# Bachelier model: number of stocks in replicating strategy

Given:

Consider a two-asset, continuous time model (B,S) where $$dB_t = B_t r dt, \quad dS_t = \mu dt + \sigma dW_t.$$ The question is:

1. Show that there exists a trading strategy which replicates the payout of a European call with strike K and maturity T.

2. Show that the number of shares of stock in the replicating portfolio is always non-negative but never greater than one.

• this is the Bachelier model. Not the BS model since that requires drift and volatility to have an S_t coefficient. Commented Nov 25, 2014 at 22:21
• This question cannot be answered without information on the claim to be replicated. What are the properties of the (I assume European) claim function? Commented Nov 26, 2014 at 1:13
• Sorry, forgot to state this: The claim is a European call with strike $K$ and maturity $T$. Actually, I don't even know why a replicating strategy even exists, since I am only taught about the Black-Scholes Model in lectures. Commented Nov 26, 2014 at 1:32
• Please edit the question and include all the details. Commented Nov 27, 2014 at 8:12

Part 1:

Show that there exists a trading strategy which replicates a European Call.

Proof: I am actually going to prove a stronger statement: that there exists an admissible trading strategy which replicates any payoff in this market. By the First Fundamental Theorem of Asset Pricing, there is no arbitrage if there exists a change of measure such that, for all assets, the following holds: $$X_t=\mathbb{\tilde{E}}[e^{-r(T-t)} X_T | \mathcal{F}_t]$$ By Girsonav's theorem, the following change of measure can be made: $$d\tilde{W}_t=dW_t+\frac{\mu-rS}{\sigma}dt$$ Substituting this into the dynamics of $$dS_t=\mu dt+\sigma dW_t$$ yields $$dS_t=rS_tdt+\sigma d\tilde{W}_t$$ The expected value of this process is $$\mathbb{\tilde{E}}[S_T | \mathcal{F}_t]=S_t+\mathbb{\tilde{E}}\left[\int_t ^ T rS_u du |\mathcal{F}_t\right]$$ Taking the expectation inside the integral on the right hand side, $$\mathbb{\tilde{E}}[S_T | \mathcal{F}_t]=S_t+\int_t ^ T r\mathbb{\tilde{E}}\left[S_u |\mathcal{F}_t\right] du$$ Letting $$f(t, T)=\mathbb{\tilde{E}}[S_T | \mathcal{F}_t]$$ And taking the differential of both sides, $$df=rfdt \implies \frac{df}{dt}=rf$$ This is an ODE with initial condition $$f(t, t)=S_t$$ This ODE has the unique solution $$S_t e^{r(T-t)}$$ Thus $$\mathbb{\tilde{E}} [S_T e^{-r(T-t)}]=S_t e^{r(T-t)} e^{-r(T-t)}=S_t$$ Hence this model does not admit arbitrage. By the Second Fundamental Theorem of Asset Pricing, there exists a unique change of measure if and only if every payoff can be replicated. In this model with a single Brownian motion, the change of measure is one such that $$dS=rS dt+\sigma(dW+\theta dt)=\mu dt+\sigma dW$$ Solving for theta, $$rS+\theta \sigma= \mu$$

Clearly this has a single solution, namely $$\theta=\frac{\mu-rS}{\sigma}$$

This proves 1.

Part 2: Show that the Delta of the option is between zero and one.

By Feynman-Kac and Ito's lemma, $$e^{-rt} g(S_t, t, T)=e^{-rT}\mathbb{E}[h(S_T)|\mathcal{F}_t]$$ implies that g has the following dynamics: $$\frac{\partial g}{\partial t} dt+\frac{\partial g}{\partial S} dS_t+\frac{\partial^2 g}{2\partial S^2} \sigma^2 dt -rg dt$$ Comparing this with the dynamics of the self-financing replicating portfolio $$X_t=\Delta S_t+\Gamma B_t$$ $$dX_t=\Delta dS_t+\Gamma dB_t$$ By the First Fundamental Theorem of Asset pricing, $$X_t=g(S_t, t, T)$$ It is thus clear that $$\Delta=\frac{\partial g}{\partial S}$$

Writing the expectation of the payoff as an integral, $$X_t=e^{-r(T-t)}\int_K ^ \infty (S_T-K) d\mathbb{\tilde{P}}$$ $$=e^{-r(T-t)}\int_{K-S} ^ \infty (S e^{r(T-t)} +y-K) p(y) dy$$ Taking the derivative with respect to S, $$\frac{\partial g}{\partial S}= \int_{K-Se^{r(T-t)}} ^ \infty p(y) dy -e^{-r(T-t)}(K-Se^{r(T-t)}+Se^{r(T-t)}-K)p(y)=\mathbb{\tilde{P}}(S_T>K)$$ Thus the delta of the option can be written as a probability, which is always between zero and one.

This proves 2.

• I edited my post to clarify the ODE and to remove all the integrals of r. Commented Nov 29, 2014 at 12:43
• I've edited for clarity around the uniqueness of the change of measure and added a proof for part 2. Commented Nov 29, 2014 at 14:43
• Thanks for your editing. But how do you get $dS= rS dt+ \sigma (dW+ \theta dt)$? Also, could you give me a source that contains more examples of the use of the second fundamental theorem of asset pricing? (I have only come across the first fund. thm. in lectures.) Commented Nov 29, 2014 at 16:05
• Shouldn't the PDE be $\frac{ \partial g}{\partial t} + \mu \frac{\partial g}{\partial s} + \frac{1}{2} \sigma^2 \frac{{\partial}^2 g}{\partial s^2} = rg$ ? The $\mu$ term seems to be missing. Also, applying Ito's lemma, I get $dg(t,S_t)= (rg-\frac{\partial g}{\partial s} \mu)dt+\frac{\partial{g}}{\partial s} dS_t,$ i.e. $\Gamma = \frac{rg-\frac{\partial g}{\partial s} \mu}{r B_t}$, $\Delta= \frac{\partial g}{\partial s}$. However, it doesn't satisfy the first equation $X_t = \Delta S_t + \Gamma B_t$. Do you also get the same results for values of gamma and delta? Commented Nov 29, 2014 at 18:57
• The PDE is done under the risk neutral measure, so $\mu$ will be missing as it doesn't enter the risk neutral stock dynamics. The following paper has a statement of the 2nd fundamental theorem of asset pricing: fm.mathematik.uni-muenchen.de/download/publications/… The equation $X_t=\Delta S_t +\Gamma B_t$ is simply the market value of a portfolio of stocks and bonds. The self financing property states that $dX_t=\Delta dS_t+\Gamma dB_t$. Substituting, $dX_t=\Delta dS_t+r(X_t-\Delta S_t)dt$. It follows that the discounted $X_t$ is a martingale. Commented Nov 30, 2014 at 0:43

The stock weight $\pi_t\geq0$ is nonnegative as $C_T=\max(S_T-K,0)$ is increasing in $S_T$ (always long).

$\pi_t\leq 1$ cannot be greater than one, because one can receive at most 1 stock from the call at maturity, so you dont pay more than 1 stock price for it. Otherwise, one could buy the (cheaper) call and sell the replicating portfolio for riskfree profit at maturity.