N-factor model European option pricing
Value European Option Put and Calls under the parameters of an N-factor model.
European_option_value( x_0, parameters, futures_maturity, option_maturity, K, r, call, verbose = FALSE )
x_0 |
Initial values of the state vector. |
parameters |
Named vector of parameter values of a specified N-factor model. Function |
futures_maturity |
Time, in years, when the underlying futures contract matures. |
option_maturity |
Time, in years, when the European option expires. |
K |
Strike price of the European Option |
r |
Risk-free interest rate. |
call |
|
verbose |
|
The European_option_value function calculates analytic expressions of the value of European call and put options on futures contracts within the N-factor model. Under the assumption that future futures prices
are log-normally distributed under the risk-neutral process, there exist analytic expressions of the value of European call and put options on futures contracts. The following analytic expression follows from that presented by Schwartz and Smith (2000) extended to the N-factor framework. The value of a European option on a futures contract
is given by calculating its expected future value using the risk-neutral process and subsequently discounting at the risk-free rate.
One can verify that under the risk-neutral process, the expected futures price at time \(t\) is:
This follows from the derivation provided within the vignette of the NFCP package as well as the details of the futures_price_forecast package.
The equality of expected futures price at time \(t\) being equal to the time-\(t\) current futures price \(F_{T,0}\) is proven by Futures prices
being given by expected spot prices under the risk-neutral process
\((F_{T,t}=E_t^\ast\left[S_T\right])\) and the law of iterated expectations \(\left(E^\ast\left[E_t^\ast\left[S_T\right]\right]=E^\ast\left[S_T\right]\right)\)
Because future futures prices are log-normally distributed under the risk-neutral process, we can write a closed-form expression for valuing European put and call options on these futures. When \(T=0\) these are European options on the spot price of the commodity itself. The value of a European call option on a futures contract maturing at time \(T\), with strike price \(K\), and with time \(t\) until the option expires, is:
Where: \[d=\frac{\ln(F/K)}{\sigma_\phi(t,T)}+\frac{1}{2}\sigma_\phi\left(t,T\right)\]
and:
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