Pricing Shifted Heston Calls by Transforming Spot and Strike

Summary

The document explains how to price a call when the underlying follows a shifted version of a Heston process. Its central method is to change coordinates so the shifted asset behaves like an ordinary Heston underlying, then adjust the option’s spot and strike before applying the original pricing formula or a Monte Carlo method.

An alternative formulation defines a transformed asset as a linear combination of the original asset and its initial value. The payoff can then be rewritten as a scaled call payoff on that transformed asset. These algebraic transformations connect the shifted-model price to standard Heston pricing. The document provides no numerical examples or validation, and the formulas rely on the stated process dynamics; the appropriate transformation depends on how the shift parameter enters the model.

Key ideas

• A shifted asset can be treated as an ordinary Heston underlying after changing coordinates.
• The transformed option uses an adjusted spot and strike.
• A linear transformation of the asset also rewrites the payoff as a scaled call payoff.
• Standard Heston pricing formulas or Monte Carlo can be applied to the transformed process.

Tags

# Shifted heston call price


# Shifted heston call price












If we take the heston model but change it slightly by introducing a new parameter $\alpha$ such that

is there a way to price the call option within this model as, maybe, a function of the call price within the original model? Or a function of $S_T$ as simulated from the original model?

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/33611

Let $X_t = a S_t+(1-a)S_0$. Then \begin{align*} dX_t &= adS_t=a\lambda X_t \sqrt{v_t} dW^S_t,\\ X_0 &= S_0. \end{align*} Moreover, \begin{align*} \max(S_T-K, 0) &= \max\left(\frac{1}{a}X_T - \frac{1-a}{a}S_0 -K, \, 0 \right)\\ &= \frac{1}{a}\max\Big(X_T-\big(aK-aS_0+S_0\big), \, 0 \Big). \end{align*} You can now value the option using the previous formula or Monte Carlo approach, assuming that the underlying asset process is represented by $\{X_t, \, t \ge 0\}$.

## Answer by Mark Joshi (score 2)

https://quant.stackexchange.com/a/33612

if we change coordinates slightly, we can regard the process as $$ d (S+\alpha) = (S+\alpha) r dt + \lambda(S+\alpha)\sqrt{V} dW_t, $$ so it's an option on $S+\alpha$ with $S+\alpha$ following the Heston process. So just take the old formula and add $\alpha$ to spot and strike.

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