Rescaling a Leveraged and Shifted Bachelier Call Payoff

Summary

The document shows how to value a call whose payoff applies leverage and a shift to the terminal underlying price. For a positive leverage factor, the payoff can be factored into the leverage multiplied by a standard call payoff. This transforms the original contract into a vanilla call with an adjusted strike: divide the original strike by leverage, then subtract the price shift. The resulting vanilla option value is scaled by the leverage factor.

This is an algebraic payoff transformation, so any pricing formula suitable for the underlying model can be used after adjusting the strike and scaling the result. The discussion is motivated by the Bachelier model, but the transformation itself does not depend on that model’s pricing formula. The key limitation is the positive-leverage condition: factoring a negative multiplier through the maximum is not valid in the same way, and zero leverage is not covered by the adjusted-strike expression.

Key ideas

• For positive leverage, a shifted and leveraged call payoff reduces to a scaled vanilla call payoff.
• The adjusted strike equals the original strike divided by leverage, less the underlying-price shift.
• Price the vanilla call at the adjusted strike, then multiply its value by the leverage factor.
• The transformation relies on leverage being positive and does not directly cover zero or negative leverage.

Tags

# Bachelier model call option pricing formula with leverage and spread


# Bachelier model call option pricing formula with leverage and spread












the call option pricing formula for the plain/vanilla payoff ($S_T-K)^+$) has been resolved, under the Bachelier model here: Bachelier model call option pricing formula

But can anyone help me with with the generalized payoff (with a leverage and a spread): $(L*(S_T+a)-K)^+$ ?

For this pay-off, what would be the call option pricing formula?

Thanks in advance for the help, and sorry if this is an obvious question (i'm new in the field).

## Answer by alexbougias (score 5, accepted)

https://quant.stackexchange.com/a/59622

\begin{align} \left( L\times(S_T+\alpha)-K \right)^ + {} & = max \{ L\times(S_T+\alpha)-K,0\} \\ {}&= max \left \{ L \times\left( S_T+\alpha-\frac{K}{L}\right),0\right \} \\ {}&\stackrel{\dagger}{=}L \times max \left \{ S_T+\alpha-\frac{K}{L} ,0\right \} \\ {}& =L \times max \left \{ S_T- \left( \frac{K}{L} -\alpha\right),0\right \} \\ \end{align} By setting $K':= \frac{K}{L} -\alpha$ you can value the option as a vanilla call with strike $K'$ and scale the resulting price by $L$, accordingly. Note that in $(\dagger)$, we have used the property:

$$max(x \times a,y \times a) = a \; max(x,y) \ \ if \ \ (a \geq 0) $$

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