A lot of research has been devoted to answering the question: do options price in the volatility risks correctly? The most noteworthy phenomenon (or bias) is called the volatility risk premium, i.e. options implied volatilities tend to overestimate future realized volatilities. Much less attention is paid, however, to the underlying asset dynamics, i.e. to answering the question: do options price in the asset dynamics correctly? Note that within the usual BSM framework, the underlying asset is assumed to follow a GBM process. So to answer the above question, it’d be useful to use a different process to model the asset price. We found an interesting article on this subject [1]. Instead of using GBM, the authors used a process where the asset returns are autocorrelated and then developed a closedform formula to price the options. Specifically, they assumed that the underlying asset follows an MA(1) process, where β represents the impact of past shocks and h is a small constant. We note that and in case β=0 the price dynamics becomes GBM. After applying some standard pricing techniques, a closedform option pricing formula is derived which is similar to BSM except that the variance (and volatility) contains the autocorrelation coefficient, From the above equation, it can be seen that
References [1] Liao, S.L. and Chen, C.C. (2006), Journal of Futures Markets, 26, 85102. Original Post Here: Is Asset Dynamics Priced In Correctly by BlackScholesMerton Model?
0 Comments
Debt instruments are an important part of the capital market. In this post, we are going to provide an example of pricing a fixedrate bond. A fixed rate bond is a long term debt paper that carries a predetermined interest rate. The interest rate is known as coupon rate and interest is payable at specified dates before bond maturity. Due to the fixed coupon, the market value of a fixedrate bond is susceptible to fluctuations in interest rates, and therefore has a significant amount of interest rate risk. That being said, the fixedrate bond, although a conservative investment, is highly susceptible to a loss in value due to inflation. The fixedrate bond’s long maturity schedule and predetermined coupon rate offers an investor a solidified return, while leaving the individual exposed to a rise in the consumer price index and overall decrease in their purchasing power. The coupon rate attached to the fixedrate bond is payable at specified dates before the bond reaches maturity; the coupon rate and the fixedpayments are delivered periodically to the investor at a percentage rate of the bond’s face value. Due to a fixedrate bond’s lengthy maturity date, these payments are typically small and as stated before are not tied into interest rates. Read more We are going to price a hypothetical bond as at October 31, 2018. We first build a zero coupon curve. Picture below shows the market rates as at the valuation date. [caption id="attachment_595" align="aligncenter" width="628"] US swap curve as at Oct 31 2018[/caption] We utilize the deposit rates (leftmost column) to construct the zero curve up to 12month maturity. We then use this zero curve to price the following hypothetical fixed rate bond: Currency: USD Maturity: 1 year Payment frequency: semiannual Coupon rate: 3% We use Python [1] to build a bond pricer. Picture below shows the result returned by the Python program. The price is $99.94 (per $100 notional). Click on the link below to download the python code.
References [1] Quantlib Python Cookbook, Balaraman and Ballabio, Leanpub, 2017 Post Source Here: Valuing a Fixed Rate BondDerivative Pricing in Python 
Archives
April 2023
