  44. Multiple Risk Factors Hull - 19

  45. Multiple Risk Factors Equivalent world can be defined as Where i* are the market prices of risk Hull - 19

  46. Multiple Risk Factors Define a world that is forward risk neutral with respect to g as a world where i*=g,i. It can be shown from Ito’s lemma, using the fact that dzi are uncorrelated, that the process followed by f/g in this world has zero drift. Hull - 19

  47. An Option to Exchange Assets Consider an option to exchange an asset worth U to an asset worth V. Assume that the correlation between assets is  and they provide no income. Setting g=U, fT=max(VT-UT,0) in 19.19 we get. Hull - 19

  48. An Option to Exchange Assets The volatility of V/U is This is a simple option. Hull - 19

  49. An Option to Exchange Assets Assuming that the assets provide an income at rates qU and qV. Hull - 19

  50. Change of Numeraire Dynamics of asset f with forward risk neutral measure wrt g and h we have When changing numeraire from g to h we update drifts by Hull - 19

More Related