Abstract
We start with the random walk on the 1-d lattice of the integers of the real line. From this simple model we derive the equations for the process of diffusion and from that, Brownian Motion. We examine the question of recurrence for random walks and in the process present Polya’s Theorem and Donsker’s Invariance Principle. Applications discussed include the derivation of option pricing in finance, self-avoiding walks, gambler’s ruin, the Kelly Criterion for risk, and Kinetic Monte Carlo. An in depth study of the random walk for analyzing electrical networks leads to develo** formulas for calculating the hitting time to a goal for a Markov Chain and its relationship with the fundamental matrix of the chain and to reversible Markov Chains.
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Notes
- 1.
For the current risk-free rate do an internet search on “U.S. 3 month treasury bill”.
- 2.
A credit spread is the option strategy of selling an option on a stock at one price and buying a lower-priced option on the same stock with the same expiration date. The cost of a call option goes down as the strike price goes up. By selling the option with the lower strike and buying the one with the higher strike, the trader brings in net income. For example, suppose IBM stock is at $114 per share. A call option for IBM with a one-month expiration sells for $3.67 if the strike price is $115 and $1.85 if it is $120. Selling the 115 option and buying the 120 earns a net credit of \(\$3.67 - \$1.85 = \$1.82\) per share. If IBM stays about the same or retreats, the $1.82 is all profit. Why buy the 120 call? If IBM advances, say to $125 (or more!), the trader must buy stock at this price per share and sell at $115 to fulfill the 115 contract, a loss of $10 per share. Buying the 120 call is insurance; invoking it to buy IBM at $120 limits the loss to $5 per share.
By duality, there is a comparable trade using put options in place of calls. (What is it?).
- 3.
A trade in which a near-term expiration option is sold and a longer term option on the same stock and for the same strike price is bought. For example, with IBM at $114, the one-month call option with strike price $115 is $3.67, while the two-month call with the same strike sells for $5.32. Selling the one-month call and buying the two-month call is for a net debit of \(\$5.32 - \$3.67 = \$1.65\). But if IBM stays about the same over the next month, the one-month call will expire worthless and entail zero obligation. Meanwhile, the two-month call (now one month from expiration) should be worth about $3.67. Altogether, a net \(\$3.67 - \$1.65 = \$2.02\) profit.
- 4.
An expanded treatment of the material in this section can be found in [KS60].
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Shonkwiler, R.W., Mendivil, F. (2024). Random Walks. In: Explorations in Monte Carlo Methods. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-031-55964-8_4
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