For a given valuation date $t$, expiry date $T$, and time remaining to expiry $\tau$ (i.e. $(T-t)$), the interest rate and carry are defined as

$$r=\frac{1}{\tau}\log(D(t,T))$$ $$b=\frac{1}{\tau}\log\Big(\frac{F(t,T)}{S(t)}\Big)$$

Asset prices are assumed to be log-normally distributed with interest rates and asset yields assumed deterministic. Under these assumptions, forwards and european options are given by closed-form formulas.

Forwards

A forward deal pays $S(T)-K$ at the maturity date $T$. where $K$ is the forward price. The value of the forward contract at time $t$ is

$$(F(t,T)-K)D(t,T)$$

European Options

A European option pays $max(\delta(S(T)-K),0)$ at the expiry date $T$, where either $\delta=+1$ for a call option or $\delta=-1$ for a put option. The value of the European option for annualized implied volatility $\sigma$ is given by Black's formula:

$$\mathcal B_\delta(F(t,T),K,\sigma\sqrt\tau)D(t,T),$$

where $\mathcal B_\delta$ is given by:

$$\mathcal B_\delta(F,K,v)=\delta(F\Phi(\delta d_1)-K\Phi(\delta d_2)),$$

and (for $F>0, K>0, v>0$):

• $d_1=\frac{\log(F/K)}{v}+\frac{v}{2}$
• $d_2=d_1-v$
• $\Phi$ is the standard normal cumulative function

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